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Role of social environment and social clustering in spread of opinions in co-evolving networks

Nishant Malik∗ and Peter J. MuchaDepartment of Mathematics, CB 3250, University of North Carolina - Chapel Hill, NC 27599, USA

Taking a pragmatic approach to the processes involved in thephenomena of collective opinion formation, weinvestigate two specific modifications to the co-evolving network voter model of opinion formation, studied byHolme and Newman [1]. First, we replace the rewiring probability parameter by a distribution of probability ofaccepting or rejecting opinions between individuals, accounting for the asymmetric influences in relationshipsamong individuals in a social group. Second, we modify the rewiring step by a path-length-based preferencefor rewiring that reinforces local clustering. We have investigated the influences of these modifications on theoutcomes of the simulations of this model. We found that varying the shape of the distribution of probability ofaccepting or rejecting opinions can lead to the emergence oftwo qualitatively distinct final states, one havingseveral isolated connected components each in internal consensus leading to the existence of diverse set ofopinions and the other having one single dominant connectedcomponent with each node within it having thesame opinion. Furthermore, and more importantly, we found that the initial clustering in network can alsoinduce similar transitions. Our investigation also bringsforward that these transitions are governed by a weakand complex dependence on system size. We found that the networks in the final states of the model have richstructural properties including the small world property for some parameter regimes.

I. INTRODUCTION

It has been widely reported in the media that online so-cial networks like Facebook, Twitter, Blackberry messenger,etc. played a key role in recent events in the world politicalsphere such as the Arab spring and London riots of 2011 [2–6]. Meanwhile, there has also been increased interest in thequantitative and analytical analysis of the mechanisms anddynamics of the spread of social contagions such as rumorsand opinions on complex networks [6–15]. In such studies,individuals in the society are represented by nodes with edgesindicating relationships between them, and then techniquesfrom statistical and nonlinear science are employed to analyzeplausible models of the dynamics of spread of social conta-gions on a network [1, 16–26].

We propose a variation of the simplest coevolving networkvoter model of opinion formation, studied by Holme and New-man [1]. In this model an edge is re-wired to connect twonodes having the same opinion, or the opinion of an individualis changed to agree with the the opinion of one of its neigh-bors based on a parameter, named the rewiring probability.We add two more simple mechanisms to this model, inspiredby a pragmatic approach to the modeling of asymmetric in-fluences and tendencies to local clustering in the phenomenonof collective opinion formation in a social group so, that wecan investigate a broader array of complex behaviors that canbe induced by these modifications to the co-evolving votermodel. For convenience of the exposition herein, we will re-fer to these additional mechanisms as : (1)Social Environmentand (2)Social Clustering. Below we describe their meaningand significance in the processes of opinion formation.

Acceptance and rejection of somebody else’s opinions orchoices by an individual depends on multitude of factors in-cluding the strength of relationship between the concernedin-dividuals and thesocial environmentthey live in. A prevail-

∗Electronic address: nmalik@email.unc.edu

ing social environment(as defined for e.g. in [27]) not onlyalters relationships among individuals but can also affecttheiropinions on different issues in a fundamental way. A highlydivisive society may be an outcome of inflexibilities in re-lationships that exist between individuals who resist accept-ing or sharing each others’ opinions, choices or views. Andthese inflexibilities themselves could be due to the prevailingnegative social environment. Other situations could involvepositive social environment leading to flexible relationshipsamong individuals hence leading to less resistance among in-dividuals to the acceptance of each others’ opinions, choicesor views. In modern times, media and advertising also play asignificant role in altering the social environment and in con-structing consent around certain opinions or choices [28].

We propose to incorporate the effect of the social environ-ment on the model of opinion formation on co-evolving net-works by a distribution of probability of accepting or rejectingopinions between individuals. The distribution for socialen-vironment replaces the constant rewiring probability thathasbeen used before in other studies on voter model with co-evolving networks [1, 16–18]. Such description of the socialenvironment becomes more plausible if we note the fact thatrelationships among individuals in a social group are inher-ently heterogeneous and asymmetric. For simplicity, we haveassumed that the social environment is modulated by externalsocial, economic or political factors and its form remains thesame over the temporal evolution of the model.

Another important aspect that has not yet been sufficientlyanalyzed in the models of opinion formation on co-evolvingnetworks has been the role of local clustering of edges inthe network and other similar preferences for new links to beformed between nodes that are already near each other in thenetwork. Indeed, in most models studied to date, the networkdistance has been considered to be independent of the pro-cesses involved in the spread of opinions. In the present modelwe have attempted to explore the complex consequences of asimple introduction of such effects, by network distances andclustering in the network, with the processes of opinion for-mation. Specifically, we replace the random rewiring step of

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other models with a step that prefers rewiring to nodes/actorswho are both already closer within the network and who havehigher probability of accepting new opinions. This way theclustering of the evolving network in the model does not van-ish in the large-network limit (as in other previous models).Clustering is a fundamental property of most network rep-resentations of social contexts, i.e., friends of friends have ahigher likelihood (relative to the rest of the network) of alsobeing friends [13, 14, 29]. However, rewiring rules for co-evolving network models that do not reinforce clustering (asin, e.g., [1, 16]) can randomize away any initial clustering,greatly simplifying the associated opinion dynamics.

The explicit incorporation of model processes forsocialenvironmentandsocial clusteringprovides a simple simula-tion for the coupled effects of opinions with clustering andhomophily, the tendency of individuals to connect with indi-viduals having similar characteristics [30].

II. DESCRIPTION OF THE MODEL

Let G(N,E) be a network ofN nodes andE edges witha predefined topology. Let{Oi} represent a set ofO num-ber of opinions uniformly distributed over theN nodes ofG(N,E) initially. Let pij be the probability of some nodej accepting an opinion from nodei. The distributionP (pij)describes thesocial environment. If an edge exists betweennodei andj then we sayEij = 1. An edge connecting twonodes with different opinions is called adiscordant edge(i.e.,whereEij = 1 butOi 6= Oj ). The total number of discordantedges inG is represented byE− andE = E+ + E− whereE+ stands for harmonious edges (i.e., edges connecting nodeswith the same opinion).

Different individuals have different probabilities of accep-tance of others’ opinions, which is here taken to be indepen-dent of the existence of a link between the individuals. Severalfactors ranging from socio-cultural affinity to the prevailingpolitical and economic situation can influence these probabil-ities. To take these features into account we have used a distri-butionP (pij) for rewiring probabilities rather than a constant.Wherepij is the probability ofjth node accepting the opinionof ith node.

We callP (pij) thesocial environment function, accountingfor the heterogeneous and asymmetric relationships amongindividuals. For the purposes of exploring a variety of set-tings, we have considered two different kinds of power lawsfor thesocial environment. We setP (pij) = pαij to represent aflexiblesocial environment, i.e., individuals are able to acceptothers’ opinion readily. Alternatively, we considerP (pij) =1−pαij to represent aninflexiblesocial environment, i.e., indi-viduals do not accept others’ opinion readily and hence morechurning happens in the society (see Fig. 1(b)). While therehas been some empirical evidence to suggest that electionresults in multi-party democracies have power law distribu-tion of votes among candidates from different parties [31–33],however our use of a power law distribution in this specificcontext is driven only by its computational simplicity to sim-ulate the qualitative kinds of social environment mentioned

Algorithm 1 A hybrid voter model of opinion formation on aco-evolving network with clustering and distributed levels of

influence.1: Generate a graphG of given topology.2: Generate a given distribution forpij i.e. P (pij).3: Populate nodes withO number of uniformly distributed opinions{Oi}.

4: CalculateE−.5: while E− 6= 0 do6: Randomly choose a discordant edgeEij .7: Generate a random numberξ between0 and18: if ξ < pij then9: Oj ← Oi

10: CalculateE−

11: else:12: Remove the link betweeni andj i.e., setEij = 0.13: Find a setN ′={j}j 6=i ∩ {k}.

⊲ Where{j}j 6=i is a set containing all the nodes such that eachelement of it haspij ≥ ξ and{k} contains all the nodes withshortest path fromi (excluding the nearest neighbours).

14: ifN ′ 6= ∅ then15: Connecti randomly to any nodel ∈ N ′

16: Ol ← Oi

17: else:18: Connecti randomly to any node j s.t.Oj = Oi

19: end if20: CalculateE−

21: end if22: end while

0 0.5 1pij

0.05

0.1

P(p

ij)

P(pij) =1−pαij(a)

0.5 3.0 6.0α

0 0.5 1pij

P(pij) =pαij(b)

FIG. 1: (Color online) Different types ofsocial environment functionP (pij), wherepij is the probability ofjth node accepting the opinionof ith node. (a) “inflexible”, when we setP (pij) = 1 − pαij , notethat in this case more links will have lower probabilities ofacceptingopinions and (b) “flexible”, when we setP (pij) = pαij , note thatin this case more links will have higher probabilities of acceptingopinions.

above [34, 35]. Other distributions such as exponential andextreme value distributions should also suffice to reproducesimilar features.

Steps 13-16 in Algorithm 1 ensure that rewiring connec-tions are mostly made according tosocial clusteringi.e., anode has higher probability of connecting to a person who iseither a friend of a friend or, if no such connections are avail-able, connecting to a person at the shortest possible distanceidentified in the network. The setN ′ in the model (see Algo-

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FIG. 2: (Color online) A visual representation of the phenomenaof formation of two distinct consensus states for two different so-cial environment. Starting with an initial Watts-Strogatznetwork(N = 1000, 〈k〉 = 4 andC = 0.1), we demonstrate the possibilitiesto end into two qualitatively different states. (a)P (pij) = 1 − pαijandα = 6 creates an “inflexible” social environment. We observedisintegration of the network into small connected components witheach in internal consensus and having its own opinion, creating com-ponents withcontrarian positions i.e.,segregated consensusoccursin the network. (b)P (pij) = pαij andα = 6 creates a “flexible”social environment. We observe formation of dominant connectedcomponent in the final consensus state, having a size comparable tothe initial network also, large number of opinions get extinct. Werefer this kind of final state as thehegemonic consensus.

rithm 1) consists of nodes/individuals who are close to someparticular nodei, both in terms of path length between themin the network and also they have higher probabilities of ac-cepting the opinion of the nodei. Hence, we call the nodeswithin the setN ′ to be socially close to the nodei. In casenodei is not able to find such individuals then it connects uni-formly at random to somebody else holding the same opinionto avoid complete social isolation. Here, we aim to study therole of clustering of the network in altering the opinion spaceand network properties of the final end state. In so doing,our emphasis will be on transitions that occur in the networkstructure (notably, sizes and clustering of connected compo-nents) rather then just the space of opinions. We will refer tothe ratio of number of opinions to nodes i.e.,O/N asdiver-sity. We have fixed the average degree〈k〉 = 4 and numberof opinionsO = 100 for the simulations, if not mentionedotherwise. We have additionally investigated other numbersof opinions and average degree to confirm the robust nature ofthe qualitative properties described in this paper. The numberof edges has been kept conserved throughout the dynamics;therefore at any timet, E(t) = 〈k〉N2 . Let the evolution ofthe system start att = t◦ with E−(t◦) the initial number ofdiscordant edges. The evolution of the system stops at the ear-liest such thatE−(tf ) = 0, i.e., the final state of this modelhas no discordant edges left in the system.

There are several levels of plausible complexity for thismodel which could provide some new insights into the co-evolving dynamics of networks, but at the price of making itanalytically harder to track. Indeed, even the limited analyt-ical tractability of graph fission in a two-opinion co-evolvingvoter model presented in [16] is undoubtedly aided by therewiring rule considered there randomizing away all non-trivial clustering. In light of the complications introduced bythe path-length influenced rewiring considered here, we haveattempted to analyze this model computationally in a compre-hensive way.

A. Basic features of the model

In this section we give a brief introduction to the basic fea-tures of this model. Firstly, we obtain two qualitatively dis-tinct final states as we vary the social environment from flex-ible to inflexible. For a flexible social environment, if we setP (pij) = pαij with α = 6.0 then in the final state of the model,we observe formation of one single large connected compo-nent with each node having the same opinion and its size iscomparable to the initial network. We call this kind of finalstate as thehegemonic consensus, because of the emergenceof one single hegemonic opinion. In the case of inflexiblesocial environment, simulated by settingP (pij) = 1 − pαijwith α = 6.0 we observe that initial network disintegrate intosmaller isolated connected components where every node ineach of these components hold the same opinion, i.e., eachcomponent is in the state of internal consensus. We will referto this kind of final state as thesegregated consensus, as thisfeature is qualitatively similar to thesegregationof individu-als in a society. A lattice based classical model of this socialphenomena was given by Thomas Schelling [20], where heshowed segregation of two groups of populations (’red’ and’white’) who move over a check board following some simplerules. Several analytical and simulation results have beenob-tained following Schelling’s model on networks as well as onco-evolving networks but not in context to the processes in-volved in collective opinion formation [36–38]. Holme andNewman [1], observe some transitions qualitatively similarto that mentioned above by changing their constant rewiringprobability parameter.

A visualization for the above observation forN = 1000nodes withO = 100 is shown in Fig. 2. The drastic transitionbetween thehegemonic consensusandsegregated consensusin the final states of the systems seems to occur somewherebetween the extreme flexible to inflexible social environment.Intuitively, it is perhaps not surprising that changing thedis-tribution of the social environment induces a transition simi-lar to that studied by Holme and Newman [1], insofar as thechange in the distribution changes the overall average level ofrewiring. Nevertheless, a priori we have no reason to expectthat change in the form of the distribution of probabilitiesofaccepting or rejecting of others’ opinions should have sim-ilar effects as the changes to the single rewiring probabilityparameter employed by Holme and Newman [1]. Also, thedetailed structural properties of the network in thehegemonic

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FIG. 3: (Color online) The effect of variation of social environmenton a network ofN = 1000 nodes withO = 100 opinions initiallypresent. The starting network is an Erdos-Renyi random network(i.e., clustering∼ 1/N ). Cif is the clustering coefficient of theithcomponent in the final consensus state. (a) Size of a marker ispro-portional to the sizes ofsi, given by the fraction of nodes in theithconnected component. The thick bold line in the middle separates thetwo types of social environment. On the right we consider flexiblesocial environment and we observe single large connected compo-nent with its size increasing with increasingα. On the left considerinflexible social environment and we observe decreasing of the sizeof largest connected component with increasingα, finally leading toits disintegration into several components of comparable sizes. (b)Sizes of two of the biggest connected components,s1 is the size (asa fraction of the nodes in the network) of the largest connected com-ponent ands2 is the size (as a fraction of the nodes in the network) ofthe second largest connected component. Simulations were carriedover100 realizations of the network and opinion distribution. Sizesof the components are estimated as the mean over these realizations.Error bars give the standard deviation of these sizes over differentrealizations.

consensusandsegregated consensusin the final state are ex-pected to be much richer as shown and discussed below insome detail. In Fig. 3 we observe the effect of varying thesocial environment, wheresi is the size (fraction of nodes)of the ith component in the final consensus state withi = 1being the largest component. A further analysis of the phasetransition involved in emergence of these two distinct statesin this system has been attempted in detail in the followingsection, as one of the two central themes of this paper.

The giant consensus community occurring in the Holmeand Newman model [1] would appear to be structurally sim-ilar to networks obtained under a configuration model withthe observed final state degree distribution. In contrast, asobserved in Fig. 4(a) the largest connected component inthehegemonic consensushas small world properties (averagepath lengths comparable to random network and high clus-

FIG. 4: (Color online) Properties of the largest connected compo-nent inhegemonic consensus: The final state reached for the flex-ible environmentP (pij) = pij with α = 6.0 when starting withan Erdos-Renyi network. (a) Different markers representdifferentinitial network sizes (see the legend). The initial networkis G0, itsinitial clustering is close to zero ands1 is the size (as a fraction of thenodes in the network) of the largest connected component in the finalstate. We observe thats1 has a significantly higher clustering coeffi-cient (0.2). Whereas it has comparable small path length to the initialErdos-Renyi networkG0, implying thats1 has small world features.Also,s1 has in general higherkmax (maximum degree) and its size iscomparable toG0. (b) Shows the the cumulative degree distributionC(k) of the initial networkG0 (dashed lines) ands1 (markers).s1does have nodes with higher degrees. In its tail, the cumulative de-gree distribution ofs1 appears to approximately follow a power lawas shown by solid grey line of exponent−8.

tering coefficients) and it also consists of nodes with highernumber of connections as apparent from the change in cumu-lative degree distribution as shown in Fig. 4(b). These fea-tures are closer to organized political or religious movements,which usually have a hierarchy of leadership and high clus-tering, thus we have pointedly not referred to this structure asa mob, because of the observed hierarchy of connectivity in-volved here. We have not observed variation indiversityO/Nto bring about any significant change to the above discussed

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FIG. 5: (Color online) The evolution of different variablesin the sys-tem with decreasing number of discordant edges. Each variable isplotted at time step when that number of discordant edges,E− waspresent for the last time in the system. The black line and panel (b)corresponds to simulations starting at the highest possible clusteringcoefficientCmax, whereas red dotted line and panel (a) correspondsto simulations starting at the negligible clustering coefficient (ran-dom network). In (a) and (b) each color corresponds to one of theopinions, width of each color gives the number nodes occupying thatopinion. In (a) note the wide width of cyan color at the end, this rep-resents the formation ofhegemonic consensus(one large connectedcomponent of size comparable to initial network and with each nodebeing at consensus with every other node.) In (b) we does not observethis transition, only difference in this simulation is the large initialclustering coefficient. (c)s1 is the size of largest connected compo-nent. Observe the abrupt drop ins1 in the case of the black line, thisindicates transition to the disintegration of network intosmaller com-ponents i.e.,segregated consensus. In contrast we do not observe anysuch transition for the red dotted line which corresponds tothe for-mation ofhegemonic consensus. (d) 〈∆t〉 is the average number ofiterations the system takes to the removal of single discordant edge.It shows a substantial increase for the black curve at the end. (e)C iscorresponding evolution of the clustering coefficient.

basic properties, while varying the values ofO from 2 to 100.

Another crucial aspect to consider in this model is the roleof initial network topology in transitions betweenhegemonicconsensusandsegregated consensusas the two distinct finalstates. Does the variation of the initial clustering coefficientchange the final state? This question have not been consideredin the previous studies of voter model on co-evolving net-works, as the previously introduced models have not treated

FIG. 6: (Color online) The effect of variation of initial clusteringC◦ on a network of N = 1000 nodes with O = 100 opinions initiallypresent when social environment is flexible withP (pij) = pαij withα = 6. Increasing of initial clusteringC◦, leading to the transitionsi.e., disintegration of network in consensus state into smaller con-nected component (segregated consensus) contrary to the expectedhegemonic consensusfor initially unclustered networks in flexiblesocial environment. (a) Size of a marker is proportional to the sizeof ith connected component in the final consensus state. Observethe disintegration of the network into several connected componentsfor higher values of initial clusteringC◦. (b) s1 is the size of largestconnected component ands2 is the size of the second largest con-nected component. Simulations were carried over100 realizationsof the network and opinion distribution. Sizes of the components areestimated as the mean over these realizations. Error bars gives thestandard deviation of these sizes over different realizations. Cif isthe clustering coefficient of theith component in the final consensusstate. Observe the higher values for connected components after thedisintegration into smaller components.

clustering as a consequence of those models, even thoughclustering is one of the essential characteristics of social net-works [13, 14, 29]. In the model considered here, the for-mation of ahegemonic consensusstate apparently does nottake place in networks with high initial clustering coefficient.To understand this feature we investigate the evolution of theclustering in this model.

Let theclustering coefficientof the network be representedby the symbolC, defined as three times the ratio of the numberof loops of length three in the network to the number of con-nected triples of nodes, also known astransitivity [39]. Sym-bolsC◦ andCf are used here for the initial clustering at startof the simulation and final clustering at the end of the simu-lation, respectively. In the Watts-Strogatz model, for examplethe maximum possible initial clusteringCmax correspondingto the ring topology, isCmax = 3(〈k〉−2)

4(〈k〉−1) . Therefore, with〈k〉 = 4, we would haveCmax = 0.5 (see e.g. [40]). TheCmax value is also the upper limit forCf . In Fig. 5 we plot theevolution of different variables of the model from a single sim-

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FIG. 7: (Color online) Variation ofs1 (size of largest connected component) ands2 (size of second largest connected component) withα.Here social environment is set to beinflexible, i.e., P (pij) = 1 − pαij . Different shapes and colors of the markers represent networks ofdifferent sizes (see legend in (g)). In (a) we observe multiple transitions ins1, first happens atα = N0.05 where all data point collapse ontothe same curve (see inset). A second transition is observed at α = 4.25 (dashed grey vertical line), where a best fit to the data changes froma polynomial to power law (see (b) and (c), whereǫ gives the error between the between fitted function and the data points.). This secondtransition also appears in an even more visually apparent form in (d), observe the abrupt decreasing ofs2 afterα = 4.25 (dashed grey verticalline). Again best fit to the data changes from a polynomial to power law (see (e) and (f)). In the figure (g) we plot Shanon’s entropyH of the10 largest connected components versusα, in this figure too, we observeH tends to saturate atα = 4.25 (dashed grey vertical line) and startto decrease after linear increase.

ulation as discordant edges are removed. The social environ-ment was set to be flexibleP (pij) = pαij with α = 6.0, i.e., theparametric regime where we expect formation of ahegemonicconsensusstate for initially unclustered networks. The size ofthe initial network wasN = 1000. When we setC◦ = 0, theopinion space does undergo a transition as expected and wesee one opinion dominating (see Fig. 5 (a)). Also to be notedat the same time there is no transition in the size of the largestconnected component (see red dotted line in Fig. 5 (c)). Forthe black curve in Fig. 5 we have setC◦ = Cmax and we ob-serve a counter intuitive and unexpected transition viz., thatthe largest connected component starts to disintegrate andbe-come smaller in size (see Fig. 5(c)) and also in opinion spacewe do not observe emergence of a single dominant opinion(see Fig. 5(b)). We also observe in the lowest panel of Fig. 5thatC saturates toCf before reaching the consensus. This isa special feature of this model and provides this opportunityto study the evolution of a clustered network topology withopinion formation. For the caseP (pij) = pαij with α = 6,Cf appears to be well approximated by a linear function ofC◦. We also see in the panel (d) of Fig. 5 that right beforethe consensus states emerges, the system start to slow down.That is, more iterations are required to decrease the numberofdiscordant edges, possibly indicative of some form of criticalslowing of the system assegregationis reached. This featureis not so apparent in case of red dotted curve, implying thatprocesses involved in formation ofhegemonic consensusdonot involve critical slowing of the system. In Fig. 6 we showthe disintegration of the network into smaller components aswe increase the initial clustering coefficient from0 to Cmax.The above discussion only briefly illustrates some of the fea-tures in the evolution of clustering in the model. Below, wewould present a systematic analysis of this transition.

III. PHASE TRANSITIONS

A. Role of social environment in transitions

As discussed above this model shows transition to two dis-tinct final states, forflexiblesocial environment we have ob-served that asα is increased, the largest connected compo-nent’s size approaches that of the whole network,s1 → 1(see Fig 3) and each node within the component hold thesame opinion, and for theinflexiblesocial environment casewe have disintegration of the network into several smallersized connected components, where nodes within each of thecomponents holds the same opinion. As we move from in-flexible to flexible social environment fewer and fewer ini-tial opinions survive, with the most extreme case being whereonly one dominant opinion survives with formation of ahege-monic consensus. Here we will attempt to infer from numer-ical simulations whether these transitions have a finite sizeeffect [41]. The complexities involved in this model makesanalytical analysis hard but it is possible to obtain a variety ofdetails using numerical simulations.

From Fig. 3 we observe that somewhere when the param-eters of the model are in theinflexible social environmentregime there is emergence of smaller sized connected com-ponents. Hence, we will examine transition within parametersetting ofinflexiblesocial environment i.e.,P (pij) = 1− pαij .The initial network for all the simulation below is Erdos-Renyi random network. In Fig. 7(a) we observe rather mul-tiple transitions in the system whenα is varied from0.5 to6.0 for the inflexible social environment. The first transitionis visible in the size ofs1, where a weak dependence on thesize of the system seems to emerge (see inset Fig. 7(a)). Allthe curves with different system sizes collapse onto one singlecurve when a small factorN−0.05 is multiplied toα that is, itappears that this transition point has dependence on the sizeof the system and it would change asα = N0.05 (see vertical

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lines in the inset of 7(a)) and this transition point would moveto infinity in the thermodynamic limit.

A second transition occurs atα = 4.25 where the best fitto the data points turns from a polynomial fit to power law fit(see Fig. 7(b-c) and (e-f)). For fitting functions we have useda least squares routine provided in SciPy’s optimize package,which uses MINPACK’s lmdif and lmder algorithms [42].This transition is more apparent in Fig. 7(d) for the size ofthe second largest connected component i.e.,s2. In Fig. 7(g)we have plotted the Shannon entropy over the sizes of the10

largest components withH =∑i=10

i=1 si ln(si). Consideringonly 10 largest components for this calculation is a reason-able approximation to the total Shannon entropy of the sizedistribution in most cases, given the rapid decrease in the tailof the size distribution. In this figure as well the transition atα = 4.25 is visibly very much apparent asH tends to saturateand start to decrease after linear increase. The polynomialfitin Fig. 7 (a)) has the following form :

s1 =aα2 + bα+ c if α < 4.25

s1 ∼f(N)α−2.4±0.02 if α ≥ 4.25 (1)

where a ≈ −0.029, b ≈ N0.052±0.001 − 1.4 and c ≈

N−0.36 log(N) andf(N) is function dependent onN . A sim-ilar analysis fors2 also yields a polynomial fit :

s2 =aα−2.1 + bα2.1 + c if α < 4.25

s2 ∼f(N)α1.42±0.12 if α ≥ 4.25 (2)

wherea ∼ N0.0027 − 1.02, b ≈ −2.68−6N − 1.54 andc ∼ N1.75 and againf(N) is function dependent onN . Thisanalysis brings out a highly complex dependence ofs1 ands2on system size for the transition occurring atα = 4.25. Butas indicated by the error to polynomial fit and power law fitsin Fig. 7(b-c) and (e-f), a polynomial fit becomes systemati-cally less erroneous asN is increased. Which means for largeN these multiple transitions might coalesce into one singlecontinuous transition.

B. Role of network structure in transitions

Social networks are generally known to have higher cluster-ing [43]. The initial definition of global and local clusteringwas in the context of social ties [13, 14, 29, 44]. In previ-ously studied coevolving voter models with random rewiringthe clustering tends to decay away to that of independentlydistributed edges (∼ 1/N ) as the system evolves with time[1, 16–18]. Whereas in the present model we observe thata net critical value is sustained throughout its evolution andnever dropping to near zero (see Fig. 5).

Such a model provides an opportunity to explore the influ-ence of variation in the clustering coefficient on transitionsbetween the formation of ahegemonic consensusandsegre-gated consensus. We are here mainly interested in knowingwhetherC◦, the initial clustering, can affect the formationof the hegemonic consensus. We know from the discussion

FIG. 8: Variation in the size of largest connected components1 withthe initial clustering coefficient. The social environmentwas fixedto be flexible i.e., P (pij) = pαij with α = 6. When log(N) ismultiplied to C◦ the data for different system sizes collapses ontosame curve. The inset curve shows the fits and the vertical lines are

1log(N)

, indicating the transition points.

above that if we setP (pij) = pαij andα = 6 (flexible so-cial environment), we will get thehegemonic consensusto bethe final state, where the size of the largest connected com-ponents1 ∼ 1 in the consensus state for an initial randomnetwork of independently distributed edges (or network withnegligible clustering coefficient). After settingP (pij) = pαijwith α = 6.0 we vary the initial clusteringC◦ of the system,employing a Watts-Strogatz model for the initial network. Weobserve in inset of Fig. 8 that with increasing initial clustering,the largest connected component does tend to disintegrate intosmaller size. For higherC◦, rather then having only one dom-inant connected component of sizes1 ∼ 1, we get smallersized connected components, i.e.,segregated consensusoc-curs in place ofhegemonic consensus. So, even in the caseof a highly flexible social environment i.e.,P (pij) = pαij andα = 6, we can still get disintegration and no single dominantopinion, if the initial clustering of the network is high enough.

To get an estimate on the valuesC◦, where we could startobserving the disintegration in the consensus state we furtheranalyze the results obtained in Fig. 8. We observe that if wemultiply a factorlog(N) to theC◦ then all the data collapsesonto one curve (see Fig. 8) implying that transition seems tobe occurring atC◦ = 1

log(N) . If we plot the transition points1

log(N) as done in the inset of Fig. 8 by means of vertical lines,we do observe spontaneous drop off in the values ofs1 aroundthese transitions. The form of the function that can be fittedtothe data in Fig. 8 is as follows :

s1 ∼

{

1 if C◦ ≤ 1log(N)

aCα◦ exp(−λC◦) if C◦ > 1

log(N)

where, λ ∼ N−0.37±0.018, a ∼ N−0.95±0.07 and α ∼N−0.13±0.012. Though the above functional form might hasa complex dependence on the system sizes, the critical values

8

FIG. 9: (Color online) The sizes of different connected componentsin the consensus state for network ofN = 1500 nodes. (a) Sizes ofconnected components v. their ordered indices. The largestcompo-nent has index1 and indices are arranged in decreasing order of sizesof the components on abscissa. As initial clustering of the networkC◦ (color bar) is increased, there is emergence of smaller compo-nents of comparable sizes. (b)β′ are the values of the exponentsof the slopes fitted to the sizes of components in the final consensusstate v. indices at each value ofC◦ (thick red line in (a) is an examplefor the same forC◦ = 0.5). In (b) observe the decrease in the slopeand error bars for higher initial clusterings, indicating the formationof several components of comparable sizes.

C◦ are clearly varying as 1log(N) (see vertical lines in inset of

Fig. 8). Hence, this transition would exist in a finite networkand the critical value ofC◦ would become zero in the thermo-dynamic limit.

A further analysis of the connected components formed insegregated consensusshows that their sizes are power law dis-tributed. In Fig. 9(b) we have plotted the slope of the linefitted to the sizes of connected components and in Fig. 9(a)there is an illustration of the same forN = 1500 nodes. Aswe increaseC◦ not only the slope becomes smaller, but alsothe error bar to the fit is reducing indicating that sizes of theconnected components are becoming comparable asC◦ is in-creased, i.e., similar sized contrarian social groups orcultsare formed. We also note from Fig. 6 that these similar sizedcomponents generally have very high clustering.

IV. CONCLUSIONS

We considered a model for the opinion formation on co-evolving networks with two additional attributes: one is thesocial environment, which is modeled by a distribution ofsusceptibilities to opinion change, and the second one is apath-length-based preference for rewiring that reinforceso-cial clustering. The social clustering component intrinsicallylinks the topological evolution of the network with the pro-cesses involved in collective opinion formation and vice versa.We observed that two qualitatively distinct final states canemerge in this model, in one where we have formation ofhegemonic consensus, a dominating large connected compo-nent with each node having the same opinion. Importantly,this dominating large connected component also maintainsnontrivial local clustering. Such clustering contrasts with theproperties of previously studied models, as random rewiring

in them leads to non-clustered random networks as the finalconsensus state.

The other outcome that emerges under the parameter set-tings of inflexible social environments is the disintegration ofthe network and formation of small isolated components con-sisting of nodes holding the same opinion. It is a feature qual-itatively similar to thesegregationof individuals in a societydue to the internal conflicts and frustrations leading to forma-tion of dysfunctional social networks. Hence, we have namedthis final state assegregated consensus.

A fundamentally key aspect we studied was the role of clus-tering in the network in the process of opinion formation onco-evolving networks using the features of this model, wherethe clustering of the network is continually re-inforced bythepreference to rewire to nodes at smaller path length in thismodel. We observed that if the initial network has clusteringabove a critical value, then even in a flexible social environ-ment we getsegregated consensusas the final state. This iscontrary to what happens when we start with a network hav-ing negligible clustering (random network). Injection of thisadditional attribute to the model makes the dynamics of thissystem richer and more interesting but at the price of makingany analytical study much more difficult than for other mod-els, such as discussed in [16–23].

One can observe similar features in the process of opin-ion formation in society, for examplehegemonic consensuscan be analogous to situations in the states with multi-partydemocratic elections, where one party wins by a landslide.In contrast, some hung elections may be similar to aseg-regated consensus[45]. A similar situation can also occurwhen choices are made on a product among the many avail-able brands, with monopoly of one brand over the product be-ing thehegemonic consensusandsegregated consensusbeingwhen there is more even competition over a product betweendifferent brands [46].

Further analysis of the transitions in numerical simulationsof different sizes indicated complex and weak dependence onthe system size. In particular, it is possible that the mul-tiple transitions induced by variations in social environmentmight coalesce into one single continuous transition for largesystems. Meanwhile, the transition induced by clustering inthe initial network only exists for a finite system. Impor-tantly, because this latter transition occurs for initial clustering∼ 1/ log(N) (cf. independently distributed edges yield clus-tering∼ 1/N ), we note that one should be careful making anyclaims about the applicability of coevolving network modelsthat lack reinforcement of clustering to real-world network sit-uations that have non-trivial transitivity.

Acknowledgments

The authors thank Mason Porter and Feng Shi for helpfulsuggestions and comments. The project described was sup-ported by Award Number R21GM099493 from the NationalInstitute of General Medical Sciences. The content is solelythe responsibility of the authors and does not necessarily rep-resent the official views of the National Institute of General

9

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