Accelerating consensus on co-evolving networks: the effect of committed individuals
aa r X i v : . [ phy s i c s . s o c - ph ] O c t Accelerating consensus on co-evolving networks: the effect of committed individuals
P. Singh,
1, 2
S. Sreenivasan ∗ ,
1, 2, 3
B.K. Szymanski,
2, 3 and G. Korniss
1, 2 Dept. of Physics, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180, USA Social and Cognitive Networks Academic Research Center, Rensselaer Polytechnic Institute,110 8 th Street, Troy, NY 12180–3590, USA Dept. of Computer Science, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180, USA
Social networks are not static but rather constantly evolve in time. One of the elements thoughtto drive the evolution of social network structure is homophily - the need for individuals to connectwith others who are similar to them. In this paper, we study how the spread of a new opinion,idea, or behavior on such a homophily-driven social network is affected by the changing networkstructure. In particular, using simulations, we study a variant of the Axelrod model on a networkwith a homophilic rewiring rule imposed. First, we find that the presence of homophilic rewiringwithin the network, in general, impedes the reaching of consensus in opinion, as the time to reachconsensus diverges exponentially with network size N . We then investigate whether the introductionof committed individuals who are rigid in their opinion on a particular issue, can speed up theconvergence to consensus on that issue. We demonstrate that as committed agents are added,beyond a critical value of the committed fraction, the consensus time growth becomes logarithmicin network size N . Furthermore, we show that slight changes in the interaction rule can producestrikingly different results in the scaling behavior of T c . However, the benefit gained by introducingcommitted agents is qualitatively preserved across all the interaction rules we consider. PACS numbers: 87.23.Ge, 89.75.Fb
I. INTRODUCTION
A dynamical process occurring on a network can bestrongly influenced by the evolution of the network’sstructure itself. Furthermore, if the dynamical process onthe network directly affects the network’s structural evo-lution, a complex feedback process arises. In the contextof the spread of opinions, behaviors, or ideas on a socialnetwork, such an interplay between individual states andthe network’s structural evolution is expected on the ba-sis of the theory of homophily. Homophily, introduced byLazarsfeld and Merton [1, 2], describes the tendency ofindividuals to form social connections with those who aresimilar to them. Complementarily, the persistence of tiesis also thought to depend strongly on the similarity of theindividuals they connect [3, 4]. If the traits of individ-uals are unchanging, then we expect that the structureof the network will stabilize when each link connects apair of individuals who are sufficiently similar. However,if individuals influence one another to adopt new behav-iors, opinions, or ideas, and thereby affect each other’sattributes, then the mutual similarities between pairs ofindividuals can be thought of as continuously evolvingentities. Thus, one can envision the structure of a socialnetwork as being in a constant state of flux: links be-tween dissimilar individuals decay with time while newties between similar individuals form at some rate. Thiscontinuous death and birth of links is presumably bal-anced in such a way that on average, at any given time,the mean number of connections ascribed to any individ- ∗ Corresponding author: [email protected] ual is roughly constant, or at least, bounded from above[5].There have been few empirical studies which track thesimultaneous co-evolution of network structure and in-dividual behaviors. Notably, Lazer et al. have recentlystudied [6] how political views of students in a publicpolicy program, and the network structure of their inter-actions evolved over a two-semester observation period.The main finding of this study was that in the process ofmaking connections to other individuals, homophilic se-lection on the basis of political views was weak, while raceand religion-based selection was comparatively strong.Furthermore, the study found that an individual’s polit-ical views at the end of the observation period were sig-nificantly correlated with the mean affiliation of his/herneighborhood at the beginning of the observation period(controlling for the individual’s own initial views), an in-dication, possibly, of social influence.In contrast, models of networks where social opin-ions and network structure co-evolve have been stud-ied extensively in previous literature [7]. Benczik et al.[8, 9] studied a two-parameter voter model that couldbe tuned to study all cases between two extremes wherenodes preferentially interacted with other nodes holdingthe same opinion, or those holding the opposite opin-ion. They demonstrated that three outcomes were pos-sible depending on the parameter values - a consensusstate, a disordered state, or a frozen, polarized state.Holme et al. [10] studied a single-parameter model ofa co-evolving network and demonstrated the existenceof a non-equilibrium phase transition between a steadystate with diverse co-existing opinions and a consensusstate. Nardini et al. [11] studied a variant of the votermodel where an individual, with a certain probability,either severs a tie with a neighbor whose state differsfrom its own and forms a new tie with another node, orotherwise adopts the neighbor’s state. They showed howsmall changes to the interaction rules, such as the orderof choosing the interacting individuals, as well as the in-troduction of an intermediate state in the voter modelcan dramatically affect the probability of consensus aswell as consensus times. Finally, Vazquez et al. [12]studied a model where nodes are assigned attributes andundergo influence as per the Axelrod model [13], while ex-isting links are rewired with a probability proportional tothe dissimilarity between the nodes they connect. Theydemonstrated that there arise three phases characterizedby differences in the steady state network structure, asthe number of possible traits per attribute is varied. Amodel similar to the one presented in [12] was studiedin [14], where the authors showed how cultural diversitycan be stably maintained despite the presence of culturaldrift.The network model we consider in this paper is similarto the latter two studies [12, 14] in its use of an Axelrod-type measure of the similarity between individuals whichin turn dictates how social influence and link rewiringoccur. However, the central motivation of our work is tounderstand how a fast consensus to a particular attributecan be ensured on such evolving networks. In particular,we investigate how the introduction of committed agents - individuals who are selectively immune to influence ona given issue, and who hold the same opinion on thatissue - affects the evolution of opinions on the network aswell as network structure itself. The effect of committedindividuals, all holding the same opinion, has been pre-viously studied on structurally static networks [15–18].The key finding in these studies was the existence of acritical committed fraction below which the deterministicevolution equations admit a mixed steady state in addi-tion to the always present consensus steady state, with asaddle point separating the two. As a consequence, on fi-nite networks [17, 18], the time to attain consensus scalesexponentially with system size when the committed frac-tion is below the critical value. This is consistent with theknown scaling of transition times between deterministi-cally stable states in stochastic bistable systems [19]. Instark contrast, above the critical value of the committedfraction, the consensus process is essentially determin-istic with consensus times logarithmically dependent onnetwork size, and where the only steady-state solution tothe deterministic equations is the consensus state. Fur-ther analysis has also revealed [18, 20] that this “critical”point or threshold is a spinodal point associated with anunderlying first-order transition [21] in the phase diagramof the model.In this article, we investigate whether a similar fastconvergence to consensus can be engineered through com-mitted agents when the network structure is evolving inresponse to the spread of opinions. More specifically,we aim to understand whether the scaling behavior ofconsensus time with network size undergoes a significant change as the committed fraction within a co-evolvingnetwork is monotonically varied. The presence of com-mitted agents holding distinct opinions in the networkobviously prevents the system from attaining consensus,a situation previously studied in [22, 23] - in the presentwork, we do not consider this case.
II. THE MODEL
In our model, individuals are represented by nodes ona network and every node is assigned a set of F indepen-dent attributes that constitute the node’s state . Eachattribute can take one of q distinct traits, represented byintegers in [0 , q − F component vector. Initially, each attribute ofeach node is assigned one of the q values randomly. Thestructure of the network connecting the individuals is ini-tialized to an Erd˝os-R´enyi (ER) random network with agiven average degree h k i . Next, we define the rules gov-erning the evolution of individual states as well as thestructure of the network connecting them. At each timestep: a node i is selected at random, and one of its neigh-bors j is selected at random. We then compute the sim-ilarity between nodes i and j , where similarity is definedas the number of attributes for which i and j possess thesame trait. Then, for a randomly chosen attribute amongthe set of attributes on1. If the similarity is found to be equal to or above agiven threshold φ , node j adopts the trait possessedby node i for a randomly chosen attribute fromamong those for which they currently do not sharethe same trait. We refer to this as the influencestep .2. Otherwise, the link between i and j is severed andnode i randomly selects a node k in the networkfrom among those to which it is currently not con-nected, and forms a link to it. We refer to this asthe rewiring step .Our model clearly is a variant of the Axelrod model withthe following distinctions: in the Axelrod model, the in-fluence step occurs with probability proportional to thesimilarity between the nodes whereas in our model influ-ence occurs only when similarity exceeds a hard thresh-old. Secondly, if influence does not occur, then a rewiringstep necessarily does. This hard threshold also distin-guishes our model from that of [12], although as shownbelow the qualitative behavior of both models is simi-lar. It is worth pointing out that the rewiring step isdesigned such that the total number of links (or averagedegree) in the network is conserved. Also, the updaterule as defined above always assumes that the node cho-sen first is the influencer and the node chosen second isthe adopter . While most of the results described in thispaper are restricted to this order in choosing the influ-encer and adopter, we discuss alternate orders of selection(motivated by the results in [11]) in Section IV.We first examine the effect of the number of traits perattribute q in determining the steady state structure ofthe network, and confirm that our results are similar tothose found in[12, 24]. We fix F = 5 and set the simi-larity threshold to be φ = 3. In particular, three phasesdiffering in the steady state network structure are ob-served, as q is varied, as shown in Fig. 1. In the firstphase, observed for low values of q , the system evolves toa state where the network structure is static and globalconsensus is achieved i.e. for each attribute, each nodepossesses the same trait as all other nodes. As the num-ber of traits is increased, at a critical value q = q c , thesystem undergoes a phase transition to phase 2 where thesteady state of the network consists of disconnected frag-ments with each fragment coming to consensus locally.While the first two phases differ in the eventual networkstructure, they are similar in that the system eventuallyreaches a frozen state in which neither the node traitsnor the network structure are evolving. However, furtherincreasing q beyond phase 2, eventually reveals a thirdphase where the initial dissimilarity among the nodes isso large that the system does not end up in a frozenstate and rewiring continues indefinitely (Fig. 1). In thisphase, at any given time, there exists a giant component;however, in the asymptotic limit of network size, the sys-tem will never reach consensus, so long as the averagedegree is not too small [12]. This three-phase behavior isexpected to be seen for different choices of F and φ . Forfixed F , as φ is increased the transitions occur at smallervalues of q . For fixed φ , as F is increased the transitionpoints move to higher values of q (Fig. 2). In order tostudy how the approach to consensus can be sped up, wecontinue with F = 5 and φ = 3 and keep the numberof traits per attribute q fixed at 2, so that the systemis confined to phase 1, and a giant component reachingglobal consensus is guaranteed. Furthermore, the choice φ = 3 results in the longest consensus times within phase1, and thus represents the most challenging case withinthis phase, in the context of our study (Fig. 3).For these parameters, when a pair of neighboring nodesis selected at time t = 0, the probability of them meet-ing the similarity threshold, and hence the probability ofinfluence, as one can see by elementary combinatorics,is exactly and equal to the probability of rewiring attime t = 0. Since for the chosen parameters the systemis in phase 1, the steady state reached is one where thenetwork consists of a single connected component andthe states of all nodes in this component are identical i.e.nodes are similar in the traits they possess for all F at-tributes. Thus the steady state is one where a consensusis reached. Shown in Fig. 4 is the scaling of the time toreach consensus (or consensus time) T c as a function ofthe system size. We contrast the behavior of a systemwhere node attributes and network structure co-evolveto the behavior of a system where the node attributesevolve on a purely static network. The latter can alsobe thought of as the case when φ = 0. As seen clearly,the effect of rewiring is detrimental to consensus times. q < S m a x / N > FIG. 1: The average size of the largest connected compo-nent (cluster), h S max i , in the final state of the system as afunction of q , starting from an ER Network with N = 200, h k i = 6 . φ = 3. The average is taken over100 realizations of network evolution. The plot shows threedifferent phases characterized by distinct steady states of net-work structure (see text). This phase diagram is analogousto the one shown in Vazquez et al. [12] for a related model.Also shown for phases 1 and 2 are initial (lower) and steady-state (upper) network snapshots for a single realization ofco-evolution. The colors on the nodes represent the traitsfor a given attribute that is being tracked in the visualiza-tion. Snapshots in phase 1 (2) have q = 2 ( q = 7) traitsper attribute, and initially begin with each node having equalprobability, 1 /q , of possessing any trait per attribute. Werepresent trait 0 for the tracked attribute by red, and allother traits by black. Therefore, in the steady-state snap-shot of phase 2, each black cluster represents that all of itsconstituent nodes have adopted the same trait for the trackedattribute, but this common trait is different from 0. q < S m a x / N > φ=0φ=1φ=2φ=3φ=4φ=5 FIG. 2: Phase diagrams of the system as in Fig. 1 for allpossible values of φ (for F = 5). As shown, the fragmentationtransition takes place at successively higher values of q as φ isdecreased, → ∞ as φ →
0. When φ is large (4 or 5) networkis fragmented even for the smallest non-trivial value of q (= 2) φ T c FIG. 3: Consensus time T c vs similarity threshold φ when theinitial network is ER with N = 200 and the average degree h k i = 6 . T c varies non-monotonically with φ and reaches itsmaximal value at φ = 3. With rewiring present, T c is exponential in N , in contrastto a linear scaling found for the static network. The di-vergence of T c with N when rewiring is introduced canbe qualitatively explained as follows. When a randomnode is chosen as the new end point of a rewired link, itis most probable that the chosen node has an attributevector that is currently the most abundant vector - themajority state - in the population. If we assume thatthe attribute vector corresponding to the majority statedoes not change too frequently, then nodes in the ma-jority state end up garnering a large number of rewiredlinks. However, this comes at the detriment of the ma-jority state, since when an influence step occurs and arandom neighbor is chosen as the adopter, this neighboris more likely to be in the majority state (due to thelarger number of links leading to a node in the major-ity state). As a result, as soon as nodes in the majoritystate accrue more links than the rest of the population,they also become more likely to get influenced. This ef-fect suppresses the growth of the majority state. Thus,the negative feedback due to rewiring strongly slows thespread of any particular state in the network. A. Committed Agents
So far we have discussed how a consensus can bereached across all attributes i.e. the state vector of everynode becomes identical. However, in a realistic scenario,it might be desirable to cause an entire population toadopt a given trait (opinion) for a given attribute (is-sue). For example, within a social network of teenagestudents in a high school, it is desirable that a consensusis reached where everyone views smoking as unhealthybehavior. Here, we study whether the introduction ofcommitted agents [18, 25] can cause fast consensus on any
50 100 150 200 250 300 N T c with rewiringwithout rewiring FIG. 4: Comparison between consensus times for two differentcases − when rewiring is present in the network (black circles)and when the network is static (red squares). These are re-sults obtained from simulations, by averaging over 100 realiza-tions of network evolution. For each realization, the startingnetwork is ER with h k i = 6 .
0. Clearly, with rewiring T c scalesexponentially with N ( T c ∼ exp( αN ) where α ≈ .
02) whilewithout rewiring the scaling is linear. given attribute characterizing the individuals. Withoutloss of generality we choose attribute 1 as the one thatcommitted agents intend to engineer a consensus on. Werefer to this attribute as the designated attribute. Also,we assume that committed agents rigidly adopt trait 1for the designated attribute. Committed agents are thusconsidered un-influencable on attribute 1, but are iden-tical in their behavior to uncommitted nodes for otherattributes. In the following, we redefine consensus time T c to mean the time taken for all nodes to adopt thetrait proselytized by committed agents (i.e. 1) for thefirst attribute. Attributes besides the designated one forcommitted nodes, as well as all attributes for uncommit-ted nodes are initialized to 1 or 0 with equal probability.In Fig. 5, we show the effect of introducing commit-ted agents into the network. In particular, we chooserandomly, a fraction p of nodes as committed agents andstudy how the consensus time scaling with N changes as p is varied. As expected, for p = 0 the exponential scal-ing of T c with N is recovered. Although consensus timevalues decrease for any given N as p is increased, the scal-ing behavior remains unchanged until a critical fractionof p = p c ≈ . N . As mentioned inthe introduction, a similar result was found in [17, 18]for a different model of social influence on a structurallystatic network. This result indicates that beyond a criti-cal value of the committed fraction, this fraction can ef-ficiently overcome the resistance to consensus generatedby the random rewiring taking place in the network. Fig.6(a) shows consensus time as a function of p , for differentnetwork sizes. N T c p=0.0p=0.05p=0.10p=0.15p=0.20p=0.25 FIG. 5: Simulation results for consensus time T c as a func-tion of network size N when the initial network is ER withaverage degree h k i = 6 . p . In these simulations, the influencer is chosen first,followed by the adopter. For low p , T c diverges exponentiallywith N . At the critical value p c ≈ .
1, the system undergoesa transition beyond which T c ∼ log( N ) III. FINITE SIZE SCALING ANALYSIS
Here we employ finite-size scaling analysis to obtain anestimate of the critical committed fraction p c from oursimulation results. In parlance with the theory of phasetransitions, we assume the following scaling ansatz for T c : T c ( p, N ) = N α f ( N β ( p − p c ))where p c is the critical value of p and α and β are crit-ical exponents. The function f is an unknown scalingfunction.At p = p c , the scaling ansatz yields T c N − α = f (0),which implies that if T c N − α is plotted as a function of p , curves for different N values must intersect at p = p c .Since α itself is unknown, we progressively increase α from values close to zero until curves of T c N − α versus p corresponding to different N , intersect at a single point.We find that a single intersection point is obtained for α = 1 and this point is at p = p c ≈ . β , we plot T c N α as a function of N β ( p − p c ), and vary β until a good scaling collapse isfound for all curves near p c . Following this procedure wefind β ≈
1. The scaling collapse is shown in Fig. 6(c).
IV. EFFECT OF INFLUENCER-ADOPTERSELECTION ORDER
In this section we study how the order of selecting theinfluencer and adopter nodes affects consensus times, andalso the transition observed in consensus times as thecommitted fraction is varied. Surprisingly, such subtle .
05 0 . .
15 0 . . p e + e + T c N= N= N= N= N= N= ( a ) p T c / N N=150N=200N=250N=300N=350N=400 (b) -40 -20 0 20 40 60 (p-p c )N T c / N N=150N=200N=250N=300N=350N=400 (c)
FIG. 6: (a) T c vs p starting from an ER network with averagedegree h k i = 6 . N . (b) T c N α with α = 1 . p for the data in (a). The critical point - the valueof p at which the curves intersect - is p c ≈ . T c N α ( α = 1 .
0) vs N β ( p − p c ) shows good scaling collapse inthe vicinity of p c for β = 1 . changes in selection order can have a profound effect onconsensus times. A. Adopter-first selection
In adopter-first selection, a random node i is selected,followed by a randomly chosen neighbor j of i . Howeverin contrast to the model studied so far, if the criterionfor an influence step to occur is met, then i adopts thetrait of j for a randomly chosen attribute from amongthose for which they currently do not share the sametrait. Thus, as the name suggests, the first chosen nodeis treated as the adopter and the second as the influ-encer. A comparison between consensus times on staticnetworks and networks with rewiring under this updat-ing scheme (Fig. 7) suggests that rewiring generates apositive feedback which aids the network in reaching con-sensus. This is in contrast to what was observed in theprevious case of influencer-first selection. Surprisingly, N T c without rewiringwith rewiring FIG. 7: The consensus times for two different cases − whenrewiring is allowed in the network (black circles) and whenthe network is static (red squares) when first chosen node isthe adopter (and p = 0 . h k i = 6 . T c on a static network scales linearly with N butwith rewiring T c follows a power law with exponent ≈ . for adopter-first dynamics, the previously observed ex-ponential divergence of T c with N is absent even in thecase when the committed fraction p = 0. Instead, with p = 0, consensus time grows as a power law in N , withan exponent ≈ .
7. Furthermore, for p > T c ∼ log N suggesting the existence of a transition in the scaling be-havior of T c occurring very close to p = 0 (Fig. 8). B. Unbiased selection
In the selection orders considered so far, the directionof influence is always fixed beforehand. Here, we investi-gate the case in which, after a pair of neighboring nodes
100 1000 N T c p=0.0p=0.05p=0.10p=0.15p=0.20p=0.25 FIG. 8: Scaling of consensus time, T c , as a function of networksize, N , for different values of the committed fraction p foradopter-first dynamics. In the absence of committed agents T c ∼ N . and is logarithmic in N for any p > is selected, a random one among them is chosen to be theinfluencer and the other, the adopter (provided the crite-rion for an influence step is met). In other words, a pairof nodes ( i, j ) is selected, and with probability node i adopts one of node j ’s traits, and otherwise node j adoptsone of node i ’s traits. In this case also, a transition inscaling behavior of T c appears when p is changed fromzero to a non-zero value. However, the transition here isfrom a linear scaling, T c ∼ N for p = 0, to T c ∼ log N for p >
100 1000 N T c p=0.0p=0.05p=0.10p=0.15p=0.20p=0.25 FIG. 9: Scaling of consensus time T c with network size N fordifferent values of committed fraction p for unbiased dynam-ics. For these simulations, a microscopic update consisted ofrandomly picking a link, and then randomly selecting one ofthe endpoint nodes of the chosen link to be the adopter andthe other, the influencer. For p = 0, T c follows a power lawwith an exponent ≈ p > T c ∼ log( N ). V. CONCLUSIONS
Using a variant of the Axelrod model with homophilicrewiring, we have studied how a small fraction of com-mitted agents can dramatically influence the scaling ofconsensus times on structurally evolving networks. Sofar, the vast majority of models studying how a targetedchange in opinion or behavior can be engineered havebeen confined to networks that are fixed in their topology.By considering the effect of persistent opinion holders -committed agents - on structurally evolving networks,we show that introducing a committed fraction p > p c represents a scalable method to cause widespread adop-tion of a given opinion on such networks. We have alsoconsidered variations to the update rule involving the se-lection order of influencers and adopters and shown thatthe transition in scaling behavior of T c across some p c isa consistent feature across these variations, even thoughthe precise scaling behavior is dependent on the selectionorder. Moreover, our results show that in the worst casescenario for consensus time - the influencer-first case -the introduction of a critical number of committed nodescan result in a dramatic reduction in consensus time. While we have employed a network model with simplebut plausible rewiring and influence rules, with data ontime-evolving networks becoming increasingly available,it may be worthwhile studying empirically how the struc-tural evolution of such networks is coupled to the at-tributes associated with their constituent nodes. VI. ACKNOWLEDGMENTS
This work was supported in part by the Army Re-search Laboratory under Cooperative Agreement Num-ber W911NF-09-2-0053, and by the Office of Naval Re-search Grant No. N00014-09-1-0607. The views and con-clusions contained in this document are those of the au-thors and should not be interpreted as representing theofficial policies, either expressed or implied, of the ArmyResearch Laboratory or the U.S. Government. The U.S.Government is authorized to reproduce and distributereprints for Government purposes notwithstanding anycopyright notation here on. We also thank C. Lim, J.Xie, A. Asztalos, A. Goscinski and D. Hunt for usefulcomments and discussions. [1] P. F. Lazarsfeld and R. K. Merton,
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