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Development of the highway network, traffic flow and the growth of settlements in interior B.C. Wills, Michael Jeffrey 1971

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DEVELOPMENT OF THE HIGHWAY NETWORK, TRAFFIC FLOW AND THE GROWTH OF SETTLEMENTS IN INTERIOR B.C. by MICHAEL JEFFREY WILLS M.A. (hons.) Soc. Sc., Un i v e r s i t y of St. Andrews, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Geography We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1971 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e rence and Study. I f u r t h e r agree that pe rmi s s i on f o r e x t e n s i v e copy ing of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copy ing or p u b l i c a t i o n of t h i s t he s i s f o r f i n a n c i a l ga in s h a l l not be a l l owed w i thou t my w r i t t e n p e r m i s s i o n . Department of Geography  The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date A p r i l 1971 Michael J e f f r e y W i l l s J ABSTRACT The o b j e c t i v e of t h i s paper i s to r e l a t e highway investment and economic growth i n a regional context and to i n v e s t i g a t e the nature of t h i s r e l a t i o n s h i p . Temporal aspects are emphasised i n so f a r as at t e n t i o n i s focussed on the way i n which economic a c t i v i t y leads or lags highway a c c e s s i b i l i t y . A s i g n i f i c a n t part of the economic develop-ment l i t e r a t u r e has been concerned with these lead-lag r e l a t i o n s h i p s . For t h i s reason i t i s remarkable that so few studies have made a serious attempt to f i n d out the nature of these r e l a t i o n s h i p s i n a given region. Three aspects of the space-economy are sing l e d out f o r a n a l y s i s . These are the lead-lag r e l a t i o n s between a c c e s s i b i l i t y and economic a c t i v i t y , between t r a f f i c flow and l i n k importance, and between economic growth i n urban centres and distance to nearest l a r g e r centre. Concepts derived from the theory of graphs are used to s i m p l i f y and o p e r a t i o n a l l y define the space-economy, and a t t e n t i o n i s paid to c r i t e r i a for the i n c l u s i o n of centres and highways i n the abstract system. Regression an a l y s i s i s used f o r c l a s s i f i c a t i o n purposes so that temporal trends i n the r e s i d u a l s can be observed. Canonical c o r r e l a t i o n analysis i s employed to reveal an underlying system of leads and lags i n the data. •Results show the existence of lagged r e l a t i o n s h i p s and the •increasing s p a t i a l i n t e g r a t i o n of the economy. Levels of a c t i v i t y are shown to have l e d highway improvement by some f i v e years, which suggests that highway investment has not played the r o l e of leading sector that i t i i i i i i s sometimes held to perform. I t i s , therefore, conjectured that the primary, export-based a c t i v i t i e s are the leading sector and that these are i d e n t i f i a b l e with the leading regions of northern B. C. Results also show that highway improvement has led the l e v e l s of t r a f f i c flow. This feedback suggests that the highway investment programme has accelerated the growth of these regions and hence has fostered the r e g i o n a l l y unbalanced growth patterns inherent i n the B. C. economy. In addi t i o n to th i s the analysis implies that t r a f f i c flow has become more i n t e r r e g i o n a l i n character. During the same period the settlement hierarchy has become r e g u l a r l y spaced. These trends are thought to be r e l a t e d to the develop-ment of a superstructure of t e r t i a r y a c t i v i t i e s d i r e c t l y upon the resource base as these are the economic functions most s e n s i t i v e to highway improve-ments. TABLE OF CONTENTS Chapter Page 1 INTRODUCTION . . . . . . 1 The Argument • 1 Assumptions . . . . . . . . . . 5 D e f i n i t i o n s 7 Methodology 9 2 A REGIONAL INTRODUCTION 12 S p a t i a l Aspects of the I n t e r i o r B. C. Economy . . . 12 S p a t i a l Aspects of Highway Improvement i n I n t e r i o r B .C. 15 3 THE SIMULATION OF TRAFFIC FLOW FROM STRUCTURAL CHARACTERISTICS 18 Defining the Highway Network 18 D e f i n i t i o n of Distance 21 C a l i b r a t i o n of the Model 26 Analysis of Residuals 27 Lead-lag Relationships 32 D i f f e r e n t i a l Impact of Highway Improvement . . . . 36 4 GRAPH THEORY AND THE MEASUREMENT OF ACCESSIBILITY. . . . 38 i v V Chapter Page 5 THE EFFECT OF INCREASED ACCESSIBILITY 48 : E f f e c t s of Increased A c c e s s i b i l i t y : The Consumer Side . . . .' 49 Ef f e c t s of Increased A c c e s s i b i l i t y : The Producer Side 54 The Transportation Component i n Growth Poles . . . 56 6 THE MEASUREMENT OF ECONOMIC GROWTH IN THE B. C. INTERIOR 59 Inadequacies of the Data 59 Advantages of Liquor Sales Data 61 7 ACCESSIBILITY AND URBAN GROWTH IN B. C. INTERIOR . . . 65 Introduction 65 Analysis of Residuals 68 Canonical C o r r e l a t i o n Analysis of Lead-lag Relationships 71 8 REGIONS OF INTERIOR B. C : THEIR EXTENT AND PERSISTENCE OVER TIME • . ' . 76 9 CONCLUSIONS AND SUMMARY . . 81 Lead-lag Relationships 81 Transformations of the Regional Economy and i t s Urban System 83 BIBLIOGRAPHY 86 APPENDIX 1 - Some Basic D e f i n i t i o n s i n Graph Theory 95 APPENDIX 2 - Shortest-Path Algorithms 99 APPENDIX 3 - Mathematical Summary of Canonical C o r r e l a t i o n . . . 105 LIST OF TABLES Table Page I. Weighting Factors Related to Highway Quality 21 I I . Performance of D i f f e r e n t Values of the Gravity Model Distance Exponent i n P r e d i c t i o n of T r a f f i c Flow, 1966 27 I I I . Performance of Gravity Model with Liquor Sales as Masses and Distance Exponent of 2.7, over time 27 IV. Results of Canonical C o r r e l a t i o n Analysis of T r a f f i c Flow and Link Importance Index: Lead-lag Relationships 34 V. Results of Canonical C o r r e l a t i o n Analysis of Liquor Sales and Route Access Index: Lead-lag Relationships 72 VI. Results of Canonical C o r r e l a t i o n Analysis of Liquor Sales and Distance to Nearest Larger Centre: Lead-lag Relationships 75 VII. T r e e - b u i l d i n g Shortest-path Algorithm: L i s t i n g s and Shortest-paths 101 V I I I . Shortest-path Matrices . 103 v i LIST OF FIGURES Figure * • Page 1 Percentage Population Change, Incorporated Places, 1956-61 14 2 B..C. Highway Network (Diagrammatic) 20. 3 B. C. Highways i n 1953 23 4 B. C. Highways i n 1969 24 5 Residuals of T r a f f i c Flow (Y) and Link Importance (X), 1953 29 6 Residuals of T r a f f i c Flow (Y) and Link Importance (X), 1960 30 7 Residuals of T r a f f i c Flow (Y) and Link Importance (X), 1966 . 31 8 Impacts of the Roger's and Yellowhead Passes on T r a f f i c Volumes and Routing 35 9 Increasing Importance of the Trans-Canada Highway (Fraser Canyon Route) R e l a t i v e to the Hope-Princeton Route 3 7 10 Some Examples of Graphs 39 11 Composition of Aggregate Demand with D i f f e r e n t Transport Rates 52 12 S u b s t i t u t i o n between Transport Costs and other Production Costs 55 13 S u b s t i t u t i o n between A c c e s s i b i l i t y and Resource Quality 55 14 Relationship between Liquor Sales and R e t a i l Sales, 1961 63 15 Residuals of Liquor Sales (Y) and A c c e s s i b i l i t y (X), 1953 • 69 v i i v i i i F igure Page 16 Residuals of L i q u o r Sales (Y) and A c c e s s i b i l i t y (X) , 1966 . . . . . . . . . . . . . . . . . . 70 17 Regions of I n t e r i o r B . C , 1953 . . . , . 78 18 Regions o f I n t e r i o r B . C , 1966 79 19 E m p i r i c a l Re la t ionsh ips Between the Highway Network and the Regional Economy of I n t e r i o r H• C• • • • • • • • 33 20 Graphs I l l u s t r a t i n g Shortes t -path Algorithms . . . . . . 100 ACKNOWLEDGEMENT S The i n i t i a l impetus to write t h i s thesis came from a paper by H. L. Gauthier (1968a). This d i r e c t i o n was reinforced by Dr. C. C. K i s s l i n g who made h i s computer programme a v a i l a b l e to me at an early stage i n my work. Later stages of th i s work owe much to the guidance of Mr. K. G. Denike and Dr. R. N. North; they helped to r a t i o n a l i s e the methodology, to remove a m u l t i p l i c i t y of ambiguities and to make the writ t e n r e s u l t readable. In ad d i t i o n , I wish to acknowledge the constant support and encouragement of Mr. Denike and to express gratitude to the Canadian Transport Commission f o r t h e i r funds. The Department of Highways in V i c t o r i a made a l l t h e i r t r a f f i c flow data r e a d i l y a v a i l a b l e to me. F i n a l l y , I have to thank Mr. W. Steinmetz and Mrs. H. Troche f o r the excellent q u a l i t y of the figures and typing r e s p e c t i v e l y . CHAPTER 1 INTRODUCTION The Argument A s i g n i f i c a n t factor i n the i n t e r n a l s tructure and economic v i a b i l i t y of an urban settlement i s the l o c a t i o n of that centre r e l a t i v e to other centres and to t h e i r routeways. Centres compete t e r r i t o r i a l l y for the trade of a region. This competition i s , to an increasing extent, f a c i l i t a t e d by the expanding network of highways. Hence the centres best located to command these highways and to minimise d i s t r i b u t i o n costs possess a competitive advantage. Changes i n t h e i r a c c e s s i b i l i t y to these routes a l t e r the competitive p o s i t i o n of the centres. A f t e r some i n t e r v a l of time has elapsed these t o p o l o g i c a l changes i n the space-economy are r e f l e c t e d i n the l e v e l of a c t i v i t y i n i n d i v i d u a l places. The d i f f i c u l t y i n determining the p r e c i s e nature of the r e l a t i o n s h i p s between a c c e s s i b i l i t y and economic v i a b i l i t y of centres i s that these tend to be c i r c u l a r . Centres that grow r e l a t i v e to other places i n a region perform an increasing proportion of t e r t i a r y a c t i v i t i e s . Hence they require a d d i t i o n a l transport f a c i l i t i e s , e s p e c i a l l y improved highways, to f a c i l i t a t e the transmission of t h e i r i n f l u e n c e and to give greater access to t h e i r functions. R e a l i s a t i o n of these f a c i l i t i e s enhances the competitive p o s i t i o n of these centres. These trends are reinf o r c e d at some stage by agglomeration economies and by m u l t i p l i e r - a c c e l e r a t o r e f f e c t s . 2 The purpose of t h i s paper i s to r e l a t e highway investment to the patterns of highway-oriented economic growth i n I n t e r i o r B. C. between i 1953 and 1966, and to i n v e s t i g a t e the nature of t h i s r e l a t i o n s h i p . The approach taken attempts, by methods based on graph theory, to quantify the increased effectiveness of the highway network owing to t h i s i n v e s t -ment. These methods enable pre c i s e measurement of changes i n the acces-s i b i l i t y of a centre and i n the importance of each l i n k to the highway network as a whole. These data are compared, for f i v e time-periods, with the growth rates of centres and with t r a f f i c flow. In each case i n t e r e s t i s focussed on the degree to which changes i n a c c e s s i b i l i t y tend to lead or lag economic growth. The nature of these lags has been the object of some conjecture i n the l i t e r a t u r e . Whereas some writ e r s have emphasized the p o s i t i v e r o l e of transport i n generating economic growth, others have assigned, to transport a more permissive r o l e i n the sense of allowing demand for economic development to be r e a l i s e d . Some of t h i s argument may be semantic since transportation, as a derived demand, can be viewed as a permissive agent rather than as a d i r e c t i n s t i g a t o r of development. What i s at issue, however, i s the sequence of events and the associated chain of causation: whether improved trans-p o r t a t i o n has f a c i l i t a t e d subsequent development or whether transportation has been improved i n response to increasing demand from e x i s t i n g economic a c t i v i t i e s . No doubt both of these forces can work concurrently i n the process of r e g i o n a l growth: the problem i s to show which i s dominant. In t h i s context three s p e c i f i c aspects of the economic growth process are examined. These are the lead-lag r e l a t i o n s , f i r s t l y , between the a c c e s s i b i l i t y of centres to the highway network and t h e i r growth, secondly, between the s t r u c t u r e of the highway network and the d i s t r i b u t i o n of t r a f f i c flow, and t h i r d l y , between the growth of communities and the i changing s t r u c t u r e of the c e n t r a l place hierarchy. These r e l a t i o n s h i p s between highway investment and r e g i o n a l economic growth also have implications f o r r e g i o n a l planning. I t i s d e s i r a b l e to a n t i c i p a t e the e f f e c t s of a given i n j e c t i o n of investment i n one sector on the other sectors. This knowledge can be used to a l l o c a t e investment i n a r e g i o n a l context to meet the p o l i c y goals determined by s o c i e t y . Consider the case of a depressed area the economy of which i s to be stimulated. How can a given amount of investment be a l l o c a t e d optimally to achieve t h i s aim? A l t e r n a t i v e l y , consider a r a p i d l y expanding area. What a l l o c a t i o n of investment w i l l achieve the desired growth-path? Owing to the great number of v a r i a b l e s involved and to the absence of e m p i r i c a l l y tested explanatory models of the m u l t i p l i e r - a c c e l e r a t o r these are v i r t u a l l y i n t r a c t a b l e problems. In t h i s context a simple d e s c r i p t i o n of lead-lag r e l a t i o n s h i p s may provide a reference point for the consideration of these questions. I t should be emphasised, however, that the r e s u l t s of the analysis i n t h i s paper would contribute only to a p a r t i a l model of the space-economy. There are many determinants of economic growth apart from highway investment. Inv e s t i g a t i o n of some of the more important f a c t o r s at work to influence r e g i o n a l growth trends demonstrates the overwhelming nature of the problem. Indeed, these factors do not influence growth i n a region i n any simple cause and e f f e c t r e l a t i o n s h i p . Rather they combine to-gether i n a complex and varying system. Isard et a l (1960) c l e a r l y demonstrate the scope of the assignment: 4 . . . a l l the c h a r a c t e r i s t i c s of a region and i t s very development path are thus intertwined i n a maze of interdependencies. This maze i n t e r l a c e s i n t e r - r e g i o n a l systems of population, resource I patterns, i n d u s t r i a l l o c a t i o n s , l o c a l economies, s o c i a l accounts, balance of payments p o s i t i o n s , markets, c e n t r a l places and urban-metropolitan areas, administrative and p o l i t i c a l structures and i n s t i t u t i o n s , and even values, motives, and s o c i a l goals. I t i n t e r l a c e s a l l these systems v i a i n t e r -r e g i o n a l systems of i n t e r - i n d u s t r y ( i n t e r - a c t i v i t y ) linkage, of commodity flows and money flows, of population movements, and of communications, and, i n general, of s o c i o - c u l t u r a l i n t e r a c t i o n i n c l u s i v e of decision-making processes. (pages 2-3) In contrast to Isard's c h a r a c t e r i s a t i o n of r e g i o n a l growth the theme of t h i s paper re s t s on the i s o l a t i o n of the highway component from other aspects of r e g i o n a l growth. At the operational l e v e l t h i s o b j e c t i v e was f a c i l i t a t e d by a considerable degree of a b s t r a c t i o n . In t e r n a l structures of the settlements were ignored and the centres were abstracted to a system of points or nodes located i n the highway network. Furthermore, only the l e v e l s of a c t i v i t y , as measured by a s i n g l e surrogate index, i n these punctiform centres were considered. Modes of transport other than highways were not"included as i t was thought that the greater complexity and comparability problems involved with mixed-mode models would outweigh the advantages of i n i t i a l s i m p l i c i t y i n s p e c i f y i n g temporal r e l a t i o n s h i p s . In any case highways are by f a r the dominant mode and the object of the greatest investment i n the time-period considered. Assumptions Given the general problem of the r e l a t i o n s h i p s between acces-••?r.-s i b i l i t y , t r a f f i c flows and economic growth i t i s necessary to make c e r t a i n assumptions regarding the behaviour of elements of the. economic system which are not e x p l i c i t l y discussed further i n th i s study. Thus r a t i o n a l behaviour of highway users i s assumed i n as much as they attempt to minimise t r a v e l l i n g distances between given o r i g i n s and destinations c e t e r i s paribus. In* p r a c t i c e t h i s problem often involves choice between a few'alternative routes. I t i s assumed i m p l i c i t l y i n t h i s that distance has negative u t i l i t y so that the desire to minimise i t i s a r a t i o n a l decision given the important c e t e r i s paribus co n d i t i o n . Distance can be measured i n numerous ways not one of which alone i s l i k e l y to be e n t i r e l y s a t i s -f a c t o r y . Time-distance was assumed to be the most relevant c r i t e r i o n but even the value of time depends on i n d i v i d u a l i s t i c values and goals, and also on the a b i l i t y of i n s t i t u t i o n s to b e n e f i t from time-savings. In a d d i t i o n , the non-stationary nature of the value of time i s apparent as the tempo of human a c t i v i t i e s has quickened even over the ti m e - i n t e r v a l considered here i n the a n a l y s i s . Although r a t i o n a l behaviour i s assumed some qua l i f y i n g , remarks are necessary concerning the 'perfect knowledge' assumption i m p l i c i t i n the concept of r a t i o n a l man. For example, i t was assumed that i n d i v i d u a l s and i n s t i t u t i o n s would attempt to minimise time-distance between o r i g i n s and destinations but the i n t e r v a l over which t h i s optimisation process takes place i s not s p e c i f i e d . Indeed any such time lags i n causal r e l a t i o n -ships are of c e n t r a l i n t e r e s t i n t h i s study. Therefore t h i s r e s t r i c t i v e assumption has been.relaxed to allow i n v e s t i g a t i o n of i n e f f i c i e n t behaviour i n the process of s p a t i a l adaptation to changing circumstances to be included i n the analysis.. These time-lags emanate from two main sources. They are due i n i t i a l l y to slowness i n perceptions and i n the access to, and evaluation of, information. Once decisions to act have been made further lags are i n e v i t a b l e i n the r e a l i s a t i o n of those decisions. These lags of the second category are response lags. They comprise reorganisation of e x i s t i n g modes of behaviour f o r i n d i v i d u a l s and i n s t i t u t i o n s , queuing where increased demands exceed short-run supply c a p a c i t i e s , and the time taken up i n the r e a l i s a t i o n of c a p i t a l investment i n technology or i n f r a s -t r u c t u r e . Among the i n s t i t u t i o n s , those of government at s e v e r a l l e v e l s are probably slower to adapt to changing circumstances and as a r e s u l t time-lags would be greater where they are involved. Long-term planning, above a l l , would involve ignoring short-run optimal decisions where they were incompatible with the long-range p o l i c i e s . As t h i s study only considers the highways instead of a l l trans-port modes i n B. C. i t i s e s s e n t i a l to assume that a large proportion of r e g i o n a l and i n t e r - r e g i o n a l changes w i l l be transmitted by, and r e f l e c t e d i n , changes i n the highway network. A c t i v i t i e s which need frequent contacts and therefore require to be a c c e s s i b l e to the system are the most s e n s i t i v e to changes i n the highway network. I t might reasonably be conjectured that these are represented by the rapid growth of t e r t i a r y a c t i v i t i e s and the emergence of d i s t r i b u t i o n centres for r a p i d l y expanding regions. Hence i t i s assumed here that highway-oriented growth i n the regional economy i s approximated by growth i n the demand f o r consumer goods. Operationally, the value of l i q u o r sales i s used as a surrogate 7 v a r i a b l e for t h i s demand and since t h i s data source i s not r e l a t e d to a r b i t r a r y a r e a l boundaries i t i s assumed that l i q u o r sales r e f l e c t the prosper i t y of centres and of t h e i r h i n t e r l a n d s . The comparative advan-tages of these data are discussed more f u l l y i n Chapter 6. D e f i n i t i o n s - :-Precise d e f i n i t i o n s of the graph-theoretic terms used are given i n Appendix 1. Some explanation, however, i s needed regarding the general nature of these concepts. The foundation of a l l the argument that follows rests on the ab s t r a c t i o n of the highway"'" system to a graph or network. This s i m p l i f i c a t i o n allows theorems or r e l a t i o n s h i p s to be s p e c i f i e d and tested far more ri g o r o u s l y and with greater c l a r i t y regarding the arguments used than would otherwise be the case. Any conclusions, however are c o n d i t i o n a l upon the assumptions being met i f those conclusions are to be translated i n t o the r e a l world. In view of this the abstract i n t e r p r e t a t i o n of r e a l world structures can now be considered. In general, a graph, or network, can be defined wherever there ex i s t s a set X of objects, s u f f i c i e n t l y w e l l d i f f e r e n t i a t e d to be defined, which are r e l a t e d i n some p a r t i c u l a r way that i s to be analysed. E m p i r i c a l l y , t h i s r e l a t i o n s h i p may be j u s t one of many sets of r e l a t i o n -ships between the set of objects. Nevertheless, i n t e r e s t can be, and often i s , l i m i t e d to a definable subset of these r e l a t i o n s h i p s . These may be defined i n graph-theoretic terms as a function F mapping the set X into i t s e l f , that i s , f o r each element i n the set X are defined "'"Note that "highways" are defined here to include a l l roads for which t r a f f i c flow data are c o l l e c t e d . This omits only logging t r a i l s and some minor, unpaved roads. 8 correspondences with c e r t a i n other elements i n the set. Each of these correspondences i s represented by a set of arcs or l i n k s j o i n i n g the objects. Thus a graph i s defined where there e x i s t the following three p r i m i t i v e s : ( i ) a set X such that X^ i s an element of X and i s c a l l e d a vertex or node, ( i i ) a set U such that U\ i s an element of U and i s c a l l e d an arc, edge or l i n k , and, ». ( i i i ) a function F with a domain i n U and a range contained i n X. I n t u i t i v e l y , a transport system i n a r e g i o n a l context can be thought of as a number of routes j o i n i n g the settlements contained i n the region. Settlements i n the r e a l world correspond to v e r t i c e s or nodes i n the abstract graph. Lines of communication connecting the nodes are recognised as arcs or l i n k s . The function F symbolises the nature of the r e l a t i o n s h i p between the nodes, the type of transportation f a c i l i t y or the manner i n which the l i n k i s weighted. This may be a function of p h y s i c a l distance, perceived distance, the time taken to traverse the l i n k or the capacity of the f a c i l i t y . The a c c e s s i b i l i t y of a place i s defined i n general as the q u a l i t y of i t s l o c a t i o n r e l a t i v e to the e n t i r e system of places and highways that i s being considered. Operationally, t h i s i s measured ei t h e r i n terms of minimising distance to a l l other places or as l o c a t i o n r e l a t i v e to routeways between the other places. Regional economic growth, on the other hand, i s defined as increasing l e v e l s of some surrogate measure of economic a c t i v i t y i n the centres contained i n the region. 9 The value of l i q u o r sales i s used throughout as t h i s surrogate. Methodology Having presented and defined the problem the question that now a r i s e s i s how to measure those r e l a t i o n s h i p s . C l e a r l y , the s t a t i s t i c a l a n a l y s i s should be set up i n such a manner as to enable the corroboration or r e j e c t i o n of hypothesized r e l a t i o n s h i p s . Yet there i s something more; the p r e c i s e meaning that can be attached to a hypothesis l a r g e l y depends on the method employed for i t s v e r i f i c a t i o n . This leads the d i s c u s s i o n to a consideration of the hypotheses, assumptions and s t a t i s t i c a l methods used i n the empirica l parts of t h i s study. A multiple-hypothesis approach was adopted i n the i n v e s t i g a t i o n of l e a d - l a g ' r e l a t i o n s h i p s . I n i t i a l l y a r e l a t i o n s h i p i s assumed between a pair of v a r i a b l e s . Given t h i s general r e l a t i o n s h i p three hypotheses are entertained simultaneously. In the case of urban economic growth and a c c e s s i b i l i t y there are three possible hypotheses: that the r e l a t i o n -ship i s balanced over time, that urban growth leads a c c e s s i b i l i t y or that urban growth lags behind a c c e s s i b i l i t y changes (Gauthier, 1968a). These hypotheses are suggested by e x i s t i n g theory. In a d d i t i o n , for aggregate an a l y s i s at l e a s t , they exhaust the e n t i r e set of outcomes and, although not independent, they are mutually exclusive. The a n a l y s i s does not purport to v e r i f y c o n c l u s i v e l y any one of these a l t e r n a t i v e hypotheses but rather to show, under s u i t a b l e assumptions, which of these c o n d i t i o n a l a l t e r n a t i v e s was a p p l i c a b l e i n I n t e r i o r B. C. Thus care should be taken not to extrapolate the findings except i n as much 10 as they c o n t r a d i c t or corroborate r e s u l t s predicted by theory. There are major s t a t i s t i c a l problems involved i n the a n a l y s i s of the type of i n v e s t i g a t i o n performed i n t h i s study. These are, i n p a r t i c u l a r , the i d e n t i f i c a t i o n or m u l t i - c o l l i n e a r i t y problem, and secondly, that of a u t o c o r r e l a t i o n . I d e n t i f i c a t i o n problems are r e l a t e d to the apparent c i r c u l a r i t y of the t o p i c : i t i s d i f f i c u l t to state a p r i o r i what aspects are causally dependent on what other aspects. Thus i t i s not p o s s i b l e to define the independent and the dependent v a r i a b l e s . This f a c t alone prevents the v a l i d use of regression a n a l y s i s for inference since the technique measures causal r e l a t i o n s h i p s . This i s suggested by the d i f f e r e n t r e s u l t s obtained i n general by r e v e r s a l of the axes (Olsson, 1970). Furthermore, where seve r a l f a c t o r s are f u n c t i o n a l l y i n t e r r e l a t e d there i s a danger of the argument becoming t a u t o l o g i c a l . D i f f i c u l t i e s r e l a t e d to the i d e n t i f i c a t i o n problem provide the motivation for two d i s t i n c t i v e approaches to the study: the use of methods of network a n a l y s i s , on one hand, and canonical c o r r e l a t i o n s , on the other. Network a n a l y s i s was adopted to provide measurement of the s t r u c t u r a l properties of the highway system independent of indices r e l a t e d to the l e v e l of economic a c t i v i t y . Measures of market p o t e n t i a l or t r a f f i c flow owing to the c i r c u l a r i t y they introduce cannot be used i f c o r r e l a t i o n techniques are to be employed. Network a n a l y s i s was therefore used i n an attempt to cut into the c i r c u l a r i t y or at l e a s t to provide r e l a t i v e l y c l e a r l y defined measures for subsequent a n a l y s i s . This analysis involved canonical c o r r e l a t i o n techniques which are designed to r e l a t e two sets of v a r i a b l e s without having to s p e c i f y dependent-independent r o l e s . One set of the v a r i a b l e s measured network structure, the other economic development. 11 A u t o c o r r e l a t i o n i s recognised where the r e s i d u a l s from regression show a strong pattern. This implies the existence of another v a r i a b l e not accounted for i n the s t a t i s t i c a l model. The r a m i f i c a t i o n s of auto-c o r r e l a t i o n are minimised i n the a n a l y s i s i n t h i s paper. This i s held to be the case for two reasons:-( i ) In the regression a n a l y s i s i n t e r e s t was confined to a c l a s s i f i c a t i o n of the r e s i d u a l s and of d i s t i n c t trends i n the strengths of r e l a t i o n s h i p s . ( i i ) Canonical c o r r e l a t i o n rather than regression was used for the lead-lag r e l a t i o n s h i p s which c o n s t i t u t e the major portion of the a n a l y s i s . CHAPTER 2 ' A REGIONAL INTRODUCTION S p a t i a l Aspects of the I n t e r i o r B. C. Economy The area selected for a n a l y s i s was the I n t e r i o r of B. C. and the i n t e r v a l of time under consideration was 1953 to 1969. Intermediate analyses, were also performed i n 1956, 1960, 1963 and 1966. The network ana l y s i s undertaken was, i n some ways at l e a s t , p a r t i c u a r l y s u i t e d to the s p a t i a l c o n f i guration of the study area, the confined routeways and i s o l a t e d settlements of which were r e a d i l y represented by a system of l i n k s and nodes. Hence leakage by minor roads from a s i m p l i f i e d represen-t a t i o n of the highways as a graph was minimised. I n t u i t i v e l y and v i s u a l l y , ^ e a c h settlement appears to be i n t i m a t e l y r e l a t e d to the highways on which i t i s s i t u a t e d . The settlements command routeways and t h e i r trade areas are elongated along them. As a r e s u l t i n t e r a c t i o n i s l i n e a r rather than a r e a l i n form. P h y s i c a l i n t e r a c t i o n i s c a r r i e d predominantly by the highways. The highways and settlements are not d i s t r i b u t e d evenly over the area. Rather they suggest a set of i n t e r l i n k e d r e g ional sub-systems such as the Okanagan, the Kootenays, the Prince George area and the Peace River area. These regions are, i n turn, l i n k e d to three metro-p o l i t a n centres: Vancouver, Calgary and Edmonton and are nested within the hinterlands of these c i t i e s . 12 13 Although the l o c a t i o n of the study area i s d i s t i n c t i v e , balanced as i t i s between the competing influences of p e r i p h e r a l c i t i e s , the pattern of growth i s even more remarkable. Indeed, as Denike has observed, i t demonstrates a case where the hi n t e r l a n d i s growing f a s t e r than the metropolitan centre.^" Taken as a whole these regional subsystems are expanding at a rate almost double that of the metropolises, which themselves are also growing r a p i d l y . The many t h r i v i n g mining, wood-processing or t o u r i s t centres contrast markedly with the stagnant and d e c l i n i n g a g r i -c u l t u r a l centres of the P r a i r i e s (Hodge, 1965). For instance, the l a t t e r have a very high average age whereas the former are t y p i f i e d by in-migration of the younger groups i n the work force. Within t h i s general aspect of r a p i d r egional expansion there e x i s t s considerable d i v e r s i t y between the d i f f e r e n t sub-regions. This i s p a r t i c u a r l y the case when considering economic prosperity and r e l a t i v e rates of growth. Indeed, growth i s so markedly d i f f e r e n t i a t e d that whereas Prince George i s now doubling i t s population every ten years, the Kootenays have stagnated to become a depressed area. Furthermore, at a s t i l l smaller scale, the fortunes of i n d i v i d u a l communities have v a c i l l a t e d enormously. Mining settlements, above a l l , have been vulnerable to the sudden disappearance of t h e i r economic v i a b i l i t y as w e l l as prone to spectacular bursts of growth. Ghost towns demonstrate the former case whereas boom towns such as Fort St. John or M e r r i t t demonstrate the l a t t e r phenomenon. Figure 1 shows the percentage change i n population between 1956 and 1961 i n the study area p a r t i t i o n e d into three major sub-areas: ^Personal communication. 1 PERCENTAGES 100" PERCENTAGE POPULATION CHANGE, INCORPORATED PLACES, 1956 - 1961 90" + 80" 70" + •. 60" -• 50" ' + • • A 40" + 30" + & • + • A • 20" + A • + • • 10" + ML 3 : + | | 0" . 1 • -10" + -20" -30" • -40 J | NORTHERN | |KOOTENAYS | |OKANAGAN-SUSWAP| Figure 1 15 the Okanagan, Kootenays and the northern region. The slower growth, and even absolute d e c l i n e , of the Kootenays i s apparent although there i s a d i v e r s i t y of growth rates. Even more v a r i a b l e , however, are the fortunes of the northern centres. Fort St. John increased i t s population by 90% whereas Kitimat declined by 14%. In contrast, the Okanagan centres are grouped around a rate of growth of 20%. Presumably t h i s v a r i a b i l i t y i n population numbers i s r e l a t e d to the r e l a t i v e i n s t a b i l i t y of l o c a l economic a c t i v i t i e s and t h i s i n turn r e f l e c t s the r e l a t i v e dependence on primary e x t r a c t i v e i n d u s t r i e s . S p a t i a l Aspects of Highway Improvement i n I n t e r i o r B. C. In general, the r e g i o n a l d i s t r i b u t i o n of economic development appears to be r e l a t e d to a vigorous programme of highway construction. Several aspects of this highway improvement can be d i s t i n g u i s h e d . The expansion of the B. C. economy has been marked by a migration of a c t i v i t y towards the North. This trend has been p a r a l l e l e d by an extension of the highway network i n the same d i r e c t i o n . Except for the P a c i f i c Great Eastern Railway (P.G.E.), the railways seem to have played a minor r o l e i n t h i s northward movement of the development f r o n t i e r . The railways are probably too i n f l e x i b l e f o r s u c c e s s f u l settlement of areas and i t has been l a r g e l y a c o l o n i s a t i o n by highway construction. Settlement and commercial investment cannot generally take place u n t i l at l e a s t a road network e x i s t s . The cost of movement i n the absence of roads confines new investment to areas where such i n f r a s t r u c t u r e already e x i s t s . I t follows that the construction of highways where none existed before has at l e a s t f a c i l i t a t e d the establishment and subsequent !6 expansion of economic a c t i v i t y i n the areas a f f e c t e d . Highways have been a necessary condition for t h i s continued growth i f not a s u f f i c i e n t one. i i Yet i t i s c l e a r that they have diminished the distance between m a t e r i a l s , : production s i t e s and p o t e n t i a l markets; they have made i n f e a s i b l e locations p o t e n t i a l l y f e a s i b l e . Improvement of the highway network has taken place by adding l i n k s w i t h i n the network and upgrading e x i s t i n g l i n k s , by paving f or example. The changes produced shocks to the e x i s t i n g pattern of i n t e r -a c t i o n and a l t e r e d the l o c a t i o n of places r e l a t i v e to each other. Most important was the routing f a c t o r . A new l i n k i n one part of the network changed r a d i c a l l y the t r a f f i c flow i n another part hundreds of miles away. The Roger's Pass shows t h i s . As soon as the Pass was opened i n August, 1962, t r a f f i c was diverted at Hope away from the Southern Trans-P r o v i n c i a l Highway to the Trans-Canada Highway. R e a l i s a t i o n that highways are not independent units i s a prime motivating force i n considering the highway network as an integrated system. In t h i s manner repercussions of road improvements are more e a s i l y appreciated and analysed. Changes i n the l o c a t i o n of places r e l a t i v e to each other can be"' expected to change t h e i r competitive r e l a t i o n s . These r e l a t i o n s are on three l e v e l s : those between neighbouring centres, those between adjacent r e g i o n a l sub-systems, and those between the p e r i p h e r a l metropolises and the B. C. I n t e r i o r as a whole. In t h i s r a p i d l y emerging region such competitive r e l a t i o n s are f a r from s t a b l e . Two neighbouring centres are not i n e f f e c t i v e competition u n t i l connected by a s u f f i c i e n t l y good road. The re g i o n a l sub-system that becomes the dominant one i n the I n t e r i o r w i l l be s t r a t e g i c a l l y located with respect to the highway network. 17 As the p e r i p h e r a l metropolises improve t h e i r e f f e c t i v e communications with the h i n t e r l a n d they compete i n c r e a s i n g l y f or the higher order functions of emerging regional centres, such as Kamloops. At the same time i n -creased proximity of these centres to a metropolis can enhance t h e i r p o s i t i o n as d i s t r i b u t i o n centres for the I n t e r i o r . CHAPTER 3 THE SIMULATION OF TRAFFIC FLOW FROM STRUCTURAL CHARACTERISTICS Defining the Highway Network The d e f i n i t i o n of the network over which the a c c e s s i b i l i t y measures are computed i s an important step. In essence, the problem co n s i s t s of approximating an open system by a closed one, and of extracting the r e l a t i v e l y simple t o p o l o g i c a l properties from the r e l a t i v e l y complex network that e x i s t s . With t h i s i n view, three c r i t e r i a were used to define the network i n i t i a l l y for subsequent a n a l y s i s . F i r s t l y , only highways that were deemed important enough by the B.. C. Dept. of Highways to warrant c o l l e c t i o n of t r a f f i c data were included. T r a f f i c flow data were required to c a l i b r a t e the gra v i t y model i n l a t e r a n a l y s i s . Secondly, the continuous nature of the highways was broken into d i s c r e t e l i n k s by the l o c a t i o n of nodes. These were defined as places possessing a l i q u o r store for the e n t i r e period of an a l y s i s , 1953-1969. A d d i t i o n a l , dummy, nodes were defined at important junctions not coincident with settlements s a t i s f y i n g t h i s condition. Cache Creek, Tete Jaune, Monte Creek and Sicamous are examples of these. The t h i r d c r i t e r i o n involved determining the s p a t i a l extent of the network. The i n i t i a l o b j e c t i v e had been to study the system of routes and c i t i e s i n the i n t e r i o r of B. C. I t seemed that t h i s system was located at a point 18 19 of balance between the impact of three metropolitan centres that en-c i r c l e d i t : Vancouver, Calgary and Edmonton. The influence of these ) c i t i e s could hardly be omitted from the a n a l y s i s . As a r e s u l t , the study area network was extended to include these centres. The network was further augmented to take i n more l i n k s and nodes along a l l through routes. A p i l o t study had shown that the impor-tance of end l i n k s and nodes of that study network were s e r i o u s l y under-predicted. In add i t i o n , r e l a t e d problems arose owing to the omission of water-transport from consideration. For instance, i n that study, Prince Rupert was located at the extremity of the highway network and no account was taken of i t s considerable port functions. Therefore, as no v a l i d measure of i t s a c c e s s i b i l i t y had been made, Prince Rupert was omitted from the study area and added to the per i p h e r a l component of the augmented network. The remaining study network was therefore bounded by Hope, 1 Terrace, Fort St. John, Tete Jaune, Golden, Fernie and the centres along the B. C. side of the Int e r n a t i o n a l Boundary. Surrounding t h i s was a per i p h e r a l network, i t s e l f bounded by Vancouver, Kitimat, Prince Rupert, 2 Wonowon, Edmonton, Calgary, Medicine Hat and seve r a l small c i t i e s along the U. S. side of the border. The s p a t i a l c o n f i g u r a t i o n of the augmented network i s shown i n Figure 2. In graph-theoretic terms, the network under study represented the a b s t r a c t i o n of the e x i s t i n g transport f a c i l i t i e s to a p a r t i a l sub-graph. I t was p a r t i a l i n that not a l l highways were included, since the most minor by-ways and logging roads were excluded. I t was p a r t i a l i n another sense, because the railways, a i r - r o u t e s and waterways were not taken into consideration. The network was a subgraph as i t was a """Equidistant between Prince George and Edmonton. 2 50 miles north of Fort St. John. .PRINCE RUPERT _,t®TERRACE B.C. HIGHWAY NETWORK (DIAGRAMMATIC) EDMONTON INCLUDED CENTRES o DUMMY NODES • MAJOR PERIPHERAL CITIES INCLUDED HIGHWAY LINKS PERIPHERAL NETWORK VANOERHOOF MEDICINE MAT F i g u r e 2 21 subset of a l l the highways, i n North America, for example. D e f i n i t i o n of Distance A l l the a c c e s s i b i l i t y measures that were used required, as a basic ingredient, the measurement of distance i n some manner. Ph y s i c a l distance, measured i n miles, would have been the most convenient. Never-theless, i n view of the great v a r i a t i o n i n road q u a l i t y between 1953 and 1969, i t was decided to weight the l i n k s i n r e l a t i o n to the q u a l i t y of the road. This was accomplished i n the following manner. I n i t i a l l y , a c e r t a i n speed l e v e l was assumed for a c e r t a i n type of road surface. Precise values are shown i n Table I. TABLE I WEIGHTING FACTORS RELATED TO HIGHWAY QUALITY Highway q u a l i t y Assumed speed ( i n m.p.h.) Minutes per mile M u l t i p l y i n g f a c t o r used 4-lane paved 60 1.0 0.8 2-lane paved 45 1.3 1 Improved gravel 35 1.7 1.3 D i r t 25 2.4 ' 2.0 The assumed speeds were converted into the time taken i n minutes to t r a v e l one mile. To save computation, these times were then expressed i n terms of the category i n which the vast majority of the roads f e l l , 22 that of two-lane paved highways. This weighting factor was f i n a l l y applied to the p h y s i c a l distance i n miles, to obtain a measurement of time-distance. Figures 3 and 4 show the extent of various types of road surface i n 1953 compared with 1969. In a d d i t i o n to weighting l i n k s by road q u a l i t y , i t was also f e l t that, the presence of f e r r i e s presented a delay f a c t o r on the l i n k s on which they operated. For t h i s reason, l i n k s containing f e r r i e s were extended by half-an-hour's t r a v e l l i n g time, according to the q u a l i t y of the road. This i s a simple a p p l i c a t i o n of the concept of opportunity cost. F e r r i e s represent more of a r e l a t i v e delay on good roads than on poor ones. The data for highway q u a l i t y were obtained from several sources. The B. C. Dept. of Highways supplied the dates and locat i o n s of a l l major road improvements.- Road maps, both the o f f i c i a l B. C. t o u r i s t maps and those sponsored by o i l companies, dating back to 1953 provided most of the data, however, and were supplemented by o f f i c i a l Survey maps. Although some e f f o r t was expended i n order to simulate the r e l a t i v e q u a l i t i e s of the l i n k s , some, inadequacies s t i l l remained. In p a r t i c u l a r , no account was taken of the number of curves, i n c l i n e s or passing places, or, indeed, of flow-capacity r e l a t i o n s h i p s as they a f f e c t t r a v e l time. These factors working together can reduce speeds considerably even when roads are f a r from reaching s a t u r a t i o n point. The author found, while d r i v i n g a small German car that speeds on two-lane roads (paved) could, i n parts, vary between 30 and 70 m.p.h. i n normal d r i v i n g conditions. Nevertheless, the overwhelming data requirements of these factors were f e l t to outweigh t h e i r u t i l i t y , e s p e c i a l l y i n view of t h e i r n o n - a v a i l a b i l i t y for the e a r l i e r of the s i x time-intervals used i n t h i s a n a l y s i s . Figure 3 Figure 4 25 Except where comprehensive transportation studies have been c a r r i e d out, o r i g i n - d e s t i n a t i o n flow data are not usually a v a i l a b l e . This i s the case i n B. C. Simple, non-directional v e h i c l e counts for J u l y and August of each year are the only data a v a i l a b l e ( B r i t i s h Columbia, Dept. of Highways). The o b j e c t i v e i n t h i s chapter i s to attempt the simulation of t h i s flow by c e r t a i n parameters. A comprehensive t r a f f i c simulation would require as b a s i c input data a l l t r a f f i c generators within and surrounding the province. This i s c l e a r l y not a f e a s i b l e p r o p o s i t i o n f o r a study of t h i s scope, p a r t i c u a r l y i n view of the emphasis here on trends over time. The method of analysis used involved three components: a l i n k importance index, masses associated with the nodes and the g r a v i t y model. Construction of the l i n k importance index was r e l a t e d to that of the route access index. Using the shortest path between a l l pairs of nodes i n the e n t i r e network and measured i n terms of time-distance, each l i n k was weighted by i t s frequency of occurrence i n these paths. In addition, each of these paths was weighted according to the gravity model: T i j = ( p i - V / D i e j where T.. = the number of t r i p s between node i and node j i l V = the population of node i D,. = the distance between i and j e = an exponent The value of l i q u o r s a l e s , a s e n s i t i v e measure of economic a c t i v i t y with the added advantage of e x c e l l e n t a v a i l a b i l i t y s p a t i a l l y and temporally was used as the population weighting (see Chapter 6 ) . 26 C a l i b r a t i o n of the Model Weighted l i n k importance indices were computed on a cross- ! s e c t i o n a l basis for 1966. In t h i s computation, the gravity model was c a l i b r a t e d by i t e r a t i o n of i t s exponent from 2.2 to 2.8. This range of values was then used to p r e d i c t the l e v e l of t r a f f i c flow for that year on each of the l i n k s . The performance of these indices i s shown i n 2 Table I I . The r increased with successive i t e r a t i o n s of the exponent u n t i l a maximum was reached at 2.7 above which i t declined. This f i g u r e i s higher than that of most gr a v i t y model studies and seems to suggest the r e l a t i v e d i f f i c u l t y of movement i n a mountainous province. On the other hand, the parameters used here, population and time-distance, have not n e c e s s a r i l y been used i n other studies so comparability i s not ensured. Having c a l i b r a t e d the model and having found that an exponent of 2.7 on the distance component gave the best f i t for 1966, the next stage i n the a n a l y s i s sought to study the behaviour of the exponent over time. As c a l i b r a t i o n of the g r a v i t y model f o r a l l the time-periods would have been expensive to the extent of being u n e t h i c a l i n terms of computing time, the exponent that gave the best f i t i n 1966 was used but population was replaced by l i q u o r sales as the masses. Link importance indices were, therefore, computed for 1953, 1956,. 1960 and 1963 i n a d d i t i o n to 1966. 2 The r e s u l t s are shown i n Table I I I . The r increased s t e a d i l y over time. This was thought to be the r e s u l t of two po s s i b l e f a c t o r s . F i r s t l y , i t seems to suggest that there was a trend from essen-t i a l l y l o c a l t r a f f i c i n 1953 for which the l i n k importance index would be expected to give a low p r e d i c t i o n , towards the highway network acting i n c r e a s i n g l y as an integrated system i n which i n t e r r e g i o n a l flows played 27 a more s i g n i f i c a n t r o l e . As the l i n k importance scores are computed from the shortest paths between a l l p a i r s of nodes i n the network they embody the i m p l i c i t assumptions that the network functions as a s i n g l e , h i g h l y - i n t e r r e l a t e d , system. Hence the better p r e d i c t i o n of flow over time by t h i s index seems s u f f i c i e n t evidence for the increasing i n t e g r a t i o n of the highway system. Secondly, the data r e l a t i n g to e a r l i e r periods may be inaccurate. TABLE II Performance of D i f f e r e n t Values of the Gravity Model Distance  Exponent i n P r e d i c t i o n of T r a f f i c Flow, 1966. Exponent 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2 r 0.62 0.66 0.70 0.75 0.79 0.81 0.80 TABLE III Performance of Gravity Model with Liquor Sales as Masses  and Distance Exponent of 2.7, over time. Year 1953 1956 1960 1963 1966 2 r 0.102 0.196 0.425 0.544 0.711 Analysis of Residuals Residuals from the ( l i n e a r ) regression are shown i n Figures 5, 6 and 7. Persistence of the Okanagan and the Kootenay regions as p o s i t i v e t r a f f i c flow r e s i d u a l s i s a s t r i k i n g feature. In contrast, negative r e s i d u a l s are associated with the highways that have undergone s i g n i f i c a n t s t r u c t u r a l improvements. Thus the opening of the Salmo-28 Creston Highway i n 1964 i s r e f l e c t e d i n the 1966 data i n which many of the Kootenay highways have become negative r e s i d u a l s . F a i l u r e of t r a f f i c flow l e v e l s to keep pace with highway improvements i s implied by these r e s u l t s . These lags are probably due to the i n a b i l i t y of«the Kootenays to respond to changing a c c e s s i b i l i t y patterns. The lack of new i n v e s t -ment i n that region may be r e f l e c t e d here. This can be argued since i t i s the r e a l i s a t i o n of new investment rather than the r e l o c a t i o n of mature c a p i t a l stock that i s most responsive to these changes. The Trans-Canada and Cariboo Highways are characterised by negative r e s i d u a l s throughout the a n a l y s i s . Perhaps the continued improvement of these routeways has led the in c r e a s i n g l e v e l s of t r a f f i c flow. The i n d i v i s i b l e nature of some types of highway investment may be a f a c t o r i n t h i s l a g . The Roger's Pass shows t h i s . This r e s u l t therefore suggests some feedback from the government p o l i c y emphasising highway investment. A l l major routes connecting regions i n I n t e r i o r B. C. have les s t r a f f i c than predicted except for the Hope-Princeton Highway. This p o s i t i v e r e s i d u a l was thought to be a r e s u l t of t r a v e l l e r s ( e s p e c i a l l y t o u r i s t s ) between Vancouver and the Okanagan tending to take a c i r c u l a r route; one t r i p would be along the Trans-Canada, the other along the Hope-Princeton Highway. The strong p o s i t i v e r e s i d u a l associated with the Okanagan implies the existence of an a t t r a c t i v e f a c t o r such as the t o u r i s t industry. Inspection of the p o s i t i v e r e s i d u a l s can help i s o l a t e where highway improvements are most needed, despite the obvious omission of capacity constraints i n t h i s a n a l y s i s . RESIDUALS OF TRAFFIC FLOW (Y) AND LINK IMPORTANCE (X), 1953 Figure 5 KEY TO LINKS1 — — — TRAFFIC GREATER THAN PREDICTED ===== TRAFFIC LESS THAN PREDICTED NO DATA NODES ARE AS GIVEN IN PREVIOUS FIGURES Figure 6 Figure 7 32 A second p o s s i b i l i t y i s that the actual t r a f f i c flow has been measured i n c r e a s i n g l y accurately over the years. It i s not inconceivable that, i n 1953 for example, data could be u n r e l i a b l e . Nevertheless, the trend i n the r e l a t i o n s h i p s between flow and l i n k importance over time seems to be s i g n i f i c a n t . Lead-lag Relationships So f a r i n the a n a l y s i s , i t was assumed i m p l i c i t l y that there existed a balanced r e l a t i o n s h i p temporally between t r a f f i c fiow and the l i n k importance index. Under t h i s assumption highway improvements, as r e f l e c t e d i n the l i n k importance index would be presumed to r e s u l t i n instantaneous readjustments i n the t r a f f i c flow strategy. In Wohl and Martin (1967), f or example, c r o s s - s e c t i o n a l approaches to the analysis of t r a f f i c flow are reviewed. None, however, looked at the temporal aspects, the lead-lag r e l a t i o n s h i p s between changes i n network str u c t u r e and the changing flows associated with them. As far as programming models are concerned, the problem becomes one of dynamic programming, the complexity of which i s f a r i n excess of that of the l i n e a r models. Yet even e m p i r i c a l l y oriented analyses seem to have ignored such aspects. An analysis designed to discover e x p l i c i t l y the lagged r e l a t i o n -ships, i f any, between changes i n the str u c t u r e of the highway network and the t r a f f i c flow was therefore performed. For t h i s purpose canonical c o r r e l a t i o n methods were employed (see Appendix 3). S u f f i c e i t to state here that the technique seeks to r e l a t e two sets of v a r i a b l e s . Inspection of the c o e f f i c i e n t s may reveal a v a r i a b l e of the f i r s t set to be strongly 33 r e l a t e d to a v a r i a b l e of the second s e t . A number of roots are extracted. Each of these shows the various ways i n which c e r t a i n v a r i a b l e s may be i i r e l a t e d to c e r t a i n other v a r i a b l e s . In t h i s a n a l y s i s the f i r s t set of v a r i a b l e s was t r a f f i c flow i n 1953, 1956, 1960, 1963 and 1966. The second set was the l i n k importance index for the same i n t e r v a l s . Results are shown i n Table IV. The general trend suggests that the l e v e l of t r a f f i c at a given period i s more highly r e l a t e d to s t r u c t u r a l properties of a preceding period than to the con-temporary one. That -this i s most marked for t r a f f i c flows between 1956 and 1963 may be a r e f l e c t i o n of s i g n i f i c a n t s t r u c t u r a l changes over those years,. The bond between 1960 and 1963 on the t h i r d root may be the r e s u l t of a time-reversal f a c t o r . Flow i n 1960 may i n f a c t be r e l a t e d to highway stru c t u r e p r i o r to 1953 but the omission of t h i s v a r i a b l e may therefore have r e s u l t e d i n the outcome shown. As the r e l a t i o n s h i p appears on t h i r d root, (canonical c o r r e l a t i o n of 0.52 only,) not too much s i g n i f i c a n c e should be attached to i t . In general, i t would appear that t r a f f i c flow became most adjusted to highway s t r u c t u r e about three years a f t e r s t r u c t u r a l change. This conclusion i s r e i n f o r c e d by the behaviour of t r a f f i c flow r e l a t e d to the Roger's Pass shown i n Figure 8. Between Kamloops and Cache Creek, and betweten Hope and Lytton i n the Fraser Canyon, increasing flow d i d not reach a maximum u n t i l about three years a f t e r opening of the Pass. There are, of course, many forces at work to influence the magnitude and routing of flows. For t h i s reason, these r e s u l t s , taken together, are not conclusive but they are suggestive. TABLE IV - RESULTS OF CANONICAL CORRELATION ANALYSIS OF TRAFFIC FLOW AND LINK IMPORTANCE INDEX: LEAD LAG-RELATIONSHIPS Canonica l Coe f f i c i en t s T r a f f i c Flow L i n k Importance Root 1953 1956 1960 1963 1966 1953 1956 1960 1963 1966 fk~ 1 -0 .33 -0.07 -0.09 0.36 0.96 0.23 -0.40 -0.10 -1 .51 2.73 0.92 2 0.24 0.20 0.96 0.39 -1.14 -1.31 2.81 . 0.90 -3.34" 1.16 0.80 3 0.60 1.71 -1 .78 -1.24 0.91 4.68 -4.00 0.41 -6 .23 5.39 0.52 4 1.06 -0.79 -0 .93 3.90 -3.36 0.58 -4.39 1.70 .1.03 1.27 0.25 A = Canonica l C o r r e l a t i o n T r a f f i c Flow L i n k Importance Index 1953 1956 1960 1963 1966 (1) = Root TRANS-CANADA n 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1952 1954 1956 1958 I960 1962 1964 1966 1968 VALUE FOR EACH YEAR IS AN AVERAGE OF JULY AND AUGUST TRAFFIC VOLUMES SOURCE: B.C. DEPT. OF HIGHWAYS, SUMMER TRAFFIC VOLUMES Figure 8 36 D i f f e r e n t i a l Impact of Highway Improvement i The graphs i n Figures 8 and 9 also show c l e a r l y the r o l e of ' highway improvement i n changing the strategy of flow through the network consequent upon the opening of the Roger's Pass and the Yellowhead Pass. Counts taken north of Hope on the Trans-Canada Highway demonstrate a dramatic increase i n flow r e l a t i v e to previous years and r e l a t i v e to the flow following the Hope-Princeton route. The d i f f e r e n t i a l d i s t r i b u t i o n of t h i s flow implies that the e f f e c t of o v e r a l l c y c l i c a l trends i n the r e g i o n a l or n a t i o n a l economy can be l a r g e l y eliminated. The lower reading f o r 1965 on the Hope-Princeton Highway presumably r e f l e c t s the deterrent e f f e c t of the l a n d s l i d e that occurred i n January of that year. T r a f f i c counts taken west and east of Kamloops show a spectacular increase i n flow. Between 1962 and 1964 the flow had increased by i d e n t i c a l amounts at the two s t a t i o n s . A f t e r 1967, trends change s i g n i f i c a n t l y . Whereas the flow west of Kamloops again rose sharply, flow east of that c i t y f e l l sharply. At this time, the Yellowhead Pass had been opened. A t r a f f i c count north of Kamloops showed that the decline i n t r a f f i c east of Kamloops was compensated for by the increased flow northwards. Presumably, t h i s i s t r a f f i c bound f o r Jasper and Edmonton that has been rerouted from the Roger's Pass. i 13-, 12 = 10. 9. DAILY SUMMER TRAFFIC 7 (IN THOUSANDS OF VEHICLES) 6, INCREASING IMPORTANCE OF THE TRANS-CANADA HIGHWAY (FRASER CANYON ROUTE) RELATIVE TO HOPE - PRINCETON ROUTE 1952 T r 1954 1956 TRANS-CANADA HIGHWAY WEST OF HOPE TRANS-CANADA HIGHWAY NORTH OF HOPE (FRASER CANYON) HOPE-PRINCETON ROUTE 1 1968 VALUE FOR EACH YEAR IS AN AVERAGE OF JULY AND AUGUST TRAFFIC VOLUMES SOURCE*- B.C. DEPT. OF HIGHWAYS. SUMMER TRAFFIC VOLUMES Figure 9 CHAPTER A GRAPH THEORY AND THE MEASUREMENT OF ACCESSIBILITY The argument of the previous chapter sought to i n v e s t i g a t e the nature of flow through a network i n r e l a t i o n to general t o p o l o g i c a l c h a r a c t e r i s t i c s of networks. As a r e s u l t a t t e n t i o n was confined to the r o l e of l i n k s i n ca r r y i n g these flows and to changes i n the network that re-routed them. Nodes were l a r g e l y ignored. I t i s apparent, however, . that flow i s also routed through the nodes and i n real-world conditions i t can have considerable influence on them. This being the case the r o l e of nodes i n the network can be studied with some relevance. This involves consideration of the r e l a t i v e l o c a t i o n or a c c e s s i b i l i t y of nodes. The f i r s t o b j e c t i v e i s to show the intimate r e l a t i o n s h i p between the t o p o l o g i c a l properties of a graph and the a c c e s s i b i l i t y of the nodes to each other and to the paths of the network. Using the perspective of graph theory i t i s po s s i b l e to derive the concept of a c c e s s i b i l i t y from the p r i m i t i v e s of a network: the l i n k s , nodes and the f u n c t i o n a l r e l a t i o n s h i p s between the nodes which are transmitted by t h e . l i n k s . Consider a f i n i t e set of nodes X = (X^, X2,..., X n) and a mapping F of X in t o X. The p a i r G = (X, F) defines a graph (G) of order n. Consider a l s o a set of l i n k s U. Then G = (X, F) = (X, U) ind i c a t e s that the functions between the nodes are i d e n t i f i e d with the l i n k s connecting the nodes. Figure 10a i l l u s t r a t e s t h i s f o r the case of dire c t e d l i n k s . 38 40 In this*example, F(X ) = (X 0, X_, X_, X.) shows that nodes X., X„, X ' 1 I j 5 6 Z 3 5 and X, are d i r e c t l y a c c e s s i b l e to X.. To reach X, requires the use of 6 1 4 i n d i r e c t l i n k s . S i m i l a r l y , F(X 2> = ( X 5 ) , F(X 3> = 0, F(X^) = (X 3>, F(X C) = (X.) and F(X,) = 0, where 0 i s the n u l l s e t . The i n t e r p r e t a t i o n 5 4 o of F ( ^ n ) as a c c e s s i b i l i t y to the given node follows from t h i s . Links may also be r e f l e x i v e , that i s , having the same value i n e i t h e r d i r e c t i o n . In the same example the l i n k (X, , X,), i f r e f l e x i v e , 1 o would give F(X^) = (X^). The concepts of d i r e c t e d and r e f l e x i v e l i n k s have an immediate empirical i n t e r p r e t a t i o n as one-way and two-way s t r e e t s r e s p e c t i v e l y . That a l i n k i s r e f l e x i v e , however, does not n e c e s s a r i l y imply that the distance function or value of the l i n k i s the same for both d i r e c t i o n s . In t h i s case a graph may be drawn as shown i n Figure 10b, allowing (X^, X..) = (X^, X ±) . Each of the l i n k s i n Figure 10b j o i n s two nodes that are imme-d i a t e l y adjacent so that the procedure shown produces an enumeration of a l l d i r e c t l i n k s . In the case of a more complex network t h i s needs to be extended to include consideration of i n d i r e c t paths as w e l l . This can.be accomplished by the method of t r a n s i t i v e closure. Following Kaufmann (1967) t r a n s i t i v e closure may be defined as follows. Given a 2 3 n f i n i t e graph G = (X, F), the mappings F , F F can be w r i t t e n as F 2 (X.) = F(F(X.)) F 3 (X.) = F(F 2(X.)) = F ( F ( F ( X . ) ) ) . The t r a n s i t i v e closure of F i s a mapping F* of X i n t o X, defined by F(X,) = (X.) U F(X.) U F 2(X.) U F 3(X.) U . . . U F n ( X . ) . i l x x x x The following example i n Figure 10c may make t h i s somewhat more trans-parent. From the graph shown i t can be seen that a c c e s s i b i l i t y v i a 41 d i r e c t l i n k s i s given by: FCXj) = (X2) • F(X 2) = (XX, X3, XA) F(X3) = (X2, X4) F(X4) = (X2, X3) and v i a paths two l i n k s i n length by: F2(X1) = F(X2) = (XR X3, X4) F2(X2) = F(X1, X3, X4) = (X2, X3, X4) F2(X3) = F(X2, X4) = (Xv X2, X3, X4) = X F2(X4) = F(X2, X3) = (X^  X2, X3, X4) = X. Via paths of three l i n k s a c c e s s i b i l i t y i s given by: F 3 ^ ) = F(XR X3, X4) = (X2, X3, X4) F3(X2) = F(X2> X3, X4) •=(X 1, X2,X3, X4) = X F3(X3) = X F3(X4) = X. F i n a l l y , a c c e s s i b i l i t y v i a paths four l i n k s i n length i s given by: F4(X1) = F(X2, X 3, X4) = (X1, X2, X3, X4) = X F4(X2) = X F4(X3) = X F4(X4) = X. * * * * Therefore, i n t h i s example F(X1) = F(X2> = F(X3) = F(X4) = X at the fourth i t e r a t i o n of the mapping. The s i g n i f i c a n c e of t h i s number of i t e r a t i o n s w i l l be conjectured i n the following argument. It i s remarkable that t h i s process can be r e p l i c a t e d by standard methods of matrix algebra (Shimbel, 1951). Shown below i s the adjacency or connection matrix.for the graph i n Figure 10c. C e l l s containing ones 42 represent the existence of a d i r e c t connection where zeros i n d i c a t e no such connection. x, x 2 x 3 • 1 0 0 1 1 0 1 1 / 0 1 0 4 0 \ 1 Following Berge (1962) there i s a theorem that i f G i s a graph and A i t s adjacency matrix, the element of the matrix P = A n (that i s , the matrix product of A with i t s e l f n times) i s equal to the number of d i s t i n c t paths of length n which go from to X_.. (For the proof see Berge, page 131). 2 2 Consider an element P .. of A . I t i s the inner product of the i j j t h row vector of A with the .th column vector of A: P I j = ( P i l ' Pi2> P i 3 ' P i 4 ) • / P l j 2j \ 3j 4 k = 1 P .P i k * r k j = (P., P „ . + P . „ P „ . + P . „ P i l ' l j 12 2j 13 3j + P., P. .) i 4 4 j ' From the example i n Figure 10c t h i s procedure gives /0 1 0 0\ / l 0 1 l \ jo 3 A = 1 0 1 1 0 1 0 1 \0 1 i o / A 2 =4 0 3 1 1 1 1 2 1 \ l 1 1 2 A 3 = / \ l 3 2 4 4 1 1\ 4 4 2 3 3 2 / 43 / 3 2 4 M 2 11 6 6 4 6 "7 6 \* 6 6 7 ) I t can e a s i l y be checked that , for example, i n matr ix A there are , for 3 X ^ , 3 paths of length 2 between and i t s e l f , that i n matr ix A there are 3 paths of length 3 between X^ and but none of that length between X^ and i t s e l f . 2 3 4 A d d i t i o n of these m a t r i c e s , A + A + A + A , produces a new matr ix (T) from which can be obtained the t o t a l number of paths up to length 4 between any two nodes i n the network. Each element t . . i s a measure of d i r e c t and i n d i r e c t c o n n e c t i v i t y between nodes X . and X , . i j The i n d i r e c t r e l a t i o n s i n c l u d e both elementary and redundant paths . Summing the t values for a p a r t i c u l a r column of the matr ix , that i s , for node X , gives an element i n a vec tor that i s a measure of the acces -s i b i l i t y of node j to a l l other nodes i n the network. I t may now be noted that the t r a n s i t i v e c lo sure and the adjacency matr ix methods were both terminated at the same stage (the fourth i t e r a t i o n , i n t h i s case ) . This r e l a t i o n s h i p seems to h o l d for graphs of vary ing c h a r a c t e r i s t i c s . I t may be conjectured therefore that the method of t r a n s i t i v e c lo sure i s s i m i l a r to that of powering a matr ix u n t i l the s o l u t i o n matr ix i s obta ined. Hence these two concepts of a c c e s s i b i l i t y are e s s e n t i a l l y the same. Indeed, i n cases where the t r a n s i t i v e c lo sure J does not terminate a non-zero s o l u t i o n matr ix does not e x i s t . Condit ions have been sought r e c e n t l y for the exis tence of the s o l u t i o n matr ix of a graph. Werner et a l (1968) found that a necessary c o n d i t i o n i s that the 44 graph should contain at l e a s t one c i r c u i t with an odd number of edges. Alao (1970) gives necessary and s u f f i c i e n t conditions for strongly connected d i r e c t e d graphs. In general, however, the existence of the s o l u t i o n matrix has not been determined and, i n a d d i t i o n , Alao's method requires, for large networks, the s o l u t i o n of a c h a r a c t e r i s t i c equation of high degree. An a l t e r n a t i v e method i s to stop the powering procedure of the adjacency matrix at the diameter of the graph. This eliminates the existence problems r e l a t e d to the s o l u t i o n matrix. I t gives a matrix s e r i e s with each term containing the number of a given length up to that of the longest short-path i n the graph. Summation of these matrices produces the non-zero matrix T described above. Katz (1953) modified t h i s method by allowing the incorporation of a distance decay parameter. This d e f i n i t i o n takes the form: . ' T = sA + s 2 A 2 + s 3 A 3 + . . . + s 1 1 A n, where s i s a s c a l a r , 0_<s<l. Unfortunately t h i s m o d i f i c a t i o n does not enhance the deductive power of the method i n that the s c a l a r value must be chosen a r b i t r a r i l y or discovered e m p i r i c a l l y i n a manner s i m i l a r to ' c a l i b r a t i o n ' of the g r a v i t y model. This could be made o p e r a t i o n a l l y f e a s i b l e by comparing the a c t u a l distance-decay function from a p a r t i c u l a r node with that predicted and adjusting the s c a l a r accordingly. Although Shimbel's adjacency matrix method produces i n t u i t i v e l y reasonable r e s u l t s as an attempt to simulate and o p e r a t i o n a l l y define the r e l a t i v e a c c e s s i b i l i t y of places i n a network there are nonetheless some problems attached to i t . In p a r t i c u l a r , the p r e c i s e method by which i t 45 computes the number of chains of length n associated with a node i s not easy to accept. The method finds not only the d i r e c t , but also the 'I c i r c u l a r and therefore redundant, chains associated with a node i n the net work and gives these the same r e l a t i v e weight. This being so i t implies that the p r o b a b i l i t y of a t r i p from node X^ to X^ by a d i r e c t route i s equal to the p r o b a b i l i t y of the most i n d i r e c t route p o s s i b l e . Whilst i t i s conceivable that such a route might be u t i l i s e d by a " t r a v e l l i n g salesman" or by t o u r i s t s looking for c i r c u l a r routes i t seems improbable that t h i s route would be as important as the d i r e c t one f o r t r a v e l between X^ and X j . Despite these shortcomings t h i s method has been used with some success by Garrison (1960) and Gauthier (1966, 1968a). I n t u i t i v e l y more acceptable methods are those of the Associated Number and the Shimbel Index. The l a t t e r i s o u t l i n e d i n Shimbel (1953) and i s defined as: n S(X^) = £ d(X^, X.), i = 1, n. where d = shortest-path 1 j = 1 ^ distance. I t measures the sum of the shortest path distances from a node X^ to a l l other nodes (X^.) i n the network. The node with the smallest t o t a l distanc i s the most a c c e s s i b l e . The Associated Number, e, (of a node X^) i s defined as: e(X.) = max d (X., X.) where d = shortest-path distance. 1 X.eX 1 J J Thus the Associated Number of a node X^ i s the maximum shortest-path distance i n the graph between X^ and X^. Accordingly a node on the periphery of the network i s l i k e l y to have a high Associated Number and thus a low a c c e s s i b i l i t y whereas a node near the centre w i l l have a smaller distance separating i t and i t s most d i s t a n t node. Some empirical 46 s tudies have shown that the Shimbel Index i s more s e n s i t i v e to network c h a r a c t e r i s t i c s than i s the Assoc ia ted Number ( G a r r i s o n , 1960; Burton, 1962; Kansky, 1963; Werner et a l , 1968). Both these ind ices are e s s e n t i a l l y measures of the c e n t r a l i t y of a node to the network. I m p l i c i t i n t h i s , however, i s a drawback. R a r e l y , i f ever, i s a t ransport network an i s o l a t e d system so that nodes on i t s per iphery may be on e i ther major i n t e r - r e g i o n a l or i n t e r n a t i o n a l routes and in f luenced by e x t r a - r e g i o n a l a c t i v i t i e s . Consequently, a node regarded as i n a c c e s s i b l e by these Indices may i n fac t be very a c c e s s i b l e i n terms of s t r a t e g i c l o c a t i o n r e l a t i v e to major routeways. For the measurement of nodal a c c e s s i b i l i t y two ind ices were used i n i t i a l l y i n the a n a l y s i s . One was the Shimbel Index, as o u t l i n e d above, and the o ther , c a l l e d the Route Access Index, was prev ious ly used by K i s s l i n g (1966). Computation of th i s measure involves s e v e r a l s teps , as fo l lows: ( i ) Determine, by the matr ix method, the shortes t -paths between a l l 1 p a i r s of nodes. ( i i ) Each time a l i n k appears i n a shor te s t -pa th i s counted. The most f requent ly used l i n k s are regarded as the most important i n that they would carry the most t r i p s or routes i f the nodes were of equal mass. (This i s the Unweighted L i n k Importance Index.) ( i i i ) Sum the scores of l i n k s to the nodes to which they are i n c i d e n t . The h igher the t o t a l score the more a c c e s s i b l e i s a node to major route -ways of the network. The l a t t e r index i s p a r t i c u l a r l y a t t r a c t i v e because i t attempts to s imulate the a c t u a l o r i g i n - d e s t i n a t i o n flows and t h e i r r o u t i n g . I t i s "'"See Appendix 2. 47 therefore s e n s i t i v e to the add i t i o n of new l i n k s that e f f e c t i v e l y by-pass nodes. S i g n i f i c a n t l y , new l i n k s may by-pass nodes i n a more general I sense than the l o c a l by-pass around a congested centre. For instance, the Kimberley by-pass constructed between Ta Ta Creek and Cranbrook i s e s s e n t i a l l y l o c a l . In contrast, the Rogers Pass rerouted along the Trans-Canada Highway many routes that would otherwise have taken the Southern T r a n s - P r o v i n c i a l , or No. 3, Highway. Owing to t h i s phenomenon, the manner i n which some nodes, and indeed, e n t i r e subgraphs of the network are by-passed i n t h i s system-wide context i s f a r from obvious i n t u i t i v e l y . For t h i s reason, the Route Access Index i s probably the best measure of a c c e s s i b i l i t y and i s the only one used i n the analysis here. Moreover, as demonstrated i n the chapter on t r a f f i c flow, i t s value i s enhanced by the p o s s i b i l i t y of weighting the routes with the masses or populations that would e f f e c t i v e l y demand them for t r a v e l . CHAPTER 5 EFFECT OF INCREASED ACCESSIBILITY Where improvements to the transport network are made, the r e s u l t i s to transform the e x i s t i n g a c c e s s i b i l i t y surface. This allows savings i n transport cost and time to be made. The nature of the impact of these savings on the space-economy depends on how they are d i s t r i b u t e d between d i f f e r e n t sectors of the economy. In order to develop some t h e o r e t i c a l understanding of these processes two a l t e r n a t i v e models may be envisaged. F i r s t l y , assume that the consumer pays the e n t i r e cost of transport (f.o.b. p r i c i n g ) . Given t h i s premise i t can be i n f e r r e d that increased a c c e s s i b i l i t y lowers the cost of goods purchased by the consumer. From t h i s and from the axioms r e l a t i n g supply and demand, i t follows that increased a c c e s s i b i l i t y r e s u l t s i n increased aggregate demand i n those centres a f f e c t e d . -Secondly, assume instead that the producer has to pay a l l transport costs. Then any improvement i n a c c e s s i b i l i t y l e v e l s w i l l lower h i s production costs. This, i t w i l l be shown, w i l l lead to a s u b s t i t u t i o n of transport inputs f o r other factors of production and to 1 increased output. Therefore the reduction of costs i n a place r e l a t i v e to other places gives that place a comparative advantage i n terms of the costs of production. The ra m i f i c a t i o n s of these models are now inves-t i g a t e d i n more d e t a i l . "4>rovided?Memand i s s u f f i c i e n t l y e l a s t i c to absorb t h i s increase. 48 49 E f f e c t s of Increased A c c e s s i b i l i t y : The Consumer Side J a n e l l e (1969) suggests that the process of s p a t i a l reorgani-s a t i o n i n the form of c e n t r a l i s a t i o n and s p e c i a l i s a t i o n w i l l accelerate most r a p i d l y at those places which stand to b e n e f i t most from increasing a c c e s s i b i l i t y . Berry (1960), reporting on h i s c e n t r a l place studies, found that as a c c e s s i b i l i t y was improved, people were more w i l l i n g to t r a v e l to higher order centres for the greater v a r i e t y of goods and ser v i c e s o f f e r e d there; he notes that c e n t r a l i s a t i o n took place through- • out the hierarchy. In part of Kentucky, Stroup and Vargha (1963) found that, as a r e s u l t of improved roads, the growth c h a r a c t e r i s t i c s of the l a r g e r centres were very d i f f e r e n t from those of the small ones. Larger centres showed, on average, a f o r t y per cent increase i n the number of operating businesses between 1950 and 1960. These increases were most marked with respect to r e t a i l and s e r v i c e businesses and coincided with the period of greatest a c t i v i t y i n road improvements. Moreover, the greatest decline i m t h e .number of open-country businesses occurred during the same period. In general, the a c t u a l number of centres and complementary t r i b u t a r y areas i s a product of the transportation that has been a v a i l -able. Many centres date from a time when communication was so poorly developed that many small settlements were needed as s e r v i c e areas. Numerous small centres are now i n a process of continuing r e l a t i v e and often absolute d e c l i n e . This phenomenon i s not confined to the postwar era. Zimmerman (1938) found i n the P r a i r i e Provinces a concentration of investment i n the l a r g e r trading centres as they grew with improved transportation f a c i l i t i e s . Some smaller centres a c t u a l l y migrated p h y s i c a l l y 50 to l o c a t i o n s on new routeways. Growth of more a c c e s s i b l e centres was observed by Kolb and Poison (1933), who found that as b e t t e r t r a n s p o r t a t i o n f a c i l i t i e s were p r o v i d e d , i t became p o s s i b l e to b e n e f i t from increased economies of s c a l e as people were able to t r a v e l fur ther tq obta in t h e i r needs. Consequently, l o c a l f a c i l i t i e s were by-passed for l a r g e r - s c a l e f a c i l i t i e s . E m p i r i c a l l y i t seems that the impact of improved t ransport has been markedly d i f f e r e n t i a l . As transport networks have developed, as t ransport technology has advanced, equal benef i t s have not been con-f e r r e d on a l l e x i s t i n g routes , but ins tead have increased the r e l a t i v e importance of a few "primary nodes." T h i s has been accomplished by producing a s u b s t a n t i a l change i n the s trategy of t r a f f i c flow or by more i n t e n s i v e use o f c e r t a i n routes and nodes. The quest ion a r i s e s as to how c e r t a i n nodes have gained a p o s i t i o n of primacy over o t h e r s . With th i s i n view, examination of a group of s tud ies wi th a s o c i o l o g i c a l perspec t ive i s i l l u m i n a t i n g . The work of Hass inger (1957a, 1957b), Brunner and Smith (1944) and Hodge (1965, 1968) has viewed the process of urban growth and d e c l i n e as a complex f u n c t i o n . Taking an evo lut ionary approach, these s tudies have thrown up the v i t a l concept of the i n i t i a l advantage of a centre . Hassinger (1957a) puts i t c l e a r l y : Centres over 5000 (populat ion) may have s t a b i l i s e d compet i t ive r e l a t i o n s i n t h e i r areas at an e a r l i e r time, and may have more near ly dominated rather than r i v a l l e d surrounding p l a c e s . In such an adjustment, the smal ler places may take on some of the c h a r a c t e r i s t i c s of suburbs, that i s , they provide housing and c e r t a i n b a s i c s e r v i c e s , and the l a r g e centres prov ide employment and s p e c i a l i s e d s e r v i c e s . 51 • Pred (1965) takes up the same theme but at a d i f f e r e n t s c a l e . The manner i n which transport a f f e c t s d i f f e r e n t l e v e l s of the I hierarchy has been suggested by Parr and Denike (1969). The scheme i s shown i n Figure 11. In the diagram, the aggregate demand curve (AR^ .) i s composed of the demand curves of i n d i v i d u a l consumers located at d i s -tances of 0, 10 and 20 miles from an urban centre. Costs of transport are included i n the a c t u a l p r i c e paid by the consumer for any good demanded. For t h i s reason, the r e a l p r i c e l i n e f o r the good, P^T, slopes upwards to the r i g h t . Given the nature of the average cost curve and the i n i t i a l absence of excess p r o f i t s the t o t a l demand f o r the good consists s o l e l y of consumers up to 10 miles away and i n the absence of e f f e c t i v e demand from there, production w i l l not take place. Parr and Denike go on to show how improved transport technology ( i n reducing the r e a l p r i c e paid by the consumer) extends the market area. In the diagram, i t i s c l e a r that, as a r e s u l t of l a r g e r demand, the profit-maximising p r i c e i s lower and output greater. This i s also due to the f a c t that the producer faces a f a l l i n g cost curve owing to the r e a l i s a t i o n of s c a l e economies over t h i s range of production. Deductive inference from t h i s model suggests that a good i n i t i a l l y supplied at one l e v e l of the hierarchy would, as a r e s u l t of transport improvement be supplied i n c r e a s i n g l y at a higher l e v e l having a larger market area. This process would r e s u l t i n "economies of s c a l e " for the consumer i n the form of multi-purpose t r i p s . Consequently, e x i s t i n g l a r g e r centres are favoured at the expense of smaller centres which are by-passed. I t i s apparent that t h i s d i f f e r e n t i a l e f f e c t does not neces-s a r i l y require the smaller centres to be p h y s i c a l l y by-passed or f o r COMPOSITION OF AGGREGATE DEMAND WITH DIFFERENT TRANSPORT RATES S = 0 S=IO S = 20 Q h Q i <Ji <"s <i2 <u q» AGGREGATE DEMAND INDIVIDUAL DEMANDS SOURCE: PARR J. a AND K.S. DENIKE (1969) Figure 11 t o 53 t r a f f i c to be r a d i c a l l y re-routed through the transport network. Never-theless, the presence of by-passes around smaller centres w i l l augment the extent of the impact. Paradoxically, the reverse trend can be t h e o r e t i c a l l y a n t i c i p a t e d through the higher l e v e l s of the hierarchy. As some of the favoured centres i n the lower echelons expand t h e i r market-areas they are i n c r e a s i n g l y able to take on higher-order functions, previously performed only by the l a r g e s t centres. Parr and Denike see t h i s as "a form of r e g i o n a l or sub-regional import s u b s t i t u t i o n . " They also point out an important feature of these functions that f a c i l i t a t e s t h i s process of downward movement through the hierarchy; there i s no p o s s i b i l i t y of multi-purpose t r i p s as these functions, such as wholesale d i s t r i b u t i o n , do not require consumer t r a v e l . In sum, the conceptual framework of this model suggests that the v i a b i l i t y of r egional centres may be enhanced by the continual up-grading of transport f a c i l i t i e s . Thus the impact of transport improvement i s s p a t i a l l y d i s c r i m i n a t i n g i n two respects. On the one hand, i t w i l l change the importance of various l e v e l s of the c e n t r a l place hierarchy r e l a t i v e to each other. On the other, the routing f a c t o r i m p l i c i t i n the s t r u c t u r a l properties of transport networks w i l l transform the r e l a t i v e l o c a t i o n s of the urban nodes. Furthermore, these two aspects are d i s -tinguishable only i n a t h e o r e t i c a l sense. In p r a c t i c e , they are h i g h l y i n t e r r e l a t e d and c o n t i n u a l l y r e i n f o r c e each other. Network change may enhance the s t r a t e g i c p o s i t i o n of a node enabling i t to extend i t s market-area at the expense of competing centres. 54 E f f e c t s of Increased A c c e s s i b i l i t y : The Producer Side I t i s axiomatic i n the micro-economic theory of production that "reduction i n the p r i c e of a factor of production i s followed by the s u b s t i t u t i o n of t h i s f a c t o r f o r other factors of production. In the previous s e c t i o n , a t t e n t i o n was focussed on the impact on consumer demand using the assumption of f.b.b. p r i c e . In f a c t , the reduction i n trans-port costs i s l i k e l y to be shared between consumer and producer. Assuming the d e l i v e r y p r i c e of a good i s a l l or p a r t l y paid by the producer, then a f a l l i n the cost, of transportation w i l l allow the producer to s u b s t i t u t e t r a n s p o r t - i n t e n s i v e forms of i n d u s t r i a l organisation f o r other forms. E x p l i c i t l y , the producer can consolidate production or d i s t r i b u t i o n into a smaller number of more e f f i c i e n t units with l a r g e r marketing areas, thereby r e a l i s i n g economies of sc a l e and r a t i o n a l i s a t i o n . For t h i s reason, transport improvement w i l l encourage manufacturing to seek lo c a t i o n s i n l a r g e r centres, that i s , to become more market-orientated i n that they minimise d i s t r i b u t i o n rather than raw m a t e r i a l costs. Mohring and Harwitz (1962) suggest how t h i s s u b s t i t u t i o n may be conceptualised diagramatically (Figure 12). From the graph, i t follows that the f i r m can reduce i t s costs below those e n t a i l e d i n operating at the point R q by s u b s t i t u t i n g T^ amounts of transportation f o r other f a c t o r s . Viewed i n t h i s framework, the e f f e c t of transportation improvement i s to make the constant expenditure l i n e s less steep. This i s shown by M^R^ and ^2^2° r a m i f i c a t i o n s of t h i s are twofold. F i r s t l y , there i s an immediate reduction i n expenditure f o r inputs from S^M^ to S^M^', saving M^M^' i n transport costs. .Secondly, by s u b s t i t u t i o n of trans-p o r t a t i o n f o r other inputs (or movement along the production i n d i f f e r e n c e curve Q Q to R ) costs are further reduced by M 'M0. SOURCE-" MOHRING AND HARWITZ (I962),p.30 Figure 12 RESOURCE QUALITY ACCESSIBILITY Figure 113^ 56 In the case of production which i s c l o s e l y t i e d to non-ubiquitous primary resources, some modification of these general s t a t e -ments i s necessary. Nevertheless, conceptually comparable processes are at work. The q u a l i t y of a c c e s s i b i l i t y i s , within l i m i t s , a s u b s t i t u t e fo r the q u a l i t y of resources (Figure 13). A c c e s s i b l e ore deposits, even though low-grade, may be worked before high-grade deposits f a r removed from e x i s t i n g transport f a c i l i t i e s . Given two deposits of equal q u a l i t y then, other things being equal, the most a c c e s s i b l e deposit w i l l be worked. The d i s t i n c t i o n between "deposit" and "resource" i s fundamental. A deposit only becomes a resource i f there is" a demand f o r i t , i f the technology i s a v a i l a b l e to process i t and i f i t i s s u f f i c i e n t l y a c c e s s i b l e to market to be able to compete e f f e c t i v e l y with s u b s t i t u t e s . Given the a c t u a l d i s t r i b u t i o n of resources, changes i n production are l i k e l y to be due to any or a l l of these three f a c t o r s : an increase i n demand, an advance i n technology and improved a c c e s s i b i l i t y . I t may, therefore, be an i n t r a c t a b l e problem to gauge the p r e c i s e e f f e c t of increased acces-s i b i l i t y i n the case of primary a c t i v i t i e s . Although the impact of transport improvement i s s p a t i a l l y d i f f e r e n t i a t e d so also i s that of demand and technological c a p a b i l i t y owing to the s p a t i a l l y heterogeneous nature of resource types. The Transportation Component i n Growth Poles The theme of t h i s chapter has contended that the most s i g n i -f i c a n t feedback from transportation improvement i s the emphasis thereby placed on c e r t a i n urban centres i n the transport network. This argument 57 i s p a r a l l e l e d by, and r e l a t e d to, the theory of growth poles or unbalanced r e g i o n a l growth. Two d e f i n i t i o n s serve to sketch the e s s e n t i a l nature I of t h i s viewpoint: Growth does not appear u n i v e r s a l l y at any one time but manifests i t s e l f at points or poles of growth with v a r i a b l e i n t e n s i t i e s and d i f f u s e s through the economy by c e r t a i n channels. (Perroux, 1955) Odeplan (1967) defined a growth pole as: an urban center s u f f i c i e n t l y large enough, showing i n c i p i e n t growth at a s t r a t e g i c transport l o c a t i o n , and with s u f f i c i e n t p o t e n t i a l f o r additions to i t s i n f r a s t r u c t u r e to' a t t r a c t modern economic a c t i v i t i e s and extend i t s dynamic inf l u e n c e to the r e s t of the region. Nichols (1969) points out that as flows between urban centres are much stronger than between a centre and i t s h i n t e r l a n d , i t seems l i k e l y that the e f f e c t s of growth i n one centre w i l l spread i n "a leap-frogging fashion" through ..the economy v i a the urban s t r u c t u r e . This being the case, i t follows that the.growth prospects of a centre w i l l depend more on i t s q u a l i t y of a s s o c i a t i o n with other centres i n the urban system, that i s , on i t s a c c e s s i b i l i t y , than on i t s absolute s i z e . In a d d i t i o n , changes i n the degree of a s s o c i a t i o n between places should be expected to change the prospects f o r growth of these centres. As has been contended, th'e impact of transport improvement on the urban system i s markedly d i f f e r e n t i a l i n as much as i t favours the l a r g e r centres to the detriment of the small ones. Nichols suggests that these centres, as growth poles, can be viewed within the conceptual framework of Keynesian economics. Accordingly, the propensity to consume 58 l o c a l l y determines the nature of the m u l t i p l i e r e f f e c t s from new i n v e s t -ment such as highway b u i l d i n g : For most i n d i v i d u a l s , the demand for low-order goods i s very i n e l a s t i c . A d d i t i o n a l increments of income are, therefore, spent on higher-order goods and i t follows that these w i l l be attained i n l a r g e r towns. The smaller the town, the smaller w i l l be the marginal propensity to consume l o c a l l y and the m u l t i p l i e r e f f e c t w i l l therefore, a l s o be smaller. (Nichols, 1969, p. 44) CHAPTER 6 THE MEASUREMENT OF ECONOMIC GROWTH IN THE B. C. INTERIOR I t i s not a simple task to f i n d a representative measure of economic growth owing to the compounding of conceptual and p r a c t i c a l d i f f i c u l t i e s . Conceptually, i t i s far from c l e a r what i s meant p r e c i s e l y by economic growth except that the term implies the increasing s i z e of some relevant v a r i a b l e ( s ) . The choice of t h i s v a r i a b l e or set of v a r i -ables i s not obvious because there are two d i s t i n c t aspects to be con-sidered. F i r s t l y , growth may be defined as the increase i n e i t h e r the productive capacity or to the actu a l volume of production. In addition, the value added to raw materials or semi-finished products by the productive factors i n the region may be estimated. A l t e r n a t i v e l y , economic growth may be defined as the increase i n the set of f i n a l commodities a v a i l a b l e to the region. Siebert (1969)-notes that t h i s concept considers the e f f e c t of trade on the a v a i l a b i l i t y of commodities since the supply of f i n a l goods i s a r e s u l t of the b e n e f i t s from i n t e r - r e g i o n a l trade. This chapter i s devoted to a consideration of the advantages and disadvantages of e x i s t i n g data. Inadequacies of the Data Data problems are p a r t i c u a r l y severe i n the study area which includes numerous small settlements i n the I n t e r i o r of B. C. which are • • • • 59 60 viewed i n t h i s a nalysis over a time span of some f i f t e e n years. The only census data of any s t r u c t u r a l d e t a i l , employment categories, are not a v a i l a b l e for places of less than 30,000 population. In 1951 none of the communities considered here was above that threshold. R e t a i l s a l e s , as a s i n g l e summary measure, are a v a i l a b l e f or many more places but only for two or three i n t e r v a l s of time. Population numbers are l i k e w i s e a v a i l a b l e subject to the same co n s t r a i n t s . C l e a r l y there i s i n s u f f i c i e n t data a v a i l a b l e from t h i s source on which to perform very much a n a l y s i s , such as a consideration of what elements Q£ economic a c t i v i t y are most a f f e c t e d by tra n s p o r t a t i o n changes. Were t h i s census data to be employed i n analysis a further d i f f i c u l t y would a r i s e owing to the manner i n which the data have been c o l l e c t e d . The values associated with each v a r i a b l e such as population or r e t a i l s a l e s , r e l a t e to contiguous areas rather than to points. These a r e a l units are administrative i n nature and usually a r b i t r a r y or -tM-at l e a s t not designed s p e c i f i c a l l y f o r t h i s p a r t i c u l a r research problem. Concern here i s with the r e l a t i v e l y a r b i t r a r y boundaries of the i n c o r -porated places, which, i n most cases, are not mutually contiguous. There are, therefore, beyond the boundary of each incorporated place, areas of unorganised land for which there i s no data or at l e a s t data i n a form that prevents i t from being associated with the adjoining i n c o r -porated places. Now t h i s unorganised land i s p r e c i s e l y the area where new development and expansion of e x i s t i n g communities might be expected to take place. Of course, the boundaries of the incorporated settlements are extended p e r i o d i c a l l y to take i n the development ( p o t e n t i a l revenue) of the unincorporated areas. This, however, i s an a d d i t i o n a l source of 61 d i f f i c u l t y i n that extension of the administrative boundaries gives the impression of r a p i d growth i n that place when i n fact e x i s t i n g a c t i v i t i e s are being recorded for the f i r s t time. F i n a l l y , some measures of economic growth generally considered meaningful such as value added and the value of i n d u s t r i a l production are not a v a i l a b l e owing to d i s c l o s u r e r e s t r i c t i o n s imposed by the census. This phenomenon i s p a r t i c u l a r l y widespread i n B. C. where t y p i c a l l y a small settlement i s dominated by a large concern accounting for a very large percentage of the work-force and the value of production. Advantages of Liquor Sales Data In view of the .conceptual and data problems associated with the measurement of economic growth i t follows that an index i s required which summarises some of the e s s e n t i a l features of growth, i s a v a i l a b l e f or a l l communities and time-periods required and i s free of administrative boun-dary problems. This i s the minimum that i s required. Fortunately, the B. C. Liquor Control Board has c o l l e c t e d data on the sales of l i q u o r throughout the province. Liquor sales are thought to be a good i n d i c a t o r of general economic growth and of the p r o s p e r i t y of the settlements of the study area f o r s e v e r a l reasons. (i). They represent the f i n a l demand for a homogeneous, low-order good by consumers l i v i n g i n the community. Hence l i q u o r . s a l e s are l i k e l y to be a good i n d i c a t o r of the p r o s p e r i t y of the community and of the strength of l o c a l market demand. High-order goods i n contrast, are offered mainly i n larger centres and thus do not give a good estimate 62 of demand in centres where they are not offered. Aggregate r e t a i l sales data suffer i : am this. Among other possible surrogate measures, the j volume of production is a poor measure in that wealth derived from local : industry may be exported to shareholders or to the parent company located elsewhere. This is often the case with the parasitic primary act i v i t i e s characteristic of the Interior. ( i i ) Value of liquor sales correlates well with other indicators of economic growth and prosperity. To show this a correlation analysis was performed, for 1961 data, and demonstrated that coefficients of correlation (r) were 0.87 with respect to population, 0.94 with r e t a i l sales and 0.83 with the number of motor vehicle licences issued. The scatter plot of liquor sales (Y) and r e t a i l sales (X) is shown in Figure 14. Inspection of the residuals showed that the trend for centres north of Cache Creek had a steeper slope than for communities in the southern part of the province. In the latter case, the trend is curvilinear especially i f Kamloops is included in the northern half. This may suggest that liquor sales in the northern centres are more sensitive to general economic' growth than is the case in the south. Doubtless this i s partly due to the contrasting composition of the population in the different areas. The overall trend is f a i r l y strong, however, and seems to approxi-mate a positive linear relationship. ( i i i ) Liquor sales, i t is argued here, give an almost immediate i n d i -cation of changes in the level of activity. The opening- of a new trans-portation f a c i l i t y , immigration of labour, and increased individual wealth would probably be reflected in liquor sales as the former took place. As a result, liquor sales are more l i k e l y than other measures of economic RELATIONSHIP BETWEEN LIQUOR SALES RETAIL SALES ($ MILLIONS) SOURCE: B.C. LIQUOR CONTROL BOARD, ANNUAL REPORT; B.C. DEPT. OF ECONOMICS AND STATISTICS (1966) j ! Figure 14 i 64 growth to show up the s i m p l i f i e d lead-lag r e l a t i o n s h i p s that t h i s paper attempts to study. In contrast, value of i n d u s t r i a l production depends on accumulated c a p i t a l investment. Had t h i s v a r i a b l e been used, f o r example, the diverse i n d u s t r i a l structures possessed by d i f f e r e n t centres would have r e s u l t e d i n the i m p l i c i t i n t r o d u c t i o n of time-lags of varying duration i n t o the a n a l y s i s . (iv ) Liquor sales data are accurately measured. At l e a s t there i s no reason to expect that any sales have not been accounted f o r . This i s a p o s s i b i l i t y with r e t a i l s a l e s , f o r example. As mentioned above, since l i q u o r s a l e s are r e l a t i v e l y free from the administrative boundary problems encountered with other data sources they probably r e l a t e w e l l to the s i z e of the market contained w i t h i n the centre's h i n t e r l a n d . In ad d i t i o n , some f i n a l goods may be brought i n by i n t e r r e g i o n a l trade. S p e c i a l i s e d goods may be purchased on t r i p s to the nearest metropolis. Others may be ordered d i r e c t from organisations outside the region. In e i t h e r case such commodities are not accounted f o r i n l o c a l r e t a i l trade data. Local l i q u o r s a l e s , on the other hand, probably include a much higher proportion of l o c a l consumption. Therefore, i t i s argued here that l i q u o r s a l e s are a very good measure of l o c a l p r o s p e r i t y and growth. (v) E x c e l l e n t a v a i l a b i l i t y i s another advantage of the data. Liquor sales data are a v a i l a b l e for each year back to 1921 and for every s e t t l e -ment of any s i z e i n the province. Furthermore, u n l i k e value of production, there are no problems of data d i s c l o s u r e . The only d i f f i c u l t i e s encountered were the absence of data f o r the e a r l i e s t time-periods. This was due to the small s i z e then of c i t i e s that are important, i n a regional context, at the present time. For t h i s reason, Dawson Creek, f o r example, could not be included i n the a n a l y s i s . Nevertheless, other data sources are far more r e s t r i c t i v e and, as has been argued above, less u s e f u l . I CHAPTER 7 ACCESSIBILITY AND URBAN GROWTH IN THE B. C. INTERIOR Introduct ion Severa l s tudies have suggested that a p o s i t i v e r e l a t i o n s h i p between the a c c e s s i b i l i t y of a l o c a t i o n and the s i z e and complexity of i t s economic a c t i v i t y does e x i s t (Kansky, 1963; K i s s l i n g , 1966; O ' S u l l i v a n , 1968). Only one study, however, has made a ser ious attempt to look at the r e l a t i o n s h i p s through time (Gauthier , 1967). Neverthe less , as that author pointed out , the conc lus ions from h i s work were not s t rong ly sub-s t a n t i a t e d owing to the f a c t that j u s t three t ime-periods were u t i l i s e d , th i s be ing the l i m i t a t i o n of the data . K i s s l i n g (1966) attempted to extend h i s d e t a i l e d c r o s s - s e c t i o n a l a n a l y s i s for 1964 over time to tes t the s t a b i l i t y of the r e l a t i o n s h i p s found for 1964. Very l i t t l e , i f any-t h i n g , could be drawn from the r e s u l t s however. Merely one fur ther t ime-p e r i o d was s e l e c t e d , 1951, the data for which were incomparable with those for 1964. In a d d i t i o n , only seventeen centres had s u f f i c i e n t popula t ion at the e a r l i e r date to be inc luded i n the a n a l y s i s . Consequently, conclus ions were based on a popula t ion of c i t i e s that changed cons iderably between the two c r o s s - s e c t i o n a l analyses . The primary o b j e c t i v e of th i s chapter i s to examine the trends through time of r e l a t i o n s h i p s between a c c e s s i b i l i t y and l i q u o r s a l e s , 65 66 the l a t t e r taken as a surrogate of urban economic p r o s p e r i t y . The measure of a c c e s s i b i l i t y used for th i s was the route access index i n i t s unweighted form (Chapter 4 ) . Weighting the index with nodal masses could not be under-taken as i t was des i red to have a measure of the t o p o l o g i c a l proper t i e s alone of the highway network. There seemed l i t t l e po int i n adding c i r c u -l a r i t y to the argument inasmuch as i n s e r t i n g nodal masses could only r e s u l t i n e x p l a i n i n g the s t r u c t u r e of the space-economy i n terms of i t s e l f . In the ana lys i s of l e a d - l a g r e l a t i o n s h i p s t h i s would prevent any conclus ions be ing a r r i v e d at regarding the highway component i n the d i s t r i b u t i o n of economic a c t i v i t y . Furthermore, the t r a f f i c flow data could not be used as a measure o f l i n k importance owing to the s trong and obvious r e l a t i o n s h i p s between the volume of t r a f f i c generated and the s i z e of the t r a f f i c generators . The i n v e s t i g a t i o n of l e a d - l a g r e l a t i o n s h i p s r e l a t e s a set of v a r i a b l e s measuring the l e v e l of economic a c t i v i t y to a second set of v a r i a b l e s measuring a c c e s s i b i l i t y to the highway network. F i v e l i q u o r sa les v a r i a b l e s correspond to f i v e t ime-per iods , 1953, 1956, 1960, 1963, 1966. S ix a c c e s s i b i l i t y v a r i a b l e s correspond to these i n t e r v a l s and 1969. As i t was not known a p r i o r i what v a r i a b l e s were most s t r o n g l y assoc iated wi th which, a m u l t i p l e - h y p o t h e s i s approach was adopted. Given the e x i s -tence of a r e l a t i o n s h i p between a c c e s s i b i l i t y and economic a c t i v i t y , three hypotheses are enter ta ined s imultaneous ly: f i r s t l y , that highway investment leads the changing growth patterns of the economy by a number of y e a r s , secondly, that highways have been b u i l t i n response to the r e l a t i v e rates of growth of urban centres , and t h i r d l y , that the r e l a t i o n -sh ip i s balanced over t ime. 67 These hypotheses have been suggested by previous i n t e r p r e t a t i o n s . The t r a d i t i o n a l view point argues that improved t r a n s p o r t a t i o n has been a j p r e r e q u i s i t e for economic growth, i n the sense of t ransport investment l ead ing other sec tors i n the development process (Rostow, 1960) . On the other hand, i t has become i n c r e a s i n g l y fashionable to regard t r a n s p o r -t a t i o n as a permiss ive agent which lags behind the growth i n other sectors (Hirschman, 1958; North , 1968). In e i t h e r case, however, t ranspor t can be viewed as a permiss ive f a c t o r ra ther than a d i r e c t i n s t i g a t o r of develop-ment. What i s at i s sue i s the sequence of events: whether improved t r a n s p o r t a t i o n has. f a c i l i t a t e d subsequent development or whether improved t r a n s p o r t a t i o n has taken p lace i n response to i n c r e a s i n g demand from e x i s t i n g a c t i v i t i e s . No doubt both of these forces work concurrent ly i n the process of r e g i o n a l growth: the problem i s to show which i s dominant. The ana lys i s that fol lows does not attempt to v e r i f y t h i s i n general but to show, under s u i t a b l e assumptions, which of these c o n d i t i o n a l a l t e r -nat ives has been a p p l i c a b l e i n I n t e r i o r B . C . In order to tes t the s trength of the r e l a t i o n s h i p between a c c e s s i b i l i t y and the l e v e l of economic a c t i v i t y , a simple c o r r e l a t i o n a n a l y s i s was performed for f i v e c r o s s - s e c t i o n s . A balanced r e l a t i o n s h i p was assumed between the route access index and l i q u o r sa les for t h i s p r e l i m i n a r y a n a l y s i s . For the years 1953, 1956, 1960, 1963 and 1966 the corresponding c o r r e l a t i o n c o e f f i c i e n t s (r) are 0.45, 0.56, 0 .53, 0.39 and 0.37. This trend seemed to imply that l e v e l s of economic a c t i v i t y and a c c e s s i b i l i t y were becoming more c l o s e l y r e l a t e d up to about 1960. Much lower c o e f f i c i e n t s for 1963 and 1966 may be the r e s u l t of major shocks to the highway system, and hence to the a c c e s s i b i l i t y sur face , 68 by the opening of the Roger's Pass i n 1962 and the Salmo Creston Highway i n 1964. A n a l y s i s of Res iduals Regression of a c c e s s i b i l i t y on l i q u o r sales showed trends over time i n the r e s i d u a l s from the ( l i n e a r ) regres s ion l i n e . The e a r l i e s t and the most recent t ime-periods used i n the ana lys i s were compared. The r e s i d u a l s are mapped i n Figures 15 and 16. They suggest some s t r u c t u r a l s t a b i l i t y w i t h i n the system as the only changes appear i n the Pr ince George and i n the Kootenay reg ions . Those areas , or set t lements , where r a p i d growth was taking p lace appear as r e s i d u a l s underpredicted by a c c e s s i b i l i t y l e v e l s and hence l i q u o r sa les are greater than pred ic ted ( p o s i t i v e r e s i d u a l s ) . With the c o n s t r u c t i o n of new highway f a c i l i t i e s the l e v e l .of a c c e s s i b i l i t y has r i s e n so a l so has the volume of l i q u o r sa les p r e d i c t e d . Owing to the i n d i v i s i b l e nature of t h i s highway investment the increased l e v e l of a c c e s s i b i l i t y that r e s u l t s from t h i s shock to the system may have overpred ic ted the l e v e l of l i q u o r sa les for some t ime. Hence they have become negat ive r e s i d u a l s . Burn's Lake and Smithers show t h i s . Both are p o s i t i v e r e s i d u a l s i n 1953 but by 1966 have become negat ive . During t h i s time, major highway improvement, and e s p e c i a l l y paving , had taken p lace along Highway 16 there . A s i m i l a r case ex i s t s i n the Kootenays with reference to Cranbrook and F e r n i e . Both p o s i t i v e i n 1953, they are negat ive r e s i d u a l s i n 1966. This i s presumably due to the increased l e v e l of a c c e s s i b i l i t y conferred upon them by the Salmo-Creston Highway completed i n 1964. In c o n t r a s t , C a s t l e g a r - K i n n a i r d , a Figure 15 ON VO RESIDUALS OF LIQUOR SALES (Y) AND ACCESSIBILITY (X), 1966 o • KEY TO NODES; LIQUOR SALES GREATER THAN PREDICTED LIQUOR SALES LESS THAN PREDICTED DUMMY NODES MAJOR PERIPHERAL CITIES F i g u r e 16 71 negat ive r e s i d u a l i n 1953 had become p o s i t i v e by 1966 owing to the con-s t r u c t i o n of a new road d i v e r t i n g many shor tes t -path routes , and hence t r a f f i c on Highway 3, away from that centre and towards T r a i l . There i s some suggest ion i n these r e s u l t s that economic growth has preceded highway c o n s t r u c t i o n i n a number of cases . To show the nature of t h i s r e l a t i o n s h i p and the d u r a t i o n of the lags invo lved more e x p l i c i t l y an a l t e r n a t i v e approach i s taken. Canonica l C o r r e l a t i o n A n a l y s i s of Lead- lag Re la t ionsh ips Canonica l c o r r e l a t i o n i s a p p l i e d to the i n v e s t i g a t i o n of l e a d -lag r e l a t i o n s h i p s . "Canonical" c o r r e l a t i o n i s the maximum c o r r e l a t i o n between l i n e a r funct ions of two sets of v a r i a b l e s . The sets may be com-bined i n more than one way, so that s e v e r a l roots may be ex trac ted . The method invo lves the computation of two sets of weights that maximise the c o r r e l a t i o n . In t h i s manner, an under ly ing system of c o r r e l a t i o n i s revealed (see Appendix 3) . A canon ica l c o r r e l a t i o n a n a l y s i s was performed wi th l i q u o r sales for 1953, 1956, 1960, 1963 and 1966 as one set of v a r i a b l e s and wi th the route access index for the same years plus 1969 as the second set . Resu l t s are shown i n Table V . The roots i n d i c a t e that the l e v e l of economic a c t i v i t y , as measured by i t s surrogate , i s a lead f a c t o r i n i t s r e l a t i o n s h i p with highway investment, as measured by a c c e s s i b i l i t y . Liquor sa les represent non-bas ic rather than b a s i c a c t i v i t i e s . These s e r v i c e a c t i v i t i e s are l i k e l y to have developed i n r e l a t i o n to some major investment. Hence i t can be i n f e r r e d from the r e s u l t s , that they TABLE V - RESULTS OF CANONICAL CORRELATION ANALYSIS OF LIQUOR SALES AND ROUTE ACCESS INDEX: LEAD-LAG RELATIONSHIPS Canonical C o e f f i c i e n t s Liquor Sales Route Access Root 1953 1956 1960 1963 1966 1953 1956 1960 1963 1966 1969 1 -0.12 -4.35 4.18 -3.63 3.87 0.08 -0.99 j 0.20 27.29 -27.75 0.57 0.76 2 -1.92 4.42 0.99 -4.01 1.13 0.22 0.20 -0.42 14.1 -15.21 1.91 0.75 3 2.87 -2.59 0.15 -2.15 2.22 1.69 -1.40 0.57 - 3.28 -0.70 2.88 0.44 4 -1.04 4.56 -0.42 -4.76 0.98 0.88 -0.33 -1.51 -9.50 8.77 1.52 0.40 5 1.10 -2.68 -6.61 7.03 1.52 -2.76 -0.43 1.99 -12.91 12.01 1.98 0.20 C o r r e l a t i o n 1969 ^ (1) = Root • 73 developed i n i t i a l l y i n r e l a t i o n to primary a c t i v i t i e s which us"ed the e x i s t i n g transport network. H i s t o r i c a l l y , pioneer settlement growth has been c l o s e l y connected with the expansion of primary a c t i v i t i e s and therefore these would seem to have been the leading sectors i n .Interior B. C. rather than highway investment. This primary investment has had a l o c a l m u l t i p l i e r - a c c e l e r a t o r e f f e c t . I n t e r p r e t a t i o n of the canonical c o e f f i c i e n t s , suggests that, a f t e r a l a g of about f i v e years, highway construction i s an important component i n t h i s subsequent investment. Another mechanism to e l i c i t t h i s response may be conjectured. P o l i t i c a l pressure has existed for the improvement of highways i n c e r t a i n l o c a l i t i e s . In terms of voting power, t h i s pressure i s l i k e l y to have been most e f f e c t i v e i n those areas experiencing rapid population growth. Lead-lag r e l a t i o n s h i p s were also investigated between s e t t l e -ment s i z e and the distance of each centre to i t s nearest larger neighbour. In both cases, value of l i q u o r sales was employed as a surrogate f o r t e r t i a r y a c t i v i t i e s . I t i s apparent that the s i z e s of centres and the distance to t h e i r nearest larger neighbours are p e r f e c t l y and p o s i t i v e l y c o r r e l a t e d i f , and only i f , they form part of a r e g u l a r l y spaced h i e r -a r c h i a l system. This c o n d i t i o n i s f a r from being f u l f i l l e d i n B. C. where centres tend to be c l u s t e r e d i n r e g i o n a l sub-systems or apparently located at random. In f a c t , where the larger centres are grouped, a negative r e l a t i o n s h i p may emerge and o v e r a l l there may be no clear r e l a t i o n -ship. Changes i n t h i s pattern may be the r e s u l t of d i f f e r e n t growth rates between one centre and another. This may have i t s o r i g i n i n changes in the economic base of the centre or i n a realignment of the patterns of s p a t i a l competition by highway construction. Centres not formerly 74 i n competition can thus be brought wi t h i n e f f e c t i v e competing range. Hence, a dynamic view of c e n t r a l place theory suggests that, as a r e s u l t } of l o c a l changes, s p a t i a l transformations throughout the system are to be a n t i c i p a t e d . I t i s reasonable a p r i o r i , and e m p i r i c a l l y substantiated, that time lags should e x i s t , during which realignments and r e l o c a t i o n s can take place. This i s the conclusion which the r e s u l t s i n Table VI support. A d i s t i n c t l a g i s shown between s i z e of centre and distance to nearest larger centre. Implications of t h i s lagged r e l a t i o n s h i p are that there e x i s t forces at work tending to make the d i s t r i b u t i o n of settlements more regular. The fourth root extracted does not follow t h i s trend. It can e f f e c t i v e l y be ignored f o r two reasons. F i r s t l y , i t could be due to a time-reversal f a c t o r , owing to the omission of l i q u o r sales observations p r i o r to 1953, and secondly, the fourth root i s u n l i k e l y to be very s i g n i f i c a n t i n any case. TABLE VI - RESULTS OF CANONICAL CORRELATION ANALYSIS OF LIQUOR SALES AND DISTANCE TO NEAREST LARGER CENTRE: LEAD-LAG RELATIONSHIPS Canonical C o e f f i c i e n t s Liquor Sales Da L S t a n c e t :o Neares 5t Largei : Centre Root 1953 1956 1960 1963 1966 1953 1956 1960 1963 1966 1969 A 1 -0.69 2.41 -0.01 -1.50 0.70 0.31 -1.61 L -0.07 1.54 -1.25 1.91 0.95 2 3.02 -4.58 2.23 -1.15 0.90 -0.15 1.75 2.12 -4.43 0.23 0.36 0.80 3 1.20 -2.80 -1.38 -1.35 4.34 -0.28 2.73 -2.52 -0.39 -1.99 2.85 0.55 4 1.03 -5.94 -1.43 6.57 0.08 -3.06 6.16 -3.92 -2.95 0.30 3.01 0.39 5 -2.30 0.12 9.91 -8.31 0.44 -1.28 -3.95 3.24 -2.99 -6.28 11.15 0.32 J\~'. = Canonical C o r r e l a t i o n Distance to Nearest Liquor Sales Larger Centre 1953 1953 I CHAPTER 8 REGIONS OF INTERIOR B. C : THEIR EXTENT AND PERSISTENCE OVER TIME Central place theory assumes that the consumer buys a good at the nearest place i n which i t i s off e r e d . Owing to t h i s , even i f allowance i s made for the e f f e c t of multiple-purpose t r i p s , i n t e r a c t i o n i s l e s s probable between d i s t a n t centres than between contiguous ones, nodal populations assumed equal. Hence f u n c t i o n a l regions may be iden-t i f i e d with groups of contiguous centres. E m p i r i c a l l y , the space-economy of B r i t i s h Columbia e x h i b i t s a set of sub-systems c o n s i s t i n g of groups of c l u s t e r e d nodes and l i n k s . These sub-systems, such as the Okanagan, the East and West Kootenays, and the Dawson Creek areas are s p a t i a l l y separated by one or two long l i n k s . Two long and c i r c u i t o u s routes, the Monashee Highway and the Southern T r a n s - P r o v i n c i a l Highway, l i n k the Okanagan and the West Kootenays. These considerations provide the motivation to recognise these sub-regions more e x p l i c i t l y . Two methods of r e g i o n a l i s a t i o n were experimented with; one produces regions of s i m i l a r a c c e s s i b i l i t y l e v e l s , the other constructs nodal or f u n c t i o n a l regions. The uniform region method used was b a s i c a l l y that outlined i n Veldman (1967) page 311, a computer program for which, i n an extended form, was made a v a i l a b l e to the author by R. A. Whitaker. Basic inputs were the Route Access Index and co n t i g u i t y c o n s t r a i n t s . The algorithm i s as follows. Each node being considered i n i t i a l l y as a group the method 76 finds the two groups whose combination makes the l e a s t a d d i t i o n to within group variance subject to the c o n t i g u i t y c r i t e r i o n . This process i s continued u n t i l one group remains or the c o n t i g u i t y c o n s t r a i n t prevents further grouping. Output from t h i s procedure produced contiguous regions of generally greater or l e s s e r a c c e s s i b i l i t y to the network as a whole. For one year, 1966, the r e s u l t s were i n t u i t i v e l y r e a l i s t i c but r e s u l t s for other years gave very d i f f e r e n t r e g i o n a l groupings. Such r a d i c a l changes from year to year were probably due to the procedure's s e n s i t i v i t y to even the smallest change i n a c c e s s i b i l i t y values. A f u r t h e r drawback was that the method produced uniform or r e l a t i v e l y homogeneous regions :in which between-group variance was greater than within-group variance. These are not n e c e s s a r i l y the. type of regions best s u i t e d to the analysis of i n t e r a c t i o n a l or f u n c t i o n a l data. For these two reasons, t h i s grouping procedure was abandoned i n favour of the following method. In the nodal region method b a s i c inputs were f u n c t i o n a l distances -between nodes and the ranking of nodes i n terms of t h e i r l i q u o r sales as a measure of r e l a t i v e importance. The i n i t i a l step requires the i d e n t i -f i c a t i o n of dominant regional centres. This was accomplished by searching, for each node, for the node with which i t had the highest score computed by the formula: 2.7 where P. = l i q u o r sales of node i 1 P. = l i q u o r sales of node j 2 7 D " = the time-distance between nodes i and j r a i s e d to the exponent i j 2 that gave the highest r i n regression with t r a f f i c flow (Y) and the l i n k importance index (X). Figure 17 80 I t e r a t i o n of t h i s procedure produced regional groupings. Each was focussed on the l a r g e s t centre w i t h i n each region. The e f f e c t s of the p e r i p h e r a l metropolises were omitted from t h i s r e g i o n a l i s a t i o n because i t was con-sidered that they were f o c a l points f o r regions of a very d i f f e r e n t order than the r e l a t i v e l y minor sub-regions of the B. C. I n t e r i o r . I d e n t i f i c a t i o n of r e g i o n a l centres by t h i s method highlighted-the following nodes: Prince George, Kamloops, Dawson Creek, Prince Rupert, Kelowna, Tra i l - R o s s l a n d , Nelson and Cranbrook. Certain l i n k s i n the network contained i n d i f f e r e n c e points between two of these regions. The network was p a r t i t i o n e d at these points and a set of eight sub-networks were formed. In essence, these are D i r i c h l e t regions. This procedure was repeated f o r each of the s i x time-periods studied. Over time i t was found that the same nodes remained i d e n t i f i e d with the same regions i n 1953, 1956 and 1960. By 1963, however, two major s t r u c t u r a l changes had taken place: the Roger's Pass and the Salmo-Creston Highway. As a r e s u l t of these shocks to systems, Kamloops was able to extend i t s influence along the trans-Canada Highway to 'capture' Golden at the expense of Cranbrook. In the south, the Cranbrook regions shrank further as T r a i l captured Creston. These trends were suggestive. Subject to the assumptions, i t followed that the l a r g e r places gained more than the smaller ones from highway improvement as the former were able to extend t h e i r f i e l d s of dominance at the expense of smaller places. The extent of these regions i s shown i n Figures 17 and 18. No further changes took place i n the composition of the regions i n 1963, 1966 or 1969. The e f f e c t s of the Prince-George-McBride and Kamloops-Tete Jaune Highways were not measured i n t h i s context as there was no competing Edmonton region, the influence of the surrounding c i t i e s having been omitted for the reasons given above. CHAPTER 9 CONCLUSIONS AND SUMMARY Lead-lag Relationships I t was found that the l i n k importance index which measured the r e l a t i v e s i g n i f i c a n c e of each l i n k to the highway system as a whole le d l e v e l s of t r a f f i c flow on those l i n k s by about three years. In ad d i t i o n , economic growth was found to lead l e v e l s of a c c e s s i b i l i t y to the routeways of the highway network by some s i x years. These two r e s u l t s may be r e l a t e d t e n t a t i v e l y as shown i n Figure 10a. The r e l a t i o n s h i p between l i n k importance and t r a f f i c flow suggests the length of time needed for the adjustment of e x i s t i n g i n t e r -a c t i o n patterns to the abruptly changed s p a t i a l r e l a t i o n s h i p s r e s u l t i n g from investment i n the highway network. E f f e c t s of generated t r a f f i c as w e l l as those of d i v e r t e d t r a f f i c are l i k e l y to be included i n flow data. The r e l a t i o n s h i p found between economic growth and a c c e s s i b i l i t y suggests that highway investment has not played the r o l e of leading sector that i t i s sometimes held to perform. I t seems f e a s i b l e i n view of t h i s to argue that the export-based primary i n d u s t r i e s are the r e a l leading sector of the economy and that highways only followed when e x i s t i n g t ransportation f a c i l i t i e s were unable to meet increasing demand, when scale-economies were to be reaped with large units of investment and when permanent communities were est a b l i s h e d . These leading sectors are 81 82 i d e n t i f i a b l e s p a t i a l l y with 'leading regions' of Northern B. C. and the highway investment programme, although lagging behind the other sectors of economic investment, has probably been instrumental i n a c c e l e r a t i n g the growth of these leading regions. Government highway-building p o l i c y has therefore fostered the r e g i o n a l l y unbalanced growth patterns already inherent i n the s p a t i a l s t r u c t u r e of the p r o v i n c i a l economy. Transformations of the Regional Economy and i t s Urban System Several r e s u l t s furnished evidence for the increasing i n t e g r a t i o n of the urban system over the t i m e - i n t e r v a l between 1953 and 1966. The attempts to simulate the d i s t r i b u t i o n of t r a f f i c flow by a s e r i e s of cross-s e c t i o n a l analyses suggested t h i s . I m p l i c i t i n t h i s aspect of the a n a l y s i s was the assumption that the e n t i r e highway network of the study area behaved as a s i n g l e integrated system. Results for 1953 gave low co-e f f i c i e n t s of c o r r e l a t i o n between t r a f f i c flow and l i n k importance whereas those for 1966 gave high c o e f f i c i e n t s suggesting that the integrated system assumption had greater empirical v a l i d i t y at the l a t e r date. Inspection of the r e s i d u a l s from regression of the v a r i a b l e s showed, moreover, that s t a b l e patterns have already emerged. The impact of new highways on these patterns was evident. This conclusion i s r e i n f o r c e d by s i m i l a r trends recognisable i n the r e s i d u a l s from regression of l i q u o r sales and a c c e s s i b i l i t y . I n v e s t i g a t i o n of another aspect of the s p a t i a l organisation of the urban system also suggested the increasing i n t e g r a t i o n of the r e g i o n a l economy over time. Levels of economic growth i n the I n t e r i o r 83 EMPIRICAL RELATIONSHIPS BETWEEN THE HIGHWAY NETWORK AND THE REGIONAL ECONOMY OF INTERIOR B.C. LINK IMPORTANCE INDEX ~ 3 YEARS W TRAFFIC FLOW ACCESSIBILITY 6 YEARS DISTANCE TO NEAREST LARGER CENTRE ECONOMIC ACTIVITY '8 YEARS Figure 19 84 communities were c o r r e l a t e d (by canonical analysis) with the distance to the nearest community of larger s i z e . Results imply that growth i n communities at orte period had an impact on the c e n t r a l place hierarchy that was greatest about eight years l a t e r . Presumably, as the influence of centres has grown with the emergence of a more extensive urban system and interconnecting highway l i n k s the centres have become i n i n c r e a s i n g l y more e f f e c t i v e competition. This would have f a c i l i t a t e d the operation of the organisation processes c h a r a c t e r i s t i c of c e n t r a l place h i e r a r c h i e s and increasing r e g u l a r i t y i n the settlement pattern. This i s not to say, however, that the centres i n the I n t e r i o r are r e g u l a r l y spaced but only that there seems to have been a tendency towards increased r e g u l a r i t y developing from the almost randomly-located resource-based pioneer s e t t l e -ments. A trend of t h i s nature may be r e l a t e d to development of a super-s t r u c t u r e of t e r t i a r y a c t i v i t i e s upon the resource base. A f t e r a l l , these are the economic a c t i v i t i e s l i k e l y to be most s e n s i t i v e to extension of the highway network. BIBLIOGRAPHY AND APPENDICES BIBLIOGRAPHY A l a o , N. (1970). A note on the s o l u t i o n matr ix of a network. Geographical  A n a l y s i s , V o l . 2(1) , 83-88. Almond, J . ( e d . ) , (1965). Proceedings of the Second I n t e r n a t i o n a l Sym- posium on the Theory of Road T r a f f i c Flow, London, 1963. P a r i s : The Organ i sa t ion for Economic Cooperat ion and Development. Beckmann, M. (1967). On the theory of t r a f f i c flow i n networks. T r a f f i c  Q u a r t e r l y , V o l . 21, 109-117. , C. B . McGuire and C. Winsten, (1956). Studies i n the Economics of T r a n s p o r t a t i o n . New Haven. Bel lman, R. (1958). On a r o u t i n g problem. Quarter ly Journa l of App l i ed  Mathematics, V o l . 16, 87-90. Berge, C. (1962). The Theory of Graphs and i t s A p p l i c a t i o n s . (Trans lated by A . D o i g ) . New Y o r k . Berge, C. and A. G h o u i l a - H o u r i (1965). Programming, Games and T r a n s - p o r t a t i o n Networks. London. B e r r y , B . J . L . (1959). Recent s tudies concerning the r o l e of t r a n s p o r t a t i o n i n the space-economy. Annals of the A s s o c i a t i o n of American  Geographers, V o l . 49(3), 328-342. , (1960). The impact of expanding metropol i tan communities upon the c e n t r a l p lace h i e r a r c h y . Annals of the A s s o c i a t i o n of American  Geographers, V o l . 50(2), 112-116. B l a l o c k , H . M . (1961). Causal Inferences i n Non-Experimental Research. Chapel H i l l , North C a r o l i n a . Boventer, E . von (1966). Land values and s p a t i a l s t r u c t u r e : a g r i c u l t u r e , urban and t o u r i s t l o c a t i o n t h e o r i e s . Papers and Proceedings , Regional Science A s s o c i a t i o n , V o l . 18, 231-242. , (1969). Walter C h r i s t a l l e r ' s c e n t r a l places and p e r i p h e r a l areas: the c e n t r a l p lace theory i n r e t r o s p e c t . J o u r n a l of Regional Sc ience , V o l . 9 (1) , 117-124. B r i t i s h Columbia, Dept. of Economics and S t a t i s t i c s (1966). Regional Index of B. C. V i c t o r i a . 86 87 B r i t i s h Columbia, Dept. of Highways. Summer T r a f f i c Volumes on P r o v i n c i a l  Highways, 1951-1968. B r i t i s h Columbia, Liquor Control Board. Annual Report, 1921-1968. B r i t i s h Columbia Research Council (1966). Population and Economic Develop- ment of the Inland Natural Gas Service Area. Vancouver. Brown, L. K. (1969). Transportation and Economic Development i n Alaska. Report, U. S. Federal F i e l d Committee for Development Planning i n Alaska. Anchorage. Brunner, E. de S. and T. L. Smith (1944). V i l l a g e growth and decline, 1930-1940. Rural Sociology, V o l . 9(2), 103-114. Burton, I. (1962). A c c e s s i b i l i t y i n Northern Ontario: an a p p l i c a t i o n  of graph theory to a regional highway network. Report, Ontario Dept. of Highways, Appendix A, Busacker, R. and T. Saaty (1965). F i n i t e Graphs and Networks: An Introduction  with A p p l i c a t i o n s . New York. . Claeson, C-K. (1968). Distance and human i n t e r a c t i o n s : review and d i s -cussion of a s e r i e s of essays on geographic model b u i l d i n g . Geografiska Annaler, Series B, V o l . 50B(2), 142-161 Constantin, J . A. (1966). P r i n c i p l e s of L o g i s t i c s Management. New York. Cooley, W. W. and P. R. Lohnes (1962). M u l t i v a r i a t e Procedures for the  Behavioural Sciences. New York. Cummings, L. P. (1967). The Structure of Networks and Network Flows. U n i v e r s i t y of Iowa, Ph.D. D i s s e r t a t i o n . Dantzig, G. E. (1960). On the shortest route through a network. In Constantin (1966). P r i n c i p l e s of L o g i s t i c s Management. New York. Dreyfus, S. E. (1969). An a p p r a i s a l of some shortest path algorithms. Operations Research, V o l . 17, 395-412. Farbey, B., A. H. Land and J . D. Murchland (1967). The Cascade Algorithm f o r f i n d i n g a l l shortest distances i n a d i r e c t e d graph. Management  Science, V o l . 14, 19-29. F l e i s c h e r , G. A. (1963). E f f e c t of highway improvement on t r a v e l time of commercial v e h i c l e s . Highway Research Record, Vol. 12, 19-47. Floyd, R. (1962). Algorithm 97: shortest path. Communications of the  A s s o c i a t i o n of Computing Machinery, Vol. 5, p.. 345. 88 Fort, D.H.S. (1965). The shortest-route problem: A l g o l programs and a discussion of computational problems i n large network a p p l i c a t i o n s . U n i v e r s i t y of B r i s t o l , Dept. of Economics, Discussion Papers, 10. Fogel, R. W. (1964). Railroads and American Economic Growth: Essays i n  Econometric H i s t o r y . Baltimore. Ford, L. R. and D. R. Fulkerson (1962). Flows i n Networks. Princeton. Fox, K. A. (1962). The study of i n t e r a c t i o n s between a g r i c u l t u r e and the non-farm economy: l o c a l , r e g i o n a l and n a t i o n a l . Journal of Farm  Economics, Vol. 44, 1-34. Friedlaender, A.F. (1965). The I n t e r s t a t e Highway System. Contributions of Economic Analysis Series, No. 38. Amsterdam. Fromm, G. (ed.), (1965). Transport Investment and Economic Development. Washington, D . C : The Brookings I n s t i t u t e . Fuguitt, G. W. (1965). The growth and d e c l i n e of small towns as a prob-a b i l i t y process. American S o c i o l o g i c a l Review, Vol. 30, 403-411. Garrison, W. L. (1959a). The s p a t i a l s t r u c t u r e of the economy I. Annals  of the A s s o c i a t i o n of American Geographers, V o l . 49, 232-239. , (1959b). The s p a t i a l s t r u c t u r e of the economy I I . Annals of the A s s o c i a t i o n of American Geographers, V o l . 49, 471-482. , (1960). The s p a t i a l s tructure of the economy I I I . Annals of the A s s o c i a t i o n of American Geographers, Vol. 50, 357-373. ' and D. Marble (1958). Analysis of highway networks: a l i n e a r programming formulation. Highway Research Board, Proceedings, No. 37, 1-14. and , (1965). A Prolegomenon to the Forecasting of Trans- po r t a t i o n Development. Research Report, The Transportation Centre at Northwestern U n i v e r s i t y , Evanston. and M. E. Marts, (1958). Geographic impact of highway improvements. Dept. of Geography and Dept. of C i v i l Engineering, U n i v e r s i t y of Washington, S e a t t l e . Gauthier, H. L. (1966). Highway Development and Urban Growth i n Sao  Paulo, B r a z i l : a Network A n a l y s i s . Northwestern U n i v e r s i t y , Ph.D. D i s s e r t a t i o n . , (1968a). Transportation and the growth of the Sao Paulo economy. Journal of Regional Science, V o l . 8(1), 77-94. 89 , (1968b). Least-cost flows i n a capacitated network: a B r a z i l i a n example. In F. Horton (ed.), Geographic Studies of Urban Trans- p o r t a t i o n and Network A n a l y s i s . Northwestern U n i v e r s i t y , Studies i n Geography, No. 16. Haggett, P. (1966). On c e r t a i n s t a t i s t i c a l r e g u l a r i t i e s i n the s t r u c t u r e  of transport networks. Mimeo. , (1967). An extension of the Horton combinational model to r e g i o n a l highway networks. Journal of Regional Science, Vol. 7(2) supp., 281-290. and R. J . Chorley (1969). Network Analysis i n Geography. London. Hamburg, J . F. (1969). The Influence of Railroads on the Processes and  Patterns of Settlement i n South Dakota. Un i v e r s i t y of North Carolina, Ph.D. D i s s e r t a t i o n . Harary, F. (1959). Graph theory and e l e c t r i c networks. Transactions  of the I n s t i t u t e of Radio Engineers, Vol. CT-6, 9'5-109. , (1969). Graph Theory. Reading, Mass. , R. Norman and D. Cartwright (1965). S t r u c t u r a l Models: An Introduction to the Theory of Directed Graphs. New York. Harvey, T. N. (1968). A method of network evaluation using the output of the t r a f f i c assignment process. Highway Research Record, V o l . 238, 46-63. Hassinger, E. (1957a). The r e l a t i o n s h i p of trade-centre population change to distance from larger centres i n an a g r i c u l t u r a l area. Rural Sociology, V o l . 22, 131-136. , (1957b). The r e l a t i o n s h i p s of r e t a i l - s e r v i c e patterns to trade-centre population change. Rural Sociology, Vol. 22, 235-240. Heggie, I. G. (1969). Are g r a v i t y - i n t e r a c t a n c e models a v a l i d technique fo r planning r e g i o n a l transport f a c i l i t i e s ? Operations Research  Quarterly, March, 93-110. Hirschman, A. 0. (1958). The Strategy of Economic Development. New Haven. Hodge, G. (1965). The p r e d i c t i o n of trade centre v i a b i l i t y i n the Great P l a i n s . Papers and Proceedings, Regional Science Association, Vol. 15, 87-115. , (1968) . Urban structure and r e g i o n a l development. Papers and  Proceedings, Regional Science Association, V o l . 21, 101-123. 90 Horwood, E . M . , C. A . Z e l l n e r and R. L . Ludwig (1965). Community Conse- quences of Highway Improvement. Nat iona l Cooperat ive Highway Research Programme, Report No. 18. Highway Research of the D i v i s i o n of Engineering and I n d u s t r i a l Research. I s a r d , W. (1956). L o c a t i o n and Space-Economy: A General Theory R e l a t i n g  to I n d u s t r i a l L o c a t i o n , Market Areas , Land Use, Trade, and Urban  S t r u c t u r e . Cambridge, Mass. , et a l (1960). Methods of Regional A n a l y s i s : An Introduct ion to Regional Sc ience . Cambridge, Mass. J a n e l l e , D. G. (1966). S p a t i a l Reorganisat ion and Time-Space Convergence. Mich igan State U n i v e r s i t y , Ph .D. D i s s e r t a t i o n . , (1969). S p a t i a l r e o r g a n i s a t i o n : a model and concept. Annals of the A s s o c i a t i o n of American Geographers, V o l . 59(2), 348-364. K a t z , W. (1953). A new status index der ived from soc iometr ic a n a l y s i s . Psychometrika, V o l . 18, 39-43. , Kanaan, N . J . (1965). S t r u c t u r e and requirements of the t ransport network of S y r i a . Highway Research Record, V o l . 115, 19-28. Kansky, K. J . (1963). S t ruc ture of Transport Networks: Re la t ionsh ips between Network Geometry and Regional C h a r a c t e r i s t i c s . U n i v e r s i t y of Chicago, Dept. of Geography, Research Papers, No. 84. Kaufmann, A. (1967). Graphs, Dynamic Programming and F i n i t e Games. Mathematics i n Science and Eng ineer ing , V o l . 36 . , New York . K i s s l i n g , C, C. (1966). T r a n s p o r t a t i o n Networks, A c c e s s i b i l i t y and  Urban Func t ions . M c G i l l U n i v e r s i t y , Ph .D . D i s s e r t a t i o n . , (1967). A c c e s s i b i l i t y and urban economic a c t i v i t y . Proceedings of the F i f t h New Zealand Geographical Conference, 143-152, New Zealand Geographical Soc i e ty , Auckland. , (1969). Linkage importance i n a r e g i o n a l highway network. Canadian Geographer, V o l . 13(2), 113-127. Ko lb , J . H . and R. A . Poison (1933). Trends i n Town-Country R e l a t i o n s . U n i v e r s i t y of Wiscons in , A g r i c u l t u r a l Experimental S t a t i o n , Research B u l l e t i n No. 117. Lachene, R. (1965). Networks and the l o c a t i o n of economic a c t i v i t i e s . Papers and Proceedings , Regional Science A s s o c i a t i o n , V o l . 14, 183-196. Landahl , H . D. (1947). O u t l i n e of a matrix ca l cu lus for n e u r a l nets . B u l l e t i n of Mathematical B i o p h y s i c s , V o l . 9, 99-108. 91 Longley, J . W. and B. T.. Goley (1962). A s t a t i s t i c a l eva luat ion of the in f l uence of highways on r u r a l land values i n the U . S . Highway  Research Board, B u l l e t i n , No. 327, 21-55. i i Maki , W. R. and Y . - i . Tu (1962). Rura l growth models for r u r a l areas development. Papers and Proceedings, Regional Science A s s o c i a t i o n , V o l . 9, 234-244. Meier , R. L . (1962). A Communications Theory of Urban Growth. Cambridge, Mass. M i l l s , G . (1966). A decomposit ion a lgor i thm for the s h o r t e s t - r o u t e problem. Operations Research Q u a r t e r l y , V o l . 14, 279-91. Mohring, H . D. and M. Harwitz (1962). Highway Bene f i t s : An A n a l y t i c a l Framework. Evanston. M o r i , M. and T. Nishimura (1967). S o l u t i o n of the rout ing problem through a network by matrix method with a u x i l i a r y nodes. Transpor ta t ion  Research, V o l . 1(2) , 165-180. Morlok, E . K. (1967). An A n a l y s i s of Transport Technology and Network  S t r u c t u r e . Evans ton . Munro, J . M. (1969). P lanning the Appalachian Development Highway:Systpm: some c r i t i c a l quest ions . Land Economics, V o l . 45(2) , 149-161. N i c h o l s , V. (1969). Growth P o l e s , an I n v e s t i g a t i o n of t h e i r P o t e n t i a l  v as a Too l for Regional Economic Development. Regional Science Research I n s t i t u t e , D i s c u s s i o n Paper S e r i e s , No. 30. Niedercorn , J . H . and B. V . Bechdol t , J r . (1969). An economic d e r i v a t i o n of the "grav i ty law" of s p a t i a l i n t e r a c t i o n . J o u r n a l of Regional  Sc ience , V o l . 9(2) , 273-282. North , R. N. (1968). T r a n s p o r t a t i o n and Economic Development i n Western  S i b e r i a . U n i v e r s i t y of B r i t i s h Columbia, Ph .D. D i s s e r t a t i o n . Nystuen, J . and M. Dacey (1961). A graph theory i n t e r p r e t a t i o n of nodal r eg ions . Papers and Proceedings, Regional Science A s s o c i a t i o n , V o l . 7, 29-42. Odeplan (1967). Las Regiones de Des a r r o l l o en C h i l e . C i t e d i n Gauthier (1966). Olsson , G. (1967). C e n t r a l p lace systems, s p a t i a l i n t e r a c t i o n and s t o c h a s t i c processes . Papers and Proceedings , Regional Science A s s o c i a t i o n , V o l . 18, 13-45. _, (1970). Exp lanat ion , p r e d i c t i o n and meaning v a r i a n c e : an assess-ment of d i s tance i n t e r a c t i o n models. Economic Geography, V o l . 46 (2, s u p p . ) , 223-233. O ' S u l l i v a n , P . M. (1968). A c c e s s i b i l i t y and the s p a t i a l s t r u c t u r e of the I r i s h economy. Regional S tudies , V o l . 2, 195-206. P a r r , J . B. and K. G . Denike (1969). T h e o r e t i c a l Problems i n C e n t r a l  P lace A n a l y s i s . Mimeo. Perroux, F . (1955). Note sur l a not ion de 'po le de c r o i s s a n c e . " Economie  Appl iquee , V o l . 8 (1-2) , 307-320. P i l l s b u r y , W. A. (1965). E f f e c t s of highway l o c a t i o n : a c r i t i q u e of c o l l a t e r a l e f f e c t a n a l y s i s . Highway Research Record, V o l . 75, 53-61. Pred, A . (1965). I n d u s t r i a l i s a t i o n , i n i t i a l advantage and American metropo l i tan growth; Geographical Review, V o l . 55(2), 158-185. Roberts , P. 0. (1969). I n t e r r e g i o n a l Transport Models . Programme on Regional and Urban Economics, D i scuss ion Paper No. 48, Harvard U n i v e r s i t y , Cambridge, Mass. Rostow, W. W. (1960). The Stages of Economic Growth. Cambridge. Rushton, G. (1969). A n a l y s i s of s p a t i a l behaviour by revealed space preference . Annals of the A s s o c i a t i o n of American Geographers, V o l . 59(2), 391-400. Sater , B . F . ( e d . ) , (1969). A r c t i c and Middle North T r a n s p o r t a t i o n . The A r c t i c I n s t i t u t e of North America, Washington, D. C. Schneider, M. (1959). G r a v i t y models and t r i p d i s t r i b u t i o n theory . Papers and Proceedings , Regional Science A s s o c i a t i o n , V o l . 5, 51-56. Seshu, S. and M. Reed (1961). L i n e a r Graphs and E l e c t r i c a l Networks. Reading, Mass. Shafran, I . and F . J . Wegmann (1969). The in f luence of the highway network s t r u c t u r e on the economic development of West V i r g i n i a . Highway Research Record, V o l . 285, 20-32. Shearer, R. A . (1968). The development of the B . C . economy. In Shearer ( e d . ) , E x p l o i t i n g Our Economic P o t e n t i a l . Shimbel, A . (1951). A p p l i c a t i o n s of matrix a lgebra to communications ne t s . B u l l e t i n of Mathematic B i o p h y s i c s , V o l . 13, 165-178. , (1953). S t r u c t u r a l parameters of communication networks. B u l l e t i n of Mathematical B i o p h y s i c s , V o l . 15, 501-507. , (1954). S t ruc ture i n communication nets . Proceedings of the  Symposium on Information Networks, Po ly technic I n s t i t u t e of Brook lyn . 93 S i e b e r t , H . (1969). Regional Economic Growth: Theory and P o l i c y . Scranton, Pa . Smith, R . H . T . (1964). Towards a measure of complementarity. Economic  Geography, V o l . 40(1), 1-8. Stroup, R. H . and L . A . Vargha (1961). Re f l ec t ions on concepts for impact research , Highway Research Board, B u l l e t i n , V o l . 311, 1-12. and (1963). Economic impact of secondary road improve-ments. Highway Research Record, V o l . 16, 1-13. Tanner, J . C. (1961). Factors A f f e c t i n g the Amount o f T r a v e l . Dept. of S c i e n t i f i c and I n d u s t r i a l Research, Road Research T e c h n i c a l Paper No. 51, H . M . S . O . , London. Ul lman, E . L . (1956) . The r o l e of t r a n s p o r t a t i o n and the bases for i n t e r -a c t i o n . In W. L . Thomas ( e d . ) , Man's Role i n Changing the Face  of the E a r t h . U n i v e r s i t y of Chicago. Veldman, D. J . (1967). F o r t r a n Programming for the Behav iora l Sc iences . New York . Voorhees, A . M. (1955). A general theory of t r a f f i c movement. Papers and Proceedings , I n s t i t u t e of T r a f f i c Engineers , V o l . 26, 46-56. Werner, et a l (1968). A research seminar i n t h e o r e t i c a l t r a n s p o r t a t i o n geography. In Horton, F . ( e d . ) , Geographic Studies of Urban  T r a n s p o r t a t i o n and Network A n a l y s i s . Northwestern U n i v e r s i t y , Studies i n Geography, No. 16. Wh eat , L . F . (1969). The e f f e c t of modern highways on urban manufacturing growth. Highway Research Record, V o l . 277, 9-24. W i l s o n , A . G . (1969). The use of entropy maximising models i n the theory of t r i p d i s t r i b u t i o n , mode s p l i t and route s p l i t . J o u r n a l of  Transport Economics and P o l i c y , V o l . 3 (1) . W i l s o n , G. W. , B . R. Bergmann, L . V . H i r s c h and M. S. K l e i n (1966). The Impact of Highway Investment on Development. The Brookings I n s t i t u t i o n , Transport Research Programme, Washington, D . C . Wohl, M. and B . V. M a r t i n (1967). T r a f f i c System A n a l y s i s For Engineers  and P lanners . New York . Wollmer, R. D. (1968). Maximising flow through a network with node and arc c a p a c i t i e s . T r a n s p o r t a t i o n Sc ience , V o l . 2(3) , 213-232. 94 Zimmerman, C. C. (1938). The Changing Community. New York. Zionts, S. (1962). Methods for s e l e c t i o n of an optimum route. In Constantin, J . A. (1966). P r i n c i p l e s of L o g i s t i c s Management. New York. APPENDIX 1: Some Basic D e f i n i t i o n s i n Graph Theory (a f t e r Berge, 1962; Kaufmann, 1967; Harary, 1969) Graph: A graph i s defined where there e x i s t s the following p r i m i t i v e s : ( i ) a set X ( c a l l e d v e r t i c e s or nodes) ( i i ) a function F mapping X into i t s e l f . Hence a graph may be w r i t t e n as G = (X, F ) . Vertex: Known v a r i o u s l y as point or node the vertex symbolises a definable object possessing r e l a t i o n s h i p s with other objects i n the same set. In a transport network v e r t i c e s represent c i t i e s or junctions. Edge: Also c a l l e d an arc or l i n k . Link i s sometimes confined to a non-d i r e c t e d or. r e f l e x i v e connection. Where the correspondence between nodes i s i n one d i r e c t i o n only the edges are termed dir e c t e d or orientated and are defined X e F(Y) or Y e F(X). Most a p p l i c a t i o n s of graph theory to transportation networks, however, have regarded the edges as non-directed. These l i n k s are generally formally defined as follows. In a graph G = (X, U) (where U = the set of l i n k s ) there e x i s t X. and X. such that. (X., X.) e U and/or (X., X . ) E U, X.J X.. i J 1 J j l l j Chain: A sequence of l i n k s (U 1, U , ..., U ) contained i n a graph i n 1 2 ft which each l i n k i s connected to ^ by one of i t s extremities and to U^ +^ by the other. A chain i s usually r e f e r r e d to by the v e r t i c e s i t contains. The term path i s , s t r i c t l y speaking, confined to d i r e c t e d arcs, although i t i s sometimes used l o o s e l y , as i n 'shortest-path algorithms. 1 96 Connected Graph: A graph i s s a i d to be connected i f for a r b i t r a r y X, and X. (X. * X.) there i s a chain going from X. to X.. i J 1 J i J Symmetrical Graph: A graph G = (X, U) i s symmetrical i f for a r b i t r a r y X i Xy (X^, X..) e U implies (Xy X/) e U. That i s , the correspondences between nodes are r e f l e x i v e or non-directed. P a r t i a l Graph: A p a r t i a l graph i s defined i f at l e a s t one l i n k i s missing but the t o t a l set of nodes remains. That i s , i n a graph (X, F) there e x i s t s a graph (X, D) such that f or a r b i t r a r y X±, D X ^ F X ± . ( B e r g e (1962) gives the example of the road map of France i n which X i s the set of towns i n France, and (X^, X^) e U i f any road j o i n s towns X^ and X . A map of the major roads only i s a p a r t i a l graph. Subgraph: In the above example, a subgraph would be, for instance, the road map of B r i t t a n y . I t can be defined as a graph (A,F ) where A c X and a r b i t r a r y X± e A, then F^ X± = (FX ±) f\ A. Planar Graph: A graph G = (X, U) i s planar i f i t can be represented i n such a way that a l l the v e r t i c e s are separate points, the l i n k s are simple curves and that no two l i n k s encounter each other except at t h e i r extremities. Tree: A f i n i t e graph without any cycles and with at l e a s t two v e r t i c e s . Cycle: A c y c l e i s a f i n i t e chain that leaves a vertex X^ and ends at the same-vertex. The term c i r c u i t i s often r e s t r i c t e d to dir e c t e d arcs. Elementary Cycle: This i s defined i f no vertex i n a cycle i s encountered more than once, except for the i n i t i a l - t e r m i n a l vertex. 97 T r a n s i t i v e Closure: Given a f i n i t e graph G = (X, F) the t r a n s i t i v e Gjosure * of F i s a mapping F of X i n X defined by F(X i) =(X ±) U F(X ±) U F 2 ( X ± ) U F 3 ( X ± ) U .... For a f u l l e r treatment of t h i s t o p i c see Chapter 4 where i t s r e l a t i o n s h i p ,to the powering of the adjacency matrix i s demonstrated. Degree of a Vertex: This i s defined as the number of l i n k s which have one extremity i n c i d e n t to X^, the other extremity not being at X/. Connected Components: Given a vertex X. and CX. t h i s i s defined as c ± i the set of v e r t i c e s which can be connected to X. by a chain to which 1 J X^ i t s e l f i s added. The subgraph formed by a set of the form CX^ i s termed the connected component of the graph. Cyclomatic Number: Let G be a graph with n v e r t i c e s , m l i n k s and r connected components. Then the cyclomatic number (v) = m - n + r . Diameter: The diameter (d) of a graph i s defined as d = max [max D(X., X.)] where D = distance. X.,eX X.eX 1 3 That i s , the longest short-path i n the graph that can be chosen to j o i n a r b i t r a r y X., X.. Matrices Related to Graphs Adjacency Matrix: This i s also c a l l e d the associated matrix or the connection matrix. Given a graph G = (X, U) which v e r t i c e s X^, X^j.-.X^, assume mij = 0 (X^, X^) Jt X mij = 1 (X,, X.) e X 98 Then the square matrix formed by the mij terms i s c a l l e d the adjacency matrix. Incidence Matrix: (non-directed case). I f U^, U^,..., are the l i n k s and X^, X^,..., X m the v e r t i c e s of a graph G = (X, F) without a loop, and i f i t i s assumed that = 1 i f X i i s an extremity of U.. = 0 i f t h i s i s not the case i l then the matrix U with elements U\_., i = 1, 2 m; j =1, 2..., p, i s c a l l e d the incidence matrix of the graph G. APPENDIX 2: Shortest-Path Algorithms There i s a great v a r i e t y of shortest-path algorithms o u t l i n e d i n the l i t e r a t u r e . Fortunately t h i s complexity may be s i m p l i f i e d as most algorithms f a l l i nto one of two contrasting methods: t r e e - b u i l d i n g methods or matrix methods. a) The t r e e - b u i l d i n g method This method seems to have been f i r s t o u t l i n e d i n Dantzig (1960). As i t s name suggests the algorithm involves the i t e r a t i v e b u i l d i n g of a tree out from the node of d e s t i n a t i o n . The end product resembles a (d e n d r i t i c ) r i v e r drainage pattern that converges on i t s o u t f a l l . The example given i n Figures 20a and 20b show t h i s g r a p h i c a l l y . Suppose i t i s desired to f i n d a l l shortest-paths between A and the other s i x nodes of the graph. The following procedure (adapted a f t e r Zionts, 1962) i s the essence of Dantzig's method. ( i ) Represent the network by l i s t s (one f o r each node) of l i n k s adjacent to each node. The values of the l i n k s i n each l i s t are arranged i n ascending order of l i n k value. ( i i ) Assign node A to the value zero, and delete from the other l i s t s a l l l i n k s adjacent to A. ( i i i ) S e lect the top l i n k of a l i s t whose sum of node value plus link value i s a minimum and add t h i s node to the tree. (iv) Assign as a l a b e l to the node on the untabled end of l i n k the minimum sum found i n step ( i i i ) . 99 Figure 20 - Graphs I l l u s t r a t i n g Shortest-Path Algorithms-101 (v) Delete a l l l i n k s adjacent to the node j u s t l a b e l l e d i n ( i v ) . (vi) Go to ( i i i ) u n t i l a l l nodes are l a b e l l e d . L i s t i n g s f o r the network i n Figure 20a and shortest-paths between node A and a l l other nodes are given below i n Table VII. TABLE VII Tree-building shortest-path algorithm:  l i s t i n g s and shortest-paths L i s t i n g s Shortest paths from A A B • • * • • I AB = 5 AD - 2 BC - 2 . . AC = 3 AC = 3 BF - 4 . . AD = 2 AB - 6 BA - 6 . . AE = 4 BE - 7 . . AF = 6 AG = 5 E s s e n t i a l l y , t h i s method involves the decomposition of a complex network into a simple tree subgraph. Haggett (1966, 1967) has developed an i n t e r e s t i n g extension of t h i s based on the a p p l i c a t i o n to networks of the concept of D i r i c h l e t regions. The algorithm has f i v e stages: i ) I d e n t i f y the dominant nodes i n the network, i i ) P a r t i t i o n the network at the " i n d i f f e r e n c e p o i n t s " between the dominant nodes to create a set of subgraphs. These are comparable to D i r i c h l e t regions. i i i ) Determine network h i e r a r c h i e s . Assume that each node dominates a l l the nodes ranked below i t , that i s , a l l the l a t t e r s 1 D i r i c h l e t regions 102 are nested i n the domain of the larger one. iv) Break the network at each of the i n d i f f e r e n c e points and remove i i • the redundant l i n k s involved from the graph. This produces the same r e s u l t as the Dantzig algorithm described above - the minimal spanning tree. v) Rank the shortest-paths by S t r a h l e r ' s stream ordering system. This gives a measure of the number of shortest-paths that use each l i n k and i s conceptually s i m i l a r to the l i n k importance index discussed e l s e -where i n t h i s paper. Empirical a p p l i c a t i o n s of t h i s algorithm(Haggett, 1966; 1969) seem to have produced i n t u i t i v e l y reasonable r e s u l t s . b) The matrix method An algorithm very d i f f e r e n t from Dantzig's uses properties of connection matrices to f i n d the shortest-paths. This method i s a t t r i b u t e d originally to Shimbel (1954) with a s e r i e s of modifications by Bellman (1958), Floyd (1962) and Farbey, Land and Murchland (1967). The most recent m o d i f i c a t i o a i s extremely e f f i c i e n t . C a l l e d the Cascade algorithm i t can also incorporate decomposition of the network i n t o subgraphs (Foot, 1965; M i l l s , 1966). The b a s i c matrix method algorithm i s as follows: i ) Construct a connection matrix (A). Diagonals are zero, whereas c e l l s with no d i r e c t connection contain a very large number, designated below by X. i i ) For a l l i j i n the matrix compare the distance i j v i a an i n t e r -mediate node k, where k runs through a l l nodes. i i i ) Retain the smallest distance i to j , put t h i s i n the corresponding c e l l of matrix B. 103 iv) Get another p a i r i j u n t i l f i n i s h e d v) Process matrix B i n the same way as matrix A. v i ) Continue to r e v i s e the matrix u n t i l two successive matrices are the same. This shows that the shortest routes have a l l been found. An example of th i s algorithm i s given i n Figure 3 and Table TABLE VIII - Shortest-Path Matrices  Matrix A Matrix B -(shortest-path) A B C D A B C D A j 0 1 4 x \ A 1 3(B) 6(C)\ B 1 0 2 X B 1 0 2 4(C) C , 4 2 0 2 C 3(B) 2 0 2 D \ x X 2 o) D \ 6(C) 4(C) 2 0 / To test whether, for example, AC i s the shortest distance between A and C step ( i i ) i s invoked as described above. Written out i n f u l l t h i s t e s t takes the form: i , k k,j and appears i n the example as: DAA + DAC = 0 + 4 = 4 °AB + DBC " 1 + 2 = 3 DAC + DCC = 4 + 0 = 4 DAD + DDC = X + 2 = X + 2. The second value (3), being the smallest, i s retained and entered i n the matrix B. 104 Several u s e f u l extensions of this matrix method have been made. Some of these are used i n t h i s paper, together with the b a s i c shortest-path algorithm. In p a r t i c u l a r , the nodes may be assigned a value which may be i n t e r p r e t e d as a delay f a c t o r or penalty f o r passing through i t ( K i s s l i n g , 1967). With reference to highway networks, an empirical i d e n t i f i c a t i o n of t h i s property of nodal centres i s immediate. The existence of a f e r r y or inadequate bridge can be simulated i n t h i s way. Another m o d i f i c a t i o n i s the storage of the routing sequence that the shortest-path took. This i s accomplished by r e t a i n i n g a l l i n t e r -mediate nodes (k) that formed the shortest-path between two other nodes. A routing matrix can be constructed from these values. Moreover, these k values give an i n d i c a t i o n of the number of times each l i n k i s used by a shortest path. The frequency with which each l i n k i s used i n a shortest-path gives a u s e f u l measure of l i n k importance. F i n a l l y , i t i s p o s s i b l e to reduce the number of computations by s p e c i f y i n g the nodes relevant to the current problem. This takes on considerable s i g n i f i c a n c e when i n t e r e s t i s r e s t r i c t e d to a few nodes i n a large network. APPENDIX 3 MATHEMATICAL SUMMARY OF CANONICAL CORRELATION Following Cooley and Lohnes (1962), the i n i t i a l step p a r t i t i o n s the c o r r e l a t i o n matrix (R) into four sub-matrices: R = / R 11 21 R 12 22 where R 11 R 22 12 21 i n t e r c o r r e l a t i o n s among the f i r s t set of v a r i a b l e s " second " " " of f i r s t set with second set transpose of R-^' These submatrices are substituted i n the canonical equation: ( R22 R21 R l l R12 " A i I ) b i = ° where Xi = l a t e n t root b^ = c h a r a c t e r i s t i c vector. This equation i s solved for A such that the determinant of the l e f t hand si d e i s zero. The vector a. i s obtained from I a i = Rii R i 2 v The canonical c o r r e l a t i o n between the two sets of va r i a b l e s i s and a. and b. are the weights associated with them, i i 105 

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