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_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