On the Influence of Committed Minorities on Social Consensus

Human behavior is profoundly affected by the influenceability of individuals and the social networks that link them together. In the sociological context, spread of ideas, ideologies and innovations is often studied to understand how individuals adopt new states in behavior, opinion, ideology or consumption through the influence of their neighbors. In this paper, we study the evolution of opinions and the dynamics of its spread. We use the binary agreement model starting from an initial state where all individuals adopt a given opinion B, except for a fraction p<1 of the total number of individuals who are committed to opinion A. In a generalization of this model, we consider also the initial state in which some of the individuals holding opinion B are also committed to it. Committed individuals are defined as those who are immune to influence but they can influence others to alter their opinion through the usual prescribed rules for opinion change. The question that we specifically ask is: how does the consensus time vary with the size of the committed fraction? More generally, our work addresses the conditions under which an inflexible set of minority opinion holders can win over the rest of the population. We show that the prevailing majority opinion in a population can be rapidly reversed by a small fraction p of randomly distributed committed agents who are immune to influence. Specifically, we show that when the committed fraction grows beyond a critical value pc=9.79%, there is a dramatic decrease in the time, Tc, taken for the entire population to adopt the committed opinion. Below this value, the consensus time is proportional to the exponential function of the network size, while above this value this time is proportional to the logarithm of the network size. This has enormous impact on stability/instability of the society opinions. We also discuss conditions under which the committed minority can rapidly reverse the influenceable majority even if the latter is supported by a small fraction of individuals committed to their opinion. The results are relevant in understanding and influencing the social perceptions of international missions operating in various countries and attitudes to the mission goals. 1.0 INTRODUCTION The propagation of social influence through the social networks that link individuals together has a profound effect on human behavior. Prior to the proliferation of online social networking, offline or interpersonal social networks have been known to play a major role in determining how societies move towards consensus in the adoption of ideologies, traditions and attitudes [1] [2]. As a consequence, the dynamics of social influence has been heavily studied in sociological, physics and computer science literature [3] [4] [5] [6]. In the sociological context, work on diffusion of innovations has emphasized how individuals adopt new states in behavior, opinion or consumption through the influence of their neighbors. Commonly used models for this process include the threshold model [7] and the Bass model [8]. A key feature in both these models is On the Influence of Committed Minorities on Social Consensus 32 2 RTO-MP-HFM-201 UNLIMITED/UNCLASSIFIED UNLIMITED/UNCLASSIFIED that once an individual adopts the new state, his state remains unchanged at all subsequent times. While appropriate for modeling the diffusion of innovation where investment in a new idea comes at a cost, these models are less suited to studying the dynamics of competing opinions where switching one’s state has little overhead. Here we address the latter case. From among the vast repertoire of models in statistical physics and mathematical sociology, we focus on one which is a 2-opinion variant [9] [10] of the Naming Game (NG) [11] [12] [13] [14] and that we refer to as the binary agreement model. The evolution of the system in this model takes place through the usual NG dynamics, wherein at each simulation time step, a randomly chosen speaker voices a random opinion from his list to a randomly chosen neighbor, designated the listener. If the listener has the spoken opinion in his list, both speaker and listener retain only that opinion; else the listener adds the spoken opinion to his list. The order of selecting speakers and listeners is known to influence the dynamics, and we stick to choosing the speaker first, followed by the listener. It serves to point out that an important difference between the binary agreement model and the predominantly used opinion dynamics models [4] is that an agent is allowed to possess both opinions simultaneously in the former, and this significantly alters the time required to attain consensus starting from uniform initial conditions. Numerical studies in [10] have shown that for the binary agreement model on a complete graph, starting from an initial condition where each agent randomly adopts one of the two opinions with equal probability, the system reaches consensus in time Tc ~ lnN (in contrast, for example, to Tc ~ N for the voter model [4]). Here N is the number of nodes in the network, and unit time consists of N speakerlistener interactions. The binary agreement model is well suited to understanding how opinions, perceptions or behaviors of individuals are altered through social interactions specifically in situations where the cost associated with changing one’s opinion is low [15], or where changes in state are not deliberate or calculated, but unconscious [16]. Furthermore, by its very definition, the binary agreement model may be applicable to situations where agents while trying to influence others, simultaneously have a desire to reach global consensus [17]. Another merit of the binary agreement dynamics in modeling social opinion change seems worth mentioning. Two state epidemic-like models of social "contagion" (examples in [18]) suffer from the drawback that the rules governing the conversion of a node from a given state to the other are not symmetric for the two states. In contrast, in the binary agreement model, both singular opinion states are treated symmetrically in their susceptibility to change. Here we study the evolution of opinions in the binary agreement model in a scenario where fractions of the network can be committed [19] [20] to one of the two opinions. Committed nodes are defined as those that can influence other nodes to alter their state through the usual prescribed rules, but which themselves are stubbornly devoted to one opinion and thus, immune to influence. The effect of having un-influencable agents has been considered to some extent in prior studies. Biswas et al. [21] considered for two-state opinion dynamics models in one dimension, the case where some individuals are "rigid" in both segments of the population, and studied the time evolution of the magnetization and the fraction of domain walls in the system. Mobilia et al. [22] considered the case of the voter model with some fraction of spins representing "zealots" who never change their state, and studied the magnetization distribution of the system on the complete graph, and in one and two dimensions. Our study differs from these not only in the particular model of opinion dynamics considered, but also in its explicit consideration of different network topologies and of finite sized networks, specifically in its derivation of how consensus times scale with network size for the case of the complete graph. Furthermore, the above mentioned studies do not explicitly consider the initial state that we care about one where the entire minority set is unOn the Influence of Committed Minorities on Social Consensus RTO-MP-HFM-201 32 3 UNLIMITED/UNCLASSIFIED UNLIMITED/UNCLASSIFIED influencable. A notable exception to the latter is the study by Galam and Jacobs [19] in which the authors considered the case of "inflexibles" in a two state model of opinion dynamics with opinion updates obeying a majority rule. While that study provides several useful insights and is certainly the seminal quantitative attempt at understanding the effect of committed minorities, its analysis is restricted to the mean-field case, and has no explicit consideration of consensus times for finite systems. Specifically, we study two different situations that involve the effect of committed agents on the eventual steady state opinion in the network. In Sec. 2 we study the situation where there exists only one kind of committed set, i.e. all committed nodes are devoted to the same opinion. The question that we specifically address in this section is: how does the consensus time vary with the size of the committed fraction p? More generally, our work addresses the conditions under which an inflexible set of minority opinion holders can win over the rest of the population. In Sec. 3 we extend our study to the case where competing committed groups are present in the population. Here the relevant question is similar to that in Sec. 2 but more formally posed as follows. Suppose the majority of individuals on a social network subscribe to a particular opinion on a given issue, and additionally some fraction of this majority consists of unshakeable in their commitment to the opinion. Then, what should be the minimal fractional size of a competing committed group in order to effect a fast reversal in the majority opinion? 2.0 EFFECT OF A SINGLE GROUP OF COMMITTED AGENTS IN COMPLETE GRAPHS We start by considering the case where a single group of committed agents is present in the population, and the connectivity of nodes within the population is defined by a complete graph. Additionally we are interested here in the situation where initially all uncommitted nodes adopt opinion B while the committed nodes are always committed to opinion A. 2.1 Infinite Network Size Limit We start along similar lines as [19] by considering the case where the social network connecting agents is a complete graph with the size of the network N→∞. We designate the densities of uncommitted nodes in states A, B as nA, nB respectively. Consequently, the density of nodes in the mixed state AB is nAB=1-p-nA-nB, where p is the fraction of the total number of nodes that are committed. Neglecting correlations between nodes, and fluctuations, one can write the following rate equations for the evolution of densities: AB A AB AB B A A pn n n n n n dt dn 1.5 2 + + + − = (1) B B AB AB B A B pn n n n n n dt dn − + + − = 2 The terms in these equations are obtained by considering all interactions which increase (decrease) the density of agents in a particular state and computing the probability of that interaction occurring. Table 1 lists all possible interactions. As an example, the probability of the interaction listed in row eight is equal to the probability that a node in state AB is chosen as speaker and a node in state B is chosen as listener (nAB nA) times the probability that the speaker voices opinion A (1/2). The fixed-point and stability analyses [23] of these mean-field equations show that for any value of p, the consensus state in the committed opinion (nA=1-p, nB=0) is a stable fixed point of the mean-field dynamics. On the Influence of Committed Minorities on Social Consensus 32 4 RTO-MP-HFM-201 UNLIMITED/UNCLASSIFIED UNLIMITED/UNCLASSIFIED However, below p=pc= 5 2− 3 2     3 5+ 24−1 2 − 3 2     3 5− 24−1 2 ≈0.09789 , two additional fixed points appear: one of these is an unstable fixed point (saddle point), whereas the second is stable and represents an active steady state where nA, nB and nAB are all non-zero (except in the trivial case where p=0). Fig. 1 shows (asterisks) the steady state density of nodes in state B obtained by numerically integrating the mean-field equations at different values of the committed fraction p and with initial condition nA=0, nB=1-p. As p is increased, the stable density of B nodes nB abruptly jumps from ≈0.6504 to zero at the critical committed fraction pc. A similar abrupt jump also occurs for the stable density of A nodes from a value very close to zero below pc, to a value of 1, indicating consensus in the A state (not shown). In the study of phase transitions, an "order parameter" is a suitable quantity changing (either continuously or discontinuously) from zero to a non-zero value at the critical point. Following this convention, we use nB the density of uncommitted nodes in state B as the order parameter appropriate for our case, characterizing the transition from an active steady state to the absorbing consensus state. Table 1: Interactions in Naming Game Figure 1. Steady-state density of nodes in state B as a function of the committed fraction p. Comparison between mean-field and simulation results. 2.2 Simulation Results for Finite Complete Graphs In practice, for a complete graph of any finite size, consensus is always reached. However, we can still probe how the system evolves, conditioned on the system not having reached consensus. Fig. 1 shows the results of simulating the binary agreement model on a complete graph for different system sizes (solid lines). For ppc). We know from the mean field equations that in the asymptotic limit, and below a critical fraction of committed agents, there exists a stable fixed point. For finite stochastic systems, escape from this fixed point is always possible, and therefore it is termed metastable. Even though consensus is always reached for finite N, limits on computation time prohibit the investigation of the consensus time, Tc, for values of p below or very close to pc. This can be overcome analytically using a quasi-stationary approximation [24]. This approximation assumes that after a short transient, the occupation probability (of different states in the configuration space of the system), conditioned on survival of allowed states excluding the consensus state, is stationary. For brevity we skip the details here which are given in [23]. Following the above method, we obtain the QS distribution, and consequently the mean consensus times Tc, for different values of committed fraction p and system size N. Fig. 3 (a) shows how the consensus time grows as p is decreased beyond the asymptotic critical point pc for finite N. For ppc, the QS approximation does not reliably provide information on mean consensus times, since consensus times themselves are small and comparable to transient times required to establish a QS state. However, simulation results show that above pc the scaling of mean consensus time with N is logarithmic (Fig. 3 (c)). The precise dependence of consensus times on p can also be obtained for p pc. On the Influence of Committed Minorities on Social Consensus RTO-MP-HFM-201 32 7 UNLIMITED/UNCLASSIFIED UNLIMITED/UNCLASSIFIED with initial conditions: P(N-Np,0,0)=1. The sequence of vectors corresponding to different T generated by Eq.3 can be thought of as a vector-valued function of T. Along a given coordinate direction, this sequence is a real-valued function of T which represents the consensus time distribution starting from the macrostate (NA,NB) corresponding to this coordinate direction: Prob(NA,NB,Tc=T)=P(NA,NB,T). Note that we calculate the consensus time distributions starting from all (NA,NB) at the same time. Figure 4 shows the consensus time distribution starting from macrostate (0,N-Np) obtained using Eq. 3 compared with simulation results. A direct consequence of Eq. 3 is that, P(T) has an eigenvector representation: P(T) =Σk Ck exp( log( λk ) T ) (k=1,..., M) where k is the index of summation, M is the total number of macrostates, Ck’s are constants, λk’s are the eigenvalues of Q in decreasing order. As λk<1, the first term C1 exp(log( λ1 )T) dominates when T is big enough. Therefore the consensus time distribution Prob(NA,NB, Tc=T) always has an exponentially decaying tail with a decay constant -log( λ1 ), where λ1 is the largest eigenvalue of Q. Using Eq. 3, we check the asymptotic behavior of the consensus time distribution as the network size grows. Figure 5 shows the normalized consensus time distribution for different p and N. When ppc, the central region of the consensus-time distribution tends to a Gaussian distribution, but always exhibits an exponentially decaying tail according to the expression shown in the previous paragraph. We also observe the concentration of the consensus time distribution for p>pc as N grows which validates the mean field assumption in calculating the consensus time. Rigorous analysis of the concentration is in [26]. 0 0.5 1 1.5 2 2.5 x 10 7 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 -7