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Voter models and external influence Majmudar, Jimit 2016

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Voter Models and External InfluencebyJimit MajmudarB.Tech. (Hons.), Indian Institute of Technology Bombay, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)June 2016c© Jimit Majmudar, 2016The undersigned certify that they have read, and recommend to the College ofGraduate Studies for acceptance, a thesis entitled: Voter Models and ExternalInfluence submitted by Jimit Majmudar in partial fulfilment of the requirements ofthe degree of Master of ScienceRebecca C. Tyson, Unit 5 (Mathematics), University of British Columbia OkanaganSupervisor, ProfessorBert O. Baumgaertner, Department of Philosophy, University of IdahoSupervisory Committee Member, ProfessorW. John Braun, Unit 5 (Mathematics), University of British Columbia OkanaganSupervisory Committee Member, ProfessorStephen M. Krone, Department of Mathematics, University of IdahoSupervisory Committee Member, ProfessorLiane Gabora, Department of Psychology, University of British Columbia OkanaganUniversity Examiner, ProfessorJune 9, 2016(Date Submitted to Grad Studies)iiAbstractOpinions, and subsequently opinion dynamics, depend not just on interactions amongindividuals, but also on external influences such as the mass media. The dependence onlocal interactions, however, has received considerably more attention. In this work, weextend the classical voter model, the biased voter model, and the threshold voter modelto include external influences. We study the new models both analytically and compu-tationally, and we show that some of the new models can be understood after employingdiffusion approximations and mean field approximations. We derive results pertainingto the probability of reaching consensus on a particular opinion and also the expectedconsensus time for the different models. We find that although including an externalinfluence leads a faster consensus in general, this effect is more pronounced in the classi-cal voter model as compared to the threshold voter model. Some of our findings suggestthe potential importance of “macro-level” phenomena such as the external influences ascompared to the “micro-level” local interactions, in modelling opinion dynamics.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Chapter 2: Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Voter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Biased Voter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Threshold Voter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Noisy Voter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 Voter Models on Finite Graphs . . . . . . . . . . . . . . . . . . . . . . . 4Chapter 3: Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1 Jump Voter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Jump Voter Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Jump Biased Voter Model . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Jump Biased Voter Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 113.5 Jump Threshold Voter Model . . . . . . . . . . . . . . . . . . . . . . . . 11Chapter 4: Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.1 Jump Voter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.1.1 Comparison with Classical Voter Model . . . . . . . . . . . . . . 174.2 Jump Biased Voter Model . . . . . . . . . . . . . . . . . . . . . . . . . . 22iiiTABLE OF CONTENTS4.3 Jump Threshold Voter Model . . . . . . . . . . . . . . . . . . . . . . . . 25Chapter 5: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31ivList of TablesTable 3.1 All model parameters and their brief meanings . . . . . . . . . . 12Table 5.1 Overview of all models and their approximations (if any) . . . . . 29vList of FiguresFigure 4.1 Fixation probability for the jump voter model . . . . . . . . . . 16Figure 4.2 Expected consensus time for the jump voter model . . . . . . . . 17Figure 4.3 Comparison between the voter model and the jump voter model 19Figure 4.4 Expected consensus time dependence on jump parameters for thejump voter model - part 1 . . . . . . . . . . . . . . . . . . . . . 20Figure 4.5 Expected consensus time dependence on jump parameters for thejump voter model - part 2 . . . . . . . . . . . . . . . . . . . . . 21Figure 4.6 Fixation probability for the jump biased voter model . . . . . . 23Figure 4.7 Expected consensus time for the jump biased voter model . . . . 24Figure 4.8 Fixation probability and expected consensus time for the jumpthreshold voter model . . . . . . . . . . . . . . . . . . . . . . . . 25Figure 4.9 Spatial comparison between the jump voter model and the jumpthreshold voter model . . . . . . . . . . . . . . . . . . . . . . . . 26viAcknowledgementsI would like to begin by thanking my advisor, Prof. Rebecca Tyson, for reasonsmore than one - giving me the chance to do mathematics and the freedom to shapemy research, putting me in touch with the right people, and so on; the list is quitelong. I am fortunate to have had Prof. Tyson as my advisor. I am also thankful tomy committee members, Prof. Steve Krone, Prof. Bert Baumgaertner, and Prof. JohnBraun, for their continued support. I am particularly grateful to Prof. Krone for theamount of time he has spent with me, and the patience he has had while I worked myway through some of the material. Without his belief in my mathematical abilities, thisthesis would not have been possible.I am incredibly grateful to my parents, Heena and Rankesh Majmudar, and mysister, Chaitasi Majmudar, for always being the solid anchor in my life. They havealways been understanding and supportive of all my academic pursuits, and providedme counsel whenever I needed it. I am highly indebted to my mother, without whoseself-sacrifices I would not have been able to get such good quality education.I am also thankful to my friends - both at UBC Okanagan and from IIT Bombay. Ithas always been a pleasure to discuss research and life, in general, with Alireza, Mayang,and Joey. I am thankful to Chad for always being there whenever I needed help withsome material. Indeed those late evening discussions with him have only fuelled myliking for mathematics. Last but not the least, Saurabh (Javelin) has been the onefriend who I can talk to at any time about pretty much anything. Conversations withhim almost always serve as pleasant reminders for my liking for fundamental research.I am also thankful to Mandy Baumgaertner and Claudia Krone for graciously givingme space to live while I visited Moscow to work on my research. I will always rememberthe warm hospitality I received while there, and the laughs we shared.Lastly, I would also like to give a special mention to some of my musical heroes -Guthrie Govan, Plini, and Warren Mendonsa. Apart from being my constant companion,their musical eloquence regularly teaches me a thing or two about creative expression.viiDedicated to my parents, Heena and Rankesh Majmudar, and my sister, ChaitasiMajmudar.viiiChapter 1Introduction1.1 BackgroundHuman opinions and collective human behaviours have now been studied for morethan a century. Prominent historic examples range from studies of irrational crowdbehaviours [Mac12, LB97], to social conformity [She36, Asc56], obedience [MF74], andherd mentality [Key06], to critical mass phenomena [Sch71]. Relatively recent is themathematical analysis of these dynamics, which has been made possible due to seminalframeworks like the voter model [CS73, HL75], DeGroot learning [DeG74], and thenaming game model [BFC+05, DBBL06]. Considerable work has been done extendingthese models so as to incorporate a combination of diverse social phenomena [PB10,YAS11, YSO+11, DGM14, FR14]. These advances, on just the quantitative side, havebeen made by mathematicians, physicists, and computer scientists, who, along withresearchers of sociology, psychology, philosophy, politics, etc. make this field immenselyinterdisciplinary. For a detailed review of the origins and evolution of this domain, thereader is referred to the comprehensive review by Xia et al [XWX11].While refining the mechanistic description of how individuals communicate has re-ceived considerable attention [CD91, Lig94, HK02], the incorporation of external in-fluences has received little modelling scrutiny, one reason for which seems to be theassociated loss of analytical tractability. There is already evidence, however, that mediacan play an important role in opinion dynamics in many different contexts, for example,climate change [BCJ12], and electoral voting, to name a few. Furthermore, commercialrelevance of this phenomenon can be found in quantitative marketing, when two brandscompete to sell their respective products via advertising. A theoretical examination ofthe effect of external influence is thus important. Existing work, however, is either notamenable to detailed mathematical analysis [QCS14], or lacks generality [FR13].1.2 ContributionsWe address herein some of the above discussed shortcomings by considering a novelapproach that incorporates an external influence that is general, but also maintains con-siderable mathematical tractability at the same time. We model the effect of externalinfluence via sporadic events that make multiple agents flip their opinion simulata-neously. To perform the mathematical analysis, we use the theory of jump diffusionprocesses, and therefore attach the word “jump” as an adjective in naming our models.Jump diffusion models are widely used in financial domains such as derivative pricing11.3. Organisationand risk management [Kou07], and also in physics, but have not before been used in thedomain of social dynamics. They were primarily introduced in finance because, moreoften than not, the quantity to be modelled exhibited nonsmooth fluctuations that aregular diffusion model failed to capture. Qualitatively speaking, in sociodynamics,external influences can generate nonsmooth effects in the evolution of opinions, henceour creation of these jump models. We explore the effects of including the jumps inthree well-established models: the classical voter model, the biased voter model, andthe threshold voter model.1.3 OrganisationChapter 2 discusses the well-established models that act as a basis for the jumpmodels. In chapter 3 we define all our models, followed by our results and their in-terpretations in chapter 4. Lastly, we include a conclusion that contains a high-leveldiscussion of our approach and our key findings in chapter 5.2Chapter 2PreliminariesIn this chapter, we review some of the existing models that inspired our models.Before doing that, we briefly present the terminology that we follow. By Zd, we meanthe d−dimensional lattice of integer coordinates, and G denotes an undirected finitegraph where S(G) denotes the set of nodes of G, and kx denotes the degree of node x.The cardinality of S(G) is N , i.e., the graph consists of N nodes. We interpret each nodeas an individual agent. A state is associated with each node, and we herein think of thatstate as the opinion of that node on a particular subject. We consider a binary opinionspace, i.e., each node must have either opinion 0 or opinion 1 at any given time t, wheret ∈ [0,∞) for continuous-time models and t ∈ {0, 1, 2, ...} for discrete-time models. Thestate of the entire system at a time t is given by s(t), and the state at a specific nodex is given by sx(t). We denote by N (x) the set of neighbours of node x; for a latticethis set consists of its four four immediate neighbours (north, south, east, west), andfor a graph it consists of all immediate nodes with which x is connected. Moreover, wedenote by c(x, s) the flip rate, i.e., the rate at which the opinion at node x flips whenthe process is in state s. Therefore, we havelim∆t→0P (sx(t+ ∆t) 6= sx(t)|s(t) = s)∆t= c(x, s) (2.1)2.1 Voter ModelThe voter model [CS73, HL75] is a continuous-time Markov process on Zd,c(x, s) = #{y ∈ N (x) : sy 6= sx} (2.2)This is a linear model in the sense that the rate of change of opinion at a node x is alinear function of the count of opposing opinions in the neighbourhood of x. Intuitively,we may interpret the process as follows: There are independent Poisson clocks at eachnode. Whenever the clock at a node x rings, the node scans its neighbourhood, uniformlyselects one of its neighbours at random, and then adopts the opinion of that chosenneighbour.2.2 Biased Voter ModelThe biased voter model [Wil72] is a modification of the voter model where,32.3. Threshold Voter Modelc(x, s) ={b ·#{y ∈ N (x) : sy 6= sx}, if sx = 0#{y ∈ N (x) : sy 6= sx}, if sx = 1(2.3)with b > 0. Notice that this is also a linear model. The difference between this modeland the voter model is only that the transition rate here is adjusted by a bias factor bfor a 0 → 1 flip, without loss of generality (WLOG). Intuitively, this adjustment maymake the opinion 0 more or less retentive, depending on the value of b. That is, a 0→ 1flip would be more frequent than a 1→ 0 flip if b > 1, and vice versa for b < 1.2.3 Threshold Voter ModelThe threshold voter model [Lig94] is a modification of the voter model where,c(x, s) ={1, if #{y ∈ N (x) : sy 6= sx} ≥ θ0, if #{y ∈ N (x) : sy 6= sx} < θ(2.4)with θ > 0. Flip rate (2.4) yields a nonlinear model in the sense that the rate of changeof opinion at a node x is a nonlinear function of the count of opposing opinions inthe neighbourhood of x. Intuitively, we may again think of independent Poisson clocksat each node. When the clock at a node x rings, the node scans its neighbourhood,and updates its opinion if and only if there are at least θ opposing opinions in itsneighbourhood.2.4 Noisy Voter ModelThe noisy voter model [GM95] is a modification of the voter model where,c(x, s) ={#{y ∈ N (x) : sy 6= sx}+ β, if sx = 0#{y ∈ N (x) : sy 6= sx}+ δ, if sx = 1(2.5)with β > 0 and δ > 0. Flip rate (2.5) yields again a linear model. Here the flipping ratesare incremented by parameters β and δ, which are aptly called the noise parameters.Intuitively, these parameters inject an additional source of flipping that is independentof the states of the neighbouring sites. As an extreme case, in this model there is apossibility of an opinion change at a node x even in the situation when all its neighbourshave the same opinion as that of x.2.5 Voter Models on Finite GraphsThe voter model and its variants discussed in the previous sections have also beenstudied on finite lattices and finite graphs. The most pertinent reference point for thatline of work is the paper by Sood et al [SR05]. They study discrete-time voter models42.5. Voter Models on Finite Graphson a heterogeneous graph in the absence of degree correlations. They define degree-weighted moment, ω1, a generalisation of the notion of density asω1(t) ≡ 1Nµ1∑x∈S(G)kxsx(t)where µ1 is the first moment of the degree distribution, orµ1 =1N∑x∈S(G)kx.Moreover, let ρk denote the subdensity of opinion 1 nodes that have degree k, given asρk(t) =∑x∈S(G),kx=ksx(t)Nkwhere Nk denotes the count of nodes that have degree k. They show that the votermodel dynamics follow a two-time-scale approach to consensus. Initially, there is arapid approach of all subdensities to ω1(0), followed by which diffusive fluctuationsdrive the model to consensus, i.e., the state where either all nodes have opinion 0 or 1.Additionally, the time to consensus, TN , is given analytically byTN (ω) = Neff[(1− ω) ln 11− ω + ω ln1ω]where ω ≡ ω1(0), and Neff denotes the effective population size defined asNeff ≡ N µ21µ2where µ2 denotes the second moment of the degree distribution. The study is furtherextended to the biased voter model on a heterogeneous graph, and it is demonstratedthat in addition to the two-time-scale dynamics, the probability of single fitter agent toflip the entire population on its side is directly proportional to its degree k.5Chapter 3ModelsFor the subsequent models, we restrict our attention to graphs G that are k−regular,where each node, as before, is considered to be an individual agent, and the links rep-resent social connections between the agents. Also as before, consider a binary opinionspace, i.e., a node x ∈ S(G) must have either opinion 0 or opinion 1 at time t, which isdenoted by sx(t). The adjacency matrix is denoted by A, with the element Axy being 1if nodes x and y are connected, and 0 otherwise. The neighbourhood of a node consistsof the immediate nodes with which it is connected. Consensus is said to be reachedwhen either all nodes have opinion 0 or opinion 1, and it is assumed that the processspends forever in that state once it is attained. Consensus on opinion 1 is herein calledfixation.3.1 Jump Voter ModelThe jump voter model is a discrete-time process that is updated according to thetwo rules given below. At each time step, one of the following occurs:Update Rule 1: With probability (1−p), a single node is randomly selected whichthen adopts the opinion of one of its neighbours chosen randomly.Update Rule 2: With probability p, either a random number of 0 opinion nodesupdate their opinions to 1 or a random number of 1 opinion nodes update their opinionsto 0. A signed form of this random variable (Z) follows the convention that it is negativein the case of the former update, and positive for the latter.The first update rule captures the node-to-node interactions of the classical votermodel. The second rule captures the more global external influence that makes severalopinions flip simultaneously, a phenomenon that we call a jump. The flexibility grantedby the model in terms of occasionality of the external influence is crucial here, becausein certain application contexts the effect of this influence may be sporadic. For example,media attention in society is observed to be irregular, with this irregularlity dependingon, among other things, the nature of the topic. Call p the jump probability, and notethat when p = 0 the model reduces to a discrete-time version of the classical voter modelon a graph structure. The jump voter model is a discrete state space Markov chain, andthe total number of opinion 1 nodes in the graph at time step t, denoted by XN (t), can63.2. Jump Voter Diffusionbe thought of as a global summary statistic of that Markov chain. More formally,XN (t) =∑x∈S(G)sx(t).The random variable Z is independent of the state of the process, and we define thescaled jump Y ≡ Z/N whose mean is zero in this version of the model. A non-zero meancould be indicative of biases in the external influence, and we cover that case within thebiased version of the jump voter model in Section 3.3. The variance of Z represents thestrength of the external influence. (A brief summary of all model parameters and theirbrief meaning is provided in Table 3.1.) This model corresponds to model B in Table5.1 that shows a global overview of all our models.3.2 Jump Voter DiffusionDiffusion approximations are an indispensable tool for inferring properties of discreteprocesses, albeit in the limiting case for large populations. We make use of a diffusionapproximation here and derive a jump-diffusion process to which the jump voter modelweakly converges. To proceed with this jump diffusion approximation, we will need thetransition probabilities corresponding to the update rule 1:• P [i → i + 1] ≡ probability that a 0 → 1 update happens at a certain time stepwhen the count of nodes with opinion 1 is i, and,• P [i → i − 1] ≡ probability that a 1 → 0 update happens at a certain time stepwhen the count of nodes with opinion 1 is i.Now,P [i→ i+ 1] =P [(selecting a node x with opinion 0) ∩(selecting a y, in the neighbourhood of the node x,with opinion 1)],and,P [i→ i− 1] =P [(selecting a node x with opinion 1) ∩(selecting a y, in the neighbourhood of the node x,with opinion 0)].73.2. Jump Voter DiffusionTherefore, we have,P [i→ i+ 1] =∑x∈S(G),sx=0 1N∑y∈S(G)AxysykP [i→ i− 1] =∑x∈S(G),sx=1 1N∑y∈S(G)Axy1− syk (3.1)We take a short detour here to explain the mean field approximation, in the spiritof that used by Sood et al [SR05]. This notion of mean field approximation is somewhatdifferent from the one that is usually used. Typically, for a mean field approximation,one simply considers the “well-mixed” case where each node has links with every othernode. For our mean field approximation, consider the scenario where the above processis run multiple times, but each time, before starting, we perform some rewiring so asto obtain a different regular graph. So, two nodes (say x and y) may be connected incertain runs (i.e. Axy = 1), but may not be connected in other runs (i.e. Axy = 0). Inthis case, in order to perform average calculations over all such realisations, we replaceAxy in (3.1) with E[Axy]. Alternatively, we may also try to find the probability thatnodes x and y are connected, sinceE[Axy] = 1 · P [Axy = 1] + 0 · P [Axy = 0].Considering all possible node pairs,E[total number of links] =(N2)P [Axy = 1]. (3.2)But the total number of links = Nk2 . Therefore, comparing with equation (3.2), weget, (N2)P [Axy = 1] =Nk2∴ P [Axy = 1] =kN − 1 = E[Axy].(3.3)(We can check that the expression P [Axy = 1] =kN−1 indeed matches the “boundarycases”. When k = 0 the probability of picking any two nodes that are connected is 0,and when k = N − 1 we have a complete graph, and the same probability is 1; both ofthese cases are consistent with the relationship derived in equation (3.3).)For large enough N , we may replace N − 1 with N which gives us83.2. Jump Voter DiffusionE[Axy] ≈ kN. (3.4)Making use of the mean field approximation (i.e., substituting the expression forE[Axy] from equation (3.4) in the equations in (3.1)), we obtain,P [i→ i+ 1] ≈(1− iN)(iN)P [i→ i− 1] ≈(iN)(1− iN).(3.5)We scale the process by dividing by N , and interpret the new state as density orproportion of nodes that have opinion 1. We also set a single time step ∆t to be equalto 1/N2. As N becomes large (i.e., the time step becomes small), the update rule 1(which is the discrete version of the classical voter model) becomes a diffusion processwhose drift µ(x), using its standard definition [KT81], can be derived as follows.µ(x) = limN→∞iN→xN2 · E[XN ([N2t] + 1)N− XN ([N2t])N∣∣∣∣X([N2t]) = i]= limN→∞iN→xN2 · E[(1N)P [i→ i+ 1] +(−1N)P [i→ i− 1]].Substituting the probabilities from equation (3.5) into equation (3.6), we obtainµ(x) = 0. Following similar arguments, the diffusion parameter is found to be,σ2(x) = 2x(1− x). (3.6)The calculations so far give us a continuous approximation for the update rule 1.If we define λ ≡ N2p and Y ≡ Z/N (scaled jump), then for a small enough p and alarge enough N , the jump voter model (both update rule 1 and update rule 2) can beapproximated by a superposition of the diffusion derived above and a compound Poissonprocess, i.e., a jump diffusion process given as,dX(t) =√2X(t)(1−X(t))dW (t) + Y dN(t) (3.7)where W (t) represents a Wiener process, and N(t) represents a rate λ Poisson pro-cess. We denote by g(x) the probability density function of the jump random variableY , and also note that g(x) will have support [−1, 1]. The mean of Y is denoted by m(which is zero here), and the variance by v. We call v the jump variance. The generatorof a stochastic process is an operator that, intuitively, encodes important informationabout that process. Mathematically, (for a time-homogeneous process) it is defined as[Øks03]93.3. Jump Biased Voter ModelL f(x) ≡ limt→0+E[f(X(t))|X(0) = x]− f(x)t.The generator of this process is the integro-differential operator L , whereL f(x) = x(1− x)f ′′(x) + λ∫ +∞−∞[f(x− y)− f(x)]g(y)dy. (3.8)The jump diffusion in equation 3.7 corresponds to model C in Table 5.1. Note herethat as a result of using the mean field approximation, the jump voter diffusion given byequation 3.7 does not have a term containing the degree parameter k. Therefore, basedon this observation, we can say that quantities related to overall opinion density do notdepend on the degree of the regular graph.3.3 Jump Biased Voter ModelWe now define a biased version of the jump voter model. For our purposes, wedivide the biases into two categories: Biases in the external influence, and biases in theinteractions between nodes. By the former, we mean that the external influence is skewedtowards a particular opinion; by the latter, we mean that the agents of a particularopinion are less likely to interact with their neighbours and subsequently update theiropinion. An example of bias in the external influence would be a biased media outlet.Bias in the node-to-node interactions, on the other hand, can stem from various sources.For example, global warming may bias the population towards an increased acceptanceof climate change. In terms of modelling, we incorporate the biases by making thefollowing adjustments to the two update rules:• External influence: The jump random variable is now allowed to have a non-zero mean, i.e., m 6= 0. If m ∈ (0, 1] then the external influence bias (or jumpbias) favours opinion 0, whereas if m ∈ [−1, 0) the external influence bias (or jumpbias) favours opinion 1. Or,• Node-to-node interactions: For mathematical simplicity, we assume, withoutloss of generality, that this bias b is in favour of opinion 1, thereby making anopinion change at a node from 1 to 0 less probable than before, while a change from0 to 1 is more probable. More specifically, we define a bias parameter b ∈ (1,∞),and adjust the probability of selecting a node as follows.P (an opinion 1 node is selected for a potential update) =ibN,P (an opinion 0 node is selected for a potential update) = 1− ibN.(Note that the higher the value of bias b, the lesser the probability of a changehappening at an opinion 1 node, making that opinion more rententive.) Or,• Both of the above.The jump biased voter model corresponds to model E in Table 5.1.103.4. Jump Biased Voter Diffusion3.4 Jump Biased Voter DiffusionThe revised transition probabilities, again incorporating the mean field approxima-tion, areP [i→ i+ 1] =(1− b−1iN)iNP [i→ i− 1] = b−1iN(1− iN) (3.9)where b−1 = 1/b is the inverse bias. Applying the jump diffusion approximation, asbefore, gives the following stochastic differential equation corresponding to the jumpbiased voter model,dX(t) =X(t)(1− b−1)dt+√X(t)[(1 + b−1)− 2b−1X(t)]dW (t)+ Y dN(t).(3.10)Note the non-zero drift in equation (3.10) owing to the bias factor k. The generatorof this process is the integro-differential operator,L f(x) =x(1− b−1)f ′(x) + x2[(1 + b−1)− 2b−1x]f ′′(x)+ λ∫ +∞−∞[f(x− y)− f(x)]g(y)dy.(3.11)The jump diffusion in equation (3.10) corresponds to model F in Table 5.1.3.5 Jump Threshold Voter ModelThe jump threshold voter model is a discrete-time process that is updated accordingto the rules given below. At each time step, one of the following occurs:Update Rule 1: With probability (1 − p), a single node is randomly selected. Ifthe number of opposing opinions in the neighbourhood of the selected node is greaterthan or equal to a threshold θ, then the opinion of the originally selected node is updated.Update Rule 2: With probability p, either a random number of 0 opinion nodesupdate their opinions to 1 or a random number of 1 opinion nodes update their opinionsto 0. A signed form of this random variable (Z) follows the convention that it is negativein the case of the former update, and positive for the latter.Similar to the jump voter model, the first update rule captures the node-to-nodeinteractions of the classical threshold voter model, whereas the second rule captures theexternal influence. Again as with the jump voter model, the mean of Z is zero in this113.5. Jump Threshold Voter Modelmodel. This model corresponds to model H in Table 5.1.Table 3.1 summarises all the parameters discussed in this chapter.Table 3.1: All model parameters and their brief meanings.Parameter Brief DescriptionN Total population.k Degree in a regular graph.p Probability of a jump occuring at a time step.Also called jump probability.Z Random variable that gives the jump or the external influence.m Jump bias, or external influence bias.v Jump variance, or strength of the external influence.b Bias in node-to-node interactions.θ Threshold parameter in the jump threshold voter model.An opinion update at a node, via node-to-node interactions,happens if and only if the number of opposing opinions inthe node’s neighbourhood is greater than or equal to θ.12Chapter 4ResultsIf X(t) is a jump diffusion with generatorL , Itoˆ’s formula for jump processes [Phi90]implies thatM(t) ≡ f(X(t))−∫ t0L f(X(s))ds (4.1)is a martingale for any C2 function f(x). Application of the Optional Stopping Theorem(OST) to this martingale will result in the formulation of “boundary” value problemsfor both fixation probability and expected value of the consensus time for both thejump voter model and the jump biased voter model. The word “boundary” is in doublequotes as this formulation does not lead to boundary value problems in the strict sense.While the solution of a traditional (one-dimensional) BVP is a function that obeys thegoverning equation in the region C, and whose value is known on ∂C, the solutions ofthe problems developed here will satisfy the governing equation in C, but their valueswill be known for the entire region R−C. Qualitatively, this is because a jump diffusionhas the potential to overshoot boundaries. Denoting by τ the time to consensus andusing the OST, we obtainEx[f(X(τ))]− Ex[∫ τ0L f(X(s))ds]= f(x). (4.2)(Here we use Ex[·] as shorthand for E[·|X0 = x], and similarly P x[·] for P [·|X0 = x].)The following two claims help us derive the fixation probability and consensus timeboundary value problems. These claims are straightforward extensions of ideas commonin a simple diffusion (i.e., a diffusion without any jumps) [Øks03]. Before proceeding,we quickly define the random times when the jump diffusion crosses the 0 boundary andthe 1 boundary as follows.T0 ≡ inf{t ≥ 0 : X(t) ∈ (−∞, 0]}T1 ≡ inf{t ≥ 0 : X(t) ∈ [1,∞)}Claim 4.1. If there exists a C2 function u : R → R such that L u(x) = 0 for allx ∈ (0, 1), and u(x) = 0 for all x ∈ (−∞, 0] and u(x) = 1 for all x ∈ [1,∞), then thefunction u gives the fixation probability for the jump voter model and the jump biasedvoter model.Proof. Since u is a C2 function, we can use equation (4.2) to get134.1. Jump Voter ModelEx[u(X(τ))]− Ex[∫ τ0L u(X(s))ds]= u(x).∴ 1 · P x(T1 < T0) + 0 · P x(T0 < T1)− 0 = u(x)=⇒ P x(T1 < T0) = u(x).The fixation probability satisfies the following BVPL u(x) = 0u(x) = 0, ∀x ∈ (−∞, 0]u(x) = 1, ∀x ∈ [1,∞).(4.3)The second, and possibly more interesting, property of the jump voter model is theexpected value of the consensus time. This is often referred to as the hitting time in thestochastic processes literature. Consensus time, here, is formally defined asτ ≡ inf{t ≥ 0 : X(t) ∈ (−∞, 0] ∪ [1,∞)}. (4.4)Claim 4.2. If there exists a C2 function v : R → R such that L v(x) = −1 for allx ∈ (0, 1), and v(x) = 0 otherwise, then the function v gives the expected consensus timefor the jump voter model and the jump biased voter model.Proof. Since v is a C2 function, we can use equation (4.2) to getEx[v(X(τ))]− Ex[∫ τ0L v(X(s))ds]= v(x).∴ 0 · P x(T1 < T0) + 0 · P x(T0 < T1)− Ex[−τ ] = v(x)=⇒ Ex[τ ] = v(x).The expected hitting time for the jump voter model and the jump biased voter modelsolves the BVPL v(x) = −1v(x) = 0, ∀x ∈ (−∞, 0] ∪ [1,∞). (4.5)4.1 Jump Voter ModelWe begin by investigating the fixation probability and the expected consensus timefor the jump voter diffusion. To determine the fixation probability, we use the generatorof the jump voter diffusion from equation (3.8) in the BVP in (4.3), to get144.1. Jump Voter Modelx(1− x)u′′(x) + λ∫ +∞−∞u(x− y)g(y)dy − λu(x) = 0u(x) = 0, ∀x ∈ (−∞, 0]u(x) = 1, ∀x ∈ [1,∞).(4.6)It is quite challenging to find an analytical solution for a variable-coefficient integro-differential equation such as equation (4.6). Even a similar constant-coefficient equationrequires imposing some structure on the function g(x) to make a closed-form solutionpossible [KW04]. We thus use numerical approaches to solve this problem.The x domain [0, 1] is discretised into m steps of size h, such that x0 = 0 and xm = 1.The difference equation corresponding to equation (4.6) thus becomesjh(1− jh)uj−1 − 2uj + uj+1h2+λh+∞∑i=−∞uj−igi − λuj = 0, ∀j ∈ {1, 2, ..., (m− 1)}uj = 0, ∀j ∈ {...,−1, 0}uj = 1, ∀j ∈ {m,m+ 1, ...}.(4.7)The infinite summation term on the LHS can be recognised as a discrete convolutionof two discrete functions u and g. Using the commutativity property of convolution, theboundary conditions on u, and the fact that g has [−1, 1] support, the infinite summationcan be truncated to obtain the simplified set of equationsjh(1− jh)uj−1 − 2uj + uj+1h2+λh2m−1∑i=1uigj−i − λuj = 0, ∀j ∈ {1, 2, ..., (m− 1)}uj = 1, ∀j ∈ {m, (m+ 1), ..., (2m− 1)}.(4.8)These equations give a system of 2m − 1 equations with the same number of un-knowns. The solution of equation (4.8) obtained numerically closely matches the resultsfrom Monte Carlo simulations (Figure 4.1).154.1. Jump Voter ModelFigure 4.1: Fixation probability for the jump voter model on a regular graph with N = 500, k =4, p = 1/(500×10), v = 0.04 (see Table 3.1 for meaning of parameters). The black curve denotesthe numerical solution of equation (4.6), black points denote simulation results based on updaterules in Section 3.1, where each point is obtained by averaging over 1000 runs. The externalinfluence, Z, has a truncated normal distribution. This match confirms our approximation ofmodel B by model C, as shown in Table 5.1.Similarly, if we expand (4.5) using the generator in (3.8), we obtainx(1− x)v′′(x) + λ∫ +∞−∞v(x− y)g(y)dy − λv(x) = −1v(x) = 0, ∀x ∈ (−∞, 0] ∪ [1,∞).(4.9)As before, rather than trying to derive a closed-form solution, we approach thisequation numerically. Discretising the integro-differential equation, and again applyingsimplification techniques similar to those used to derive equation (4.8), we obtain asystem of m− 1 equations, that approximates equation (4.9),jh(1− jh)vj−1 − 2vj + vj+1h2+λhm−1∑i=1vigj−i − λvj = −1, ∀j ∈ {1, 2, ..., (m− 1)}v0 = vm = 0.(4.10)The solution of equation (4.10) closely matches the results of the Monte Carlo sim-164.1. Jump Voter Modelulations (Figure 4.2). Note that the solution is symmetric about x = 0.5, which makesqualitative sense, since the mean external influence is 0.Figure 4.2: Expected consensus time for the jump voter model on a 25× 25 lattice, with p =1/(625×5), v = 0.04 (see Table 3.1 for meaning of parameters). The black curve denotes the exactnumerical solution of equation (4.9), black points denote simulations results based on updaterules in Section 3.1, where each point is obtained by averaging over 2000 runs. The externalinfluence, Z, has a truncated normal distribution. This match confirms our approximation ofmodel B by model C, as shown in Table 5.1.4.1.1 Comparison with Classical Voter ModelBased on the previous work by Sood et al [SR05], we know that the diffusion ap-proximation of the classical voter model (discrete-time version) is given by the diffusionprocess,dX(t) =√2X(t)(1−X(t))dW (t) (4.11)whose generator is given by the differential operator,L f(x) = x(1− x)f ′′(x). (4.12)As mentioned earlier, if the jump probability is set to p = 0 in our jump votermodel, we retrieve the classical voter model. We derive the expected consensus timesolution corresponding to that case, and find that it matches with the results in Sood174.1. Jump Voter Modelet al [SR05]. (Results not shown.) The two parameters, jump probability p and jumpvariance v, together determine the overall impact of the external influence, and wecollectively refer to them as jump parameters. We find that even for fairly small jumpparameter values, the expected consensus time differs dramatically between the classicalvoter model and the jump voter model (Figure 4.3(b)).184.1. Jump Voter Model(a) (b)(c)Figure 4.3: (a) Fixation probability and (b) expected consensus time comparison between theclassical voter diffusion and the jump voter diffusion (and hence the classical voter model andthe jump voter model), and (c) Difference in fixation probability between the classical voterdiffusion and the jump voter diffusion (and hence the classical voter model and the jump votermodel). N = 500 and for the jump voter model, p = 1/(500 × 10), v = 0.03 (see Table 3.1 formeaning of parameters). This comparison corresponds to comparing model A and model D (ormodel B and model E), as shown in Table 5.1.If we consider just the maximum value of consensus time (i.e. the consensus timewhen the initial density of 1s is 0.5) (Figure 4.3), we can plot it as a function of thetwo jump parameters. We see that the consensus time decreases rapidly as jumps are194.1. Jump Voter Modelintroduced.Figure 4.4: Maximum consensus time as a function of jump variance v and jump probabilityp, N = 500. (The maximum consensus time axis has been restricted to 100 time steps fordiagrammatic clarity.)We also isolate the dependence of the consensus time on the individual jump param-eters. The surface in Figure 4.4 appears symmetric about p = v. However, examinationof two curves on the surface (Figure 4.5) indicates that the symmetry is only approxi-mate. The parameter p has a slightly stronger effect on consensus time than parameterv.204.1. Jump Voter ModelFigure 4.5: Maximum consensus time for the jump voter diffusion as a function of the jumpparameters individually, N = 500. (The maximum consensus time axis has been restricted to10000 time steps for diagrammatic clarity.) The maximum consensus time decreases rapidly asp and v increase. For plotting the dependence on p, v = 0.001, and for plotting the dependenceon v, p = 0.001 (see Table 3.1 for meaning of parameters). We notice that the jump probability(p) has a slightly stronger effect than the jump variance (v) on the consensus time.We notice in Figure 4.3(b) that although our solution for the consensus time hasproperties that are qualitatively similar to those of a classical voter model, the quanti-tative difference between the two is considerable. We therefore make a few observations:1. Since the mean external influence is 0, the effect of jumps on the fixation prob-ability (the probability of reaching consensus on opinion 1), is mostly minimal(Figure 4.3(a) and Figure 4.3(c)). In other words, the tendency of the process toreach consensus at a particular state is mostly unaltered (except at very low initialminority densities) by the external influence due to symmetry in the latter.2. Since jumps can have a significant impact on the consensus time of the process,the jumps may be included in opinion dynamics models based on the voter model.(Figure 4.3)3. The dependence of consensus time on both jump parameters, jump variance andjump probability, is qualitatively very similar. (Figure 4.5)We also note how the effect of the jumps begins to appear as the values of thejump parameters are gradually increased (Figure 4.4 and Figure 4.5). Consensus time214.2. Jump Biased Voter Modelis decreasing in both jump parameters (v and p). The rate of decrease of the consensustime is very high at low jump parameter values, and drops as parameter values increase.Therefore, it is primarily the presence of jumps that appears to be a key factor for theconsensus time. In other words, the voter model is very sensitive to the presence ofjumps as far as consensus time is concerned.Overall, the jumps have a key role in driving the dynamics. Moreover, the jumps alsointroduce little skew in addition to that inherently present due to the initial densities.4.2 Jump Biased Voter ModelIn this section, we turn to investigating the fixation probability and the consensustime for the jump biased voter diffusion. This model, discussed in Section 3.3, incor-porates a bias in the external influence and/or a bias in the node-to-node interactions.The fixation probability BVP based on the generator in equation (3.11) is,x(1− b−1)u′(x) + x2N[(1+b−1)− 2b−1x]u′′(x)+λ∫ +∞−∞u(x− y)g(y)dy − λu(x) = 0u(x) = 0, ∀x ∈ (−∞, 0]u(x) = 1, ∀x ∈ [1,∞).(4.13)We solve this problem numerically, and again find close match between the solutionof equation (4.13) and Monte Carlo simulations. (Figure 4.6)224.2. Jump Biased Voter ModelFigure 4.6: Fixation probability for the jump biased voter model on a regular graph withN = 500, k = 100, p = 1/(500 × 5),m = 0.1, v = 0.03, b = 1.1 (see Table 3.1 for meaning ofparameters). The black curve denotes the numerical solution of equation (4.13), black points de-note simulation results based on update rules in Section 3.1 including the adjustments discussedin Section 3.3. Each point is obtained by averaging over 1000 runs. The external influence, Z,has a truncated normal distribution. This biased model corresponds to the case where the ex-ternal influence biases opinions towards 0, whereas the bias in node-to-node interactions makes1 retentive. This match confirms our approximation of model E by model F, as shown in Table5.1.The consensus time BVP is,x(1− b−1)v′(x) + x2N[(1+b−1)− 2b−1x]v′′(x)+λ∫ +∞−∞f(x− y)g(y)dy − λv(x) = −1v(x) = 0, ∀x ∈ (−∞, 0] ∪ [1,∞).(4.14)The numerical solution of equation (4.14), along with the results from Monte Carlosimulations, is shown in Figure 4.7.234.2. Jump Biased Voter ModelFigure 4.7: Expected consensus time for the jump biased voter model on a regular graph withN = 500, k = 50, p = 1/(500×2),m = 0.1, v = 0.04, b = 1.1 (see Table 3.1 for meaning of param-eters). The black curve denotes the numerical solution of equation (4.14), black points denotesimulation results based on update rules in Section 3.1 including the adjustments discussed inSection 3.3. Each point is obtained by averaging over 100 runs. The external influence, Z, hasa truncated normal distribution. This biased model corresponds to the case where the exter-nal influence biases opinions towards 0, whereas the bias in node-to-node interactions makes 1retentive. This match confirms our approximation of model E by model F, as shown in Table5.1.We make some observations for the fixation probability and consensus time resultsfor the jump biased voter model. Firstly, there is an increase in the fixation probabilityacross all initial conditions (Figure 4.6). This implies that when the two biases arecomparable, the jump bias seems to have a weaker effect than the bias embedded inthe node-to-node interactions. Therefore, we see a net bias towards 1, which is the biasin the node-to-node interactions. Secondly, the expected consensus time for the jumpbiased voter model is reduced across all initial conditions, and it no longer remains asymmetric function (Figure 4.7). (The reduction for initial densities greater than 0.5 isgreater than the reduction for the densities lesser than 0.5.) The asymmetry may beexplained as follows: First, consider the case where the initial density of 1s is greaterthan 0.5. In the absense of a net bias, the process for this case would terminate atopinion 1 more than 50% of the times. With the introduction of a net bias towards 1,this tendency will be further reinforced, resulting in an increase in the relative numberof instances where the process terminates at 1. This behaviour can be thought of as areduction in the “distance” to termination. Moreover, the net bias can also be thought244.3. Jump Threshold Voter Modelas increasing the “speed” of the process, i.e., higher the net bias, higher the speed of theprocess and more quickly it approaches termination. The combined effect of reduceddistance and increased speed reduces the consensus time for this case. Now considerthe case where the initial density of 1s is less than 0.5. Due to the overall increasein the fixation probability, we again have higher number of instances terminating at 0than before. However, since the initial density in this case is less than 0.5, we now havea larger distance to termination. Therefore, here we have a combination of increaseddistance and increased speed. It appears that the effect of the latter is stronger than theformer, thereby still decreasing the consensus time, but not as much as the case wherethe initial density is greater than 0.5.4.3 Jump Threshold Voter ModelIn this section, we study the fixation probability and the consensus time for the jumpthreshold voter model. The results are obtained through Monte Carlo simulations, andare shown in Figure 4.8.(a) (b)Figure 4.8: (a) Fixation probability and (b) expected consensus time for the jump thresholdvoter model in comparison with the jump voter model and the threshold voter model, on a 25×25lattice with p = 1/(625 × 10), v = 0.03, θ = 2 (see Table 3.1 for meaning of parameters). Eachpoint is based on 1000 runs, and error bars indicate standard error of the mean. The externalinfluence, Z, has a truncated normal distribution. The comparison corresponds to comparingmodel B, model G, and model H, as shown in Table 5.1.Based on the results in Figure 4.8, we make the general observation here that thefixation probability and the consensus time for the threshold voter model and the jumpthreshold voter model become noticeably different only when the minority opinion at254.3. Jump Threshold Voter Modelthe start of the process is greater than approximately 0.4. For the current choice ofmodel parameters, we may think of the initial density value of 0.4 as a “critical” densityin the sense that the behaviour of the jump threshold voter model begins to differ fromthe threshold voter model, if the initial density is higher than the critical density.For another comparison – between jump threshold voter model and jump voter model– the two models were run in parallel and the spatial results are shown in Figure 4.9.This direction of spatial analysis was motivated by the current understanding of spatialdifferences between the threshold voter model and the voter model.(a)(b)Figure 4.9: Snapshots of evolution of (a) the jump threshold voter model, and (b) the jumpvoter model on a 100 × 100 lattice, with p = 1/(10000 × 5), v = 0.03, θ = 2 (see Table 3.1 formeaning of parameters). Initial density of 1s is 0.5 for both models. The numbers in the topright corner denote the time step for the respective panel. The external influence, Z, has atruncated normal distribution. This comparison corresponds to comparing model B and modelH, as shown in Table 5.1.It is known that the evolution of clusters in the threshold voter model is characterisedby motion by mean curvature [CFL09, DC07]. As a result, a random or disordered initialopinion distribution is rapidly arranged in the form of clusters or blobs first, and thesubsequent dynamics is governed by boundary forces akin to surface tension. That is,convex shaped regions of any arbitrarily shaped opinion clusters will get encroachedupon by the opposing opinion. Spatially, the introduction of jumps to this model thenplays the role of disrupting the clustering sporadically. This can be seen in Figure 4.9(a).264.3. Jump Threshold Voter ModelThe jump threshold voter model is similar, for the fixation probability and the consen-sus time, to the threshold voter model at low initial minority densities. As noted before,the differences between the threshold voter model and the jump threshold voter modelbecome prominent only at initial minority densities greater than the critical density 0.4.The fixation probability for the threshold voter model exhibits behaviour similar to astep function (Figure 4.8), where consensus is almost always reached on the opinion inmajority at the start of the process. The threshold voter model thus amplifies the ad-vantage held by a particular opinion type due to a higher initial density. Comparing thefixation probability in this case to that for the jump threshold voter model, we noticethat the probabilities become less extreme at initial minority densities greater than thecritical density. Therefore, at initial minority densities greater than the critical density,the jumps help in moderating the advantage amplification inherent to the thresholdvoter model. Next we assess the effect of the jumps on the consensus time. The jumpshave an effect of increasing variability in the opinion density. This effect appears morepronounced at initial minority densities higher than the critical density, as we get aquicker consensus in that regime.Overall, for initial minority densities lower than the critical density, the clusteringeffect is too strong to be disrupted by the jumps. However, once the initial minoritydensity exceeds the critical density, the disrupting effects of the jumps begin to counterthe clustering effect inherent to the threshold voter model.Jumps expedite the threshold voter model less than the voter model. We next comparethe consensus time reduction caused by the jumps in threshold voter model with thatin the voter model. We notice, based on comparing Figure 4.3 and Figure 4.8, that thejumps reduce the voter model consensus time significantly more than the threshold votermodel consensus time. This suggests that the threshold voter model is comparativelymore robust to an external influence. This is again attributable to motion by meancurvature in the threshold voter model. Consider (WLOG) the extreme case, for bothjump voter model and jump threshold voter model, where the jump causes a 0 → 1flip at a node in the interior of a 0 cluster. In the jump voter model, there is a non-zero probability that the 1 in the interior can in turn cause a 0 → 1 flip at one ofits neighbouring nodes. But in the threshold voter model, the probability that thisopinion 1 node can flip any of its opinion 0 neighbours is zero. Therefore, the clusterpatterns of the voter model would tend to more vulnerable to jumps as compared tothose of the threshold voter model. This serves as a possible explanation for the higherrobustness of the threshold voter model to jumps with regards to the consensus time.This phenomenon can also be observed in Figure 4.9: Notice the similarity in the clusterpattern in the second and fourth panels of Figure 4.9(a) despite the occurence of a jumpin the third panel. On the contrary, no such trend is seen in Figure 4.9(b).27Chapter 5ConclusionIn this work, I have developed extensions of the classical voter model, the biasedvoter model, and the threshold voter model, in order to incorporate an external influ-ence that causes many opinions to shift in the same direction simultaneously. This typeof influence could occur, for example, through mass media. I approximated the exten-sions of the classical voter model and the biased voter model, i.e., the jump voter modeland the jump biased voter model, by means of jump diffusion processes. This approachallowed me to analytically determine the probability of reaching consensus on opinion 1(fixation probability), and the consensus time. For the extension of the threshold votermodel, the jump threshold voter model, I chiefly relied on simulations to determine fixa-tion probability and consensus time. Most existing literature on opinion dynamics onlystudies opinion evolution under influences internal to the system. This work provides asystematic study of opinion dynamics under an additional external influence. A sum-mary of all the models with their diffusion approximations (if any), along with relevantexisting models is shown in Table 5.1.The noisy voter model is another contemporary variant of the voter model, and it isinstructive to compare the same with the jump voter model as both the models introducean additional source of opinion flipping. In the (discrete-time) noisy voter model, anagent is chosen, following which, with some probability the agent adopts one of itsneighbours’ opinion, and with the remaining probability flips its opinion irrespectiveof its neighbourhood. In the noisy voter model, a spontaneous 0 → 1 or 1 → 0 fliphappens only at one site at a given time. Our jump voter model can be thought of asa “noisier” voter model where the spontaneous flips happen simultaneously at multiplesites. Another subtle difference is the connection between the opinion density at a giventime, and the spontaneous flip that occurs at that time. In a noisy voter model, aspontaneous 0 → 1 flip, for example, is positively correlated with the density of 0s atthat point of time. This correlation is because agents with opinion 0 will have a relativelyhigher probability of getting sampled. However, in our jump voter model, there is noexplicit or implicit connection between the opinion density and the opinion flipping dueto the jumps. Therefore, our model may be used over the noisy voter model in scenarioswhere a larger external influence (or noise) is required, and where the external influenceis completely independent of the state of the system.The diffusion approximations for the voter model and the biased voter model do notdepend on the degree of the regular graph, and we note that this property is retained evenfor the jump voter model and the jump biased voter model. This means that the fixationprobability and consensus time results for the jump voter model and the jump biased28Chapter 5. ConclusionTable 5.1: A global overview of the models formulated in this work, along with relevant existingmodels. Also shown are the approximations used to derive theoretical results, wherever possible.Note that the jump version models (column 2) are defined on a regular graph, meaning that thedegree k is a parameter among those models. Their approximate models (column 3), derivedusing diffusion approximation and mean field approximation, have no degree dependence. In thecomparison graphs (column 4), black dots are obtained from the jump versions of the modelsand the black curves are obtained from their corresponding approximate models.ExistingModelFormulated JumpVersion ModelApproximateModelComparison of JumpModel with its Ap-proximate ModelVoter Model(Model A)Jump Voter Model(Model B) (On anyregular graph)Jump Voter Diffu-sion(Model C) (No de-gree dependence)(See Figure 4.1)Biased VoterModel(Model D)Jump Biased VoterModel(Model E) (On anyregular graph)Jump Biased VoterDiffusion(Model F) (No de-gree dependence)(See Figure 4.6)ThresholdVoter Model(Model G)Jump ThresholdVoter Model(Model H)- -29Chapter 5. Conclusionvoter model (discussed in Chapter 4) will remain true for all regular graphs ranging froma complete graph (where k = N − 1) to a lattice (where k = 4) to even a cycle graph(where k = 2). Thus, in a society where everyone has the same neighbourhood size,fixation probability and consensus time are independent of the neighbourhood size aslong as agents update their opinions by randomly sampling from their neighbourhood.Another key observation is that the jumps expedite consensus in both the votermodel and the threshold voter model, but more so in the voter model. Thus, in asociety where agents update their opinion if and only if the pressure from their neigh-bourhood is enough (based on a threshold parameter), external influence has a lessereffect on consensus time than in a society where agents update their opinions by ran-domly sampling from their neighbourhood. In a real-world system, agents most likelyupdate their opinions by a combination of the two update rules, i.e., opposing pressurefrom the neighbourhood and random sampling. Since the jumps expedite consensus inboth the models, we may expect them to also have a similar effect in a combination ofthe two models which might better reflect reality.This work opens up multiple interesting directions that may be further explored.The domain of network science is currently expanding very rapidly, and one naturalextension of our work is to study our models on a heterogeneous graph structure suchas a scale-free network. Such work could lead to pragmatic insights since scale-freenetworks have been shown to be ubiquitous in various real-world social systems [BA99].For the jump threshold voter model, this work mainly relied on simulation analyses,which may be extended further to rigorous mathematical analyses. 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