Committed activists and the reshaping of status-quo social consensus
aa r X i v : . [ phy s i c s . s o c - ph ] O c t Committed activists and the reshaping of status-quo social consensus
Dina Mistry, Qian Zhang, Nicola Perra, and Andrea Baronchelli Laboratory for the Modeling of Biological and Socio-technical Systems,Northeastern University, Boston MA 02115 USA Centre for Business Network Analysis, University of Greenwich, Park Row, London SE10 9LS, United Kingdom Department of Mathematics, City University London, London EC1V 0HB, UK
The role of committed minorities in shaping public opinion has been recently addressed withthe help of multi-agent models. However, previous studies focused on homogeneous populationswhere zealots stand out only for their stubbornness. Here, we consider the more general case inwhich individuals are characterized by different propensities to communicate. In particular, wecorrelate commitment with a higher tendency to push an opinion, acknowledging the fact thatindividuals with unwavering dedication to a cause are also more active in their attempts to promotetheir message. We show that these activists are not only more efficient in spreading their messagebut that their efforts require an order of magnitude fewer individuals than a randomly selectedcommitted minority to bring the population over to a new consensus. Finally, we address the roleof communities, showing that partisan divisions in the society can make it harder for committedindividuals to flip the status-quo social consensus.
PACS numbers: 89.75.-k,89.65.-s,64.60.aq
I. INTRODUCTION
Social change is often produced by committed groupsthat challenge the status-quo [1, 2]. Sometimes thetransformation is significant, and the new social orderis a-posteriori considered an improvement over the oldregime. Examples include universal suffrage [3], the abo-lition of the transatlantic slave trade, and racial deseg-regation in the public sphere[4]. More often, the changeconsists of the emergence of a new social consensus onopinions and behaviors that do not seem to be betterthan the ones they replaced. This is the case we willconsider in the present paper. A prominent example con-cerns shifts in the adoption of social conventions, whichby definition are arbitrary behaviors or rules of action[5]. This is the case of the constant renewal of currentday slang [6], cultural fads, and fashion [7]. In general,the reshaping of consensus can be extremely fast, andthe coexistence between supporters of the old and newstatus-quo short-lived.A natural question concerns the dynamics leading aminority opinion backed by committed supporters to be-come dominant. Insight into this problem was recentlyobtained by theoretical work and multi-agent modeling[8–15]. In particular, refs [8–10] introduced commitmentin the context of the Naming Game model of conventionformation [16, 17], which has recently been shown to re-produce accurately experimental results on the sponta-neous emergence of conventions [18]. The model allowsagents to hold more than one opinion at the same time,and thus describes ‘undecided’ or ‘neutral’ agents natu-rally. In the model [16], agents interact in pairs, chosenuniformly at random; one of them playing as speaker andthe other as hearer, or listener. The speaker randomlyselects one of her opinions and transmits it to the hearer.If the hearer holds it in her list, then both speaker andlistener retain only that opinion. Otherwise, the listener adds the opinion to her inventory. Thus, when the num-ber of opinions is constrained to two, agents can be di-vided into three groups, namely those who hold opinion A , those who hold opinion B and those who hold bothopinions, AB [19]. A committed minority of individu-als that only retain and propagate one opinion can easilyflip a majority of individuals initially holding the otheropinion provided that its size exceeds a critical value p c ≃
10% [8, 9]. Interestingly, this value also holds forthe general Naming Game case of O ( N ) opinions in thesystem [20] and for a different model on interdependentnetworks [21].However, the majority of previous studies did not con-sider that committed individuals are not only less proneto abandon the opinion they have, but they are usuallyalso more active in trying to convince other people [22–25]. Activity-driven networks appear particularly suit-able to take these activists into account, as they attachto each node a variable, called “activity”, that describesthe propensity of the node to establish new connectionsat a given time [26]. Committed activists are then easilydescribed by assigning a large value of activity to them.It is important to notice that in the classic Naming Gameeach node is selected uniformly at random, a scenariothat corresponds to a homogenous distribution of activ-ity.In this paper, we consider the Naming Game modelwith committed activists on activity driven networks. Tothis end, we first study a variant of the Naming Game inwhich only the listeners update their opinion post inter-action [27]. The listener only model allows us to separatethe role of speaker and listener, which is useful in activitydriven networks where multiple nodes can be speakers atthe same time. In the first stage we consider a systemcharacterized by homogenous activity. We show analyti-cally that the threshold of the minimal required commit-ted minority p c for the listener only variant is around 7%of the population (as estimated in [28]), i.e. smaller thanthe ≃
10% obtained when both agents negotiate their po-sition [8, 9]. Then, we extend our analysis considering apopulation of agents described by a heterogeneous distri-bution of activity. If the activity of the individuals is notcorrelated to their role (i.e., to be committed or not),the threshold p c turns out to be the same as found inthe previous case where each node is selected uniformlyat random. Interestingly, we show that a much smallerminority can quickly influence the whole system whencommitted individuals are more active than the rest ofthe population. Finally, we consider how a polarized so-cial network, where individuals thinking alike tend to bemore connected with each other than to individuals re-taining a different opinion, can hinder the effectivenessof a committed minority [8, 29].The rest of the paper is structured as follows: Westart by introducing the Naming Game with commit-ted minorities on temporal networks in Section II. Nextwe investigate the case of activists as the proponents ofthe new social convention and the extent of their advan-tage in Section III. In Section IV we consider collectiveopinion flipping with committed agents on a real net-work with community structures built from a network oftwo weakly connected political blog communities. We areable to show that activists hold a clear advantage over acommitted minority chosen at random regardless of net-work structures present. Finally we discuss our resultsin the Conclusion and provide an Appendix for a thor-ough explanation of the approach taken to arrive at ouranalytical results. II. COMMITTED MINORITIES IN THENAMING GAME
The microscopic rules of the Naming Game (NG)model are simple [16]. At every time step two agentsare selected to interact uniformly at random, one as aspeaker and the other as a listener. The speaker ran-domly selects one of her opinions and shares it with thelistener. If the listener has the opinion in her inventory,then both agents retain that opinion and forget all oth-ers. If not, the listener adds the opinion to her inventory.This process repeats until all agents agree upon one opin-ion and only that opinion. Agents begin the game withno prior knowledge of any opinions and new opinions arecreated by speakers who have none to share.Here, we will focus on the case where only 2 opinionsare possible [19], also known as the binary NG[9, 27].Thus agents belong to one of three possible groups; thosewith opinion A , m A , those with opinion B , n B , or thosewith both A and B , n AB , such that m A + n B + n AB =1. With A as the opinion of the committed minority,the group holding opinion A can be further split intotwo; those who can be influenced and persuaded to adoptopinion B , n A , and zealots committed to opinion A , p ,so that m A = n A + p . In the NG, and also in previous studies with committedindividuals, both speakers and listeners negotiate theiropinion state for each interaction. In this paper, we willfocus on the listener only or hearer only NG a variantwhere only the listener updates their opinion after aninteraction, which yields the same scaling of convergencetime with population size N as observed in the usual NG[27].With our interests lying in the study of collective opin-ion flipping, we consider the case in which only the com-mitted minority know of the new convention ( A ) at thebeginning, while everyone else agrees on the old conven-tion ( B ). Previous studies of the NG with committedminorities considered this scenario resulting in a criticalsize of the committed minority p c ≃
10% [8, 9]. In thehearer only NG variant (hereafter referred to as the NG)we find the critical size becomes smaller with p c ≃ Committed minority threshold for the Hearer-OnlyNaming Game
With the density of committed agents fixed to p and allagents having a priori knowledge of at least one opinion,the fraction of the population knowing A and B is givenby n AB = 1 − n A − n B − p . The mean field rate equationsof the states are then easily obtained by considering thepossible interactions listed in Table I. They read:˙ n A = − n A n B + 12 n A n AB + 12 n AB + pn AB (1)˙ n B = − n A n B + 12 n B n AB + 12 n AB − pn B (2)The terms on the right of (1) describe the number ofagents leaving and joining the state n A according to dif-ferent interaction pairs. In particular, the first considersindividuals who leave n A after hearing from a speakerwith only B in their memory. The second is a combinedterm describing those who join n A from n AB after hear-ing from a speaker with only A in their memory and thosewho leave n A after hearing B from a speaker with A and B , where both have equal probability to be transmitted.The terms on the right of (2) are similar but describingthe number of agents leaving and joining state n B .Following the analysis of Xie, et al [9], we determinethe conditions of existence for the fixed points of theabove mean field rate equations. Simplifying our nota-tion by replacing n A with x and n B with y , we get thefollowing: Speaker Listener Listenerbefore interaction after interaction
A, A c A, AB AB ABA c A c B A ABB, AB BA c A c AB A −→ A, AB AB ABA c A c AB B −→ A ABB, AB BA c A c TABLE I. Interaction outcomes for different speaker-listenerpairs in the 2 state (hearer only) NG. For speakers in thestate AB there are two possible opinions to share; the opin-ion shared is indicated above the arrow. A c refers to agentscommitted to opinion A . ˙ x = − xy + 12 x (1 − x − y − p ) + 12 (1 − x − y − p ) (3)+ p (1 − x − y − p )˙ y = − xy + 12 y (1 − x − y − p ) + 12 (1 − x − y − p ) − py (4)The fixed points of this system correspond to (x,y) pointswhich satisfy ˙ x = ˙ y = 0, giving us: x = (1 − y ) − p y + p (5) y = (1 − x − p ) x + p (6)With (3) substituted into (4) we get:2 y (1 + y + p ) (cid:0) y + 4( p − y + ( p + 1) (cid:1) = 0 (7)The fixed point values of y , y ∗ , are given by solutions ofthis expression.Neglecting trivial solutions, the third factor in (7) givesone or two additional fixed points depending on the pa-rameter p : y ∗ = − p − ± p p − p + 13 (8)The only physical solution gives p c = 7 − √ ≈ . p c = 0 . ± FIG. 1.
Numerical results for convergence time T conv in the bi-nary NG with a randomly selected committed minority for a pop-ulation of N = 10 agents. Only two opinions are possible: A or B , with a state of consensus reached when everyone knows onlyopinion A . Committed agents begin with knowledge of A and in-troduce this new opinion to the rest of the population who initiallyknow opinion B . The median of 100 simulations show that con-vergence time diverges at the analytically derived critical thresholdof p c ≈ . .
01 of [28]. At p c the additional fixed points affordedby (8) collapse to a single point to give the fixed state( n A , n B ) = (0 . , . ≤ p ≤ p c . The existence offixed points where y ∗ 6 = 0 indicates the possibility of sta-ble states for the population in which opinion B persistsand convergence is never met. Then p c also represents acritical threshold around which convergence time T conv diverges (see Figure 1).Figure 1 shows that as p → p c , average convergencetime begins to diverge in simulations of the NG for apopulation of 10 agents. At minimum then ∼
7% of thepopulation must be committed to opinion A in order tostrike an effective mutiny against the prevailing opinion B . Committed minorities in heterogeneous populations
Thus far we have considered the NG in homogeneouspopulations, where all individuals have the same proba-bility of speaking. However, in real life social networksare comprised of individuals heterogeneous in their ten-dency to connect and communicate. To take this intoaccount, we consider activity-driven networks[26, 30, 31],in which nodes are assigned an activity rate a which de-scribes their propensity to communicate. At the start ofeach time step t nodes activate with a probability a ∆ t and connect to m random neighbours, where ∆ t is theduration of the step. At the next time step t + ∆ t alllinks are broken and the network is built anew. For ourmodel we only consider the case where active nodes con-nect to inactive nodes. Activity rates are values in theinterval ( ǫ, ǫ <<
1, and are given according to aprobability distribution function P ( a ). For a wide set oflarge datasets describing social interactions the function P ( a ) is shown to be a power law function of a such that P ( a ) ∝ a − γ [26, 30, 32]. This results in a heavy-tailed ac-tivity distribution with a lower activity cut-off ǫ . We willuse with this functional form for P ( a ) to produce pop-ulations with wide varying rates of activity from whichwe will select our activists , where γ = 2 . ǫ = 10 − , andeach active agent connects to m = 1 inactive neighborsper time step ∆ t = 1.We now consider the NG on these time varying net-works where speakers are active nodes and listeners aretheir m inactive neighbours. In this setting, listeners up-date their opinion at the end of the time step. After allspeakers have communicated each listener will retain themode of their opinions. However, nodes have a proba-bility ∝ N − of being a listener for s speakers duringa given time step, so that in large populations listenerstypically receive only one opinion in a given time step.To determine the critical threshold p c it is convenientto write down equations for each activity rate or class a . Thus, n ay represents the fraction of population whoseactivity is a which holds opinion y , and p a is the frac-tion of committed individuals with activity a . The rateequations for the density of states A and B at each class a are:˙ n aA = − n aA X a ′ n a ′ B a ′ + n aAB X a ′ n a ′ A a ′ − n aA X a ′ n a ′ AB a ′ + 12 n aAB X a ′ n a ′ AB a ′ + n aAB X a ′ p a ′ a ′ (9)˙ n aB = − n aB X a ′ n a ′ A a ′ + n aAB X a ′ n a ′ B a ′ − n aB X a ′ n a ′ AB a ′ + 12 n aAB X a ′ n a ′ AB a ′ − n aB X a ′ p a ′ a ′ (10)where each summation is an average over all activityclasses for speakers in different states. For each termin the above the left side describes the state of the lis-tener in class a , the right side is the average speakercommunicating with them who can cause them to changetheir position, while the factor in front is the probabilityfor this change. We can simplify these equations withthe use of the following notation: n A = x, n B = y,n AB = z = 1 − x − y − p, e X = P a n aA a, e Y = P a n aB a, e Z = P a n aAB a. Summing (9) and (10) over all classes of a then givesus the full rate equations:˙ x = − x e Y + z e X − x e Z + 12 z e Z + zp (cid:10) a (cid:11) (11)˙ y = − y e X + z e Y − y e Z + 12 z e Z − yp (cid:10) a (cid:11) (12) where P a p a a = p h a i . This comes from committedagents being randomly distributed among activity classesresulting in p a being proportional to the probability ofagent having activity class a , thus p a = pP ( a ) so thatwhen summed over all classes the total density of com-mitted agents is p .With some consideration on the definition for z (seeAppendix), we find that e Z = h a i− e X − e Y − p h a i . We notethe existence of solutions for x and y relies only on theexistence of valid solutions for e X and e Y . Three steadystate solutions are found for e X and e Y through a fixedpoint stability analysis of their rate equations (Appendix:Activity-driven networks). Adopting the same line ofreasoning seen for homogeneously mixing populations, wefind p c = 7 − √ p c obtainedfor the case of a homogeneous activity distribution, andnumerical simulations support this result, once time isrescaled so as to take into account multiple speakers ateach time step rather than a single speaker ( I conv nowinstead of T conv for the number of interactions neededto reach convergence; see Figure 2). Thus, the introduc-tion of varying rates of communication through activitydriven networks has no effect on p c when our committedminority is selected at random while I conv increases byless than two orders of magnitude near p c .From Figure 3 we see that increasing m - the numberof neighbors speakers communicate with at each step - byan order of magnitude also has no effect on p c nor I conv with committed agents selected at random. III. ACTIVISTS
Individuals committed to a cause are not only lesslikely to leave their position but are also far more activein recruiting others for their cause [22–24]. To describethe presence of activists, we select the committed minor-ity to be the most active agents in the population, thuscorrelating the agent’s activity rate a with commitment.These activists will communicate the most often in ourpopulation and have many more opportunities to sharetheir message. Concretely, activists are now chosen tobe the N p agents with the highest activity rates a . Thisresults in a different definition of a committed agent ofactivity class a . Only a range of high activity classesare now selected according to p a ∝ P ( a ), while p a = 0holds for the rest of the population since by definition noactivist comes from a low activity class. Hence, p a = ( P ( a ) a c ≤ a ≤
10 a < a c (13)where a c is the lower limit of activity classes for activistsdefined from the integral: I c o n v Random CMp c =0.0718 I c o n v Activistsp c =0.001 density p committed to opinion A FIG. 2. (Color online) Average number of interactions required toreach convergence I conv vs p in the NG in activity driven networksfor a population of N = 10 agents for 100 simulations (Top). I conv diverges at the same p c for random committed minoritiesas in systems with homogenous activity distributions (Figure 1).(Bottom) Activists (commitment correlated with high probabilityto speak) on the other hand show considerable advantage in needingfewer interactions to reach consensus and a numerical threshold of p c ≈ × − , agreeing well with the analytically derived p c ≈ × − . I c o n v p c =0.0718 density p committed to A m =1m =10 FIG. 3.
Comparison of average number of interactions required toreach convergence I conv vs p in the NG in activity driven networkswith N = 10 agents for m = 1 and 10 listeners per speaker pertime step. I conv and the critical threshold p c remains the samedespite a tenfold increase in communication per step. Error barsare not visible at the scale of the plot. p = Z a c P ( a ) da, (14)as the integral of all activists of the higher activity classesmust equal the total density of activists p in the popu-lation. The lower limit of activity classes for activistsis: a c = (cid:0) − p (1 − ǫ − γ +1 ) (cid:1) − γ +1 (15)The rate equations (11) and (12) need to be modifiedaccordingly, resulting in X a p a a = Z a c P ( a ) ada = (cid:18) − γ + 1 − γ + 2 (cid:19) (cid:18) − a − γ +2 c − ǫ − γ +1 (cid:19) (16)We will refer to this term as h a c i ; it is the average activityof activists. Using this e Z becomes: e Z = h a i − e X − e Y − h a c i (17)and the rate equations for e X and e Y become:˙ e X = − e X e Y + 12 e X e Z + 12 e Z + e Z (cid:10) a c (cid:11) (18)˙ e Y = − e X e Y + 12 e Y e Z + 12 e Z − e Y (cid:10) a c (cid:11) (19)Real solutions for e X and e Y yield the value p c ≈ . × − for the critical threshold. Below p c ≈ . × − thenwe can expect to find stable states for the populationwhere opinion B never dies out and convergence to A is never reached. Figure 2 shows that numerical simula-tions for the case of activists agree well this theoreticalprediction. Convergence occurs for a significantly lowerrange of p compared to a random committed minoritywhere p c ≈ . p ≈ × − . IV. COMMUNITY STRUCTURE
In the real world, the effectiveness of activism might behindered by the tendency of individuals to communicatepreferentially with like-minded peers [33, 34]. To investi-gate this point, we consider the simple case of two weaklyconnected communities by examining the network of on-line political blogs analysed in Adamic and Glance’s workon the polarized political blogosphere two months beforethe 2004 U.S. Presidential Election [29], which shows twodistinct communities: Democratic and Republican lean-ing blogs.The original dataset constructs a directed network sowe will focus on the greatest weakly connected compo-nent which contains 1222 blogs and 19089 links out of thetotal 1490 blogs and 19090 links and use an undirectedrepresentation of this network. The remaining network issplit nearly into two equal sized communities: 586 liberalblogs and 636 conservative blogs. The two communitystructure of this network then provides a natural divi-sion for knowledge of the two opinions in the binary NGwith one community being the source for the new socialconvention. For this network then we will focus on twoscenarios of the NG:1. In the first scenario agents begin with knowledge ofopinion A or B dependent on the community theyare a part of, with a select group from the firstcommunity committed to A ;2. In the second scenario everyone knows opinion B except for the group committed to A in the firstcommunity.The activity is defined as the propensity of each nodeto engage in social interactions. For each node i , it is pro-portional to the ratio between the number of interactionsinvolving the node and the total number of interactionsin the system. Formally, the activity is then a i = η s i P j s j where s i is the strength of i , i.e. total number of inter-action of i , and η is a rescaling factor. In the dataset weconsider here, we have information just about the num-ber of different peers, the degree k i , in contact with i .For this reason, we generate the activity rates from thereal data considering the normalized degree for individ-ual blogs or agents in the network a i = η k i P i k i . At eachtime step active agents connect to an inactive neighborchosen among those for which they have an existing linkto in the original dataset. This preserves the commu-nity structure, allows for an activity-driven creation oflinks in the network and produces a ranking of agentsby which we can select activists . In contrast to the ac-tivity driven networks previously considered, where thedistribution of activity rates were given by a power lawwith exponent γ = 2 .
5, this political blogosphere networkexhibits a power law distribution of normalized degreeswith γ ≈ . ± . η = 50 to raise the ac-tivities so that the number of active agents per time stepin this network is at least 1.Figure 4 shows the results of the NG played out in thiscommunity structured network in comparison to a ran-dom rewiring of the network (shown in the inset) in thefirst scenario detailed above. Agents begin with knowl-edge of opinion A or B depending on the community theyare a part of and we select a group within community A to be committed. I conv is now the number of interactionsrequired to get 95% or more of the network to hold onlyopinion A . We lower the conditions for convergence from100% of the network holding only opinion A to 95% toguarantee a faster yet accurate estimation of the thresh-old. I c o n v Random CM I c o n v Activists density p committed to A FIG. 4. (Color online) I conv in the NG on a real network of N = 1222 agents containing two weakly connected communitiesvs a random rewiring of the network (inset). Agents begin withknowledge of opinion A or B dependent upon the community theyare originally a part of with a select group in community A re-maining committed to A . I conv is the number of interactions atwhich 95% or more of the network agrees upon opinion A . In boththe original community based network and the rewired network,shown in the insets, activists (bottom) show considerable advan-tage over randomly selected committed minorities (top) in needingfewer committed agents. Rewired networks are obtained by reshuffling the endpoints of the links, which destroys the community struc-ture without altering the degree, and therefore activity, ofeach blog. The comparison of the results obtained in thereal network against those obtained in the rewired net-work is crucial to highlight and isolate the effects of thecommunity structure. Numerical results, shown in Fig-ure 4, indicate that once again activists demonstrate aconsiderable advantage over a randomly selected commit-ted minority in both the original two community network( p ca = 0 .
04 vs p cr = 0 .
15, thresholds for activists and arandomly selected committed minority, respectively) andthe rewired network shown in the inset ( p ca = 4 . × − vs p cr = 0 .
05; 5 blogs vs 62 blogs) with activists need-ing a significantly smaller minimal committed group toconvince the rest of the population to their position. I c o n v Random CM original network I c o n v Activists density p committed to A original network FIG. 5. (Color online) I conv in the NG on a real network of N = 1222 agents containing two weakly connected communitiesvs a random rewiring of the network (inset). All agents beginwith knowledge of opinion B , regardless of their original commu-nity membership except for a select group in community A whoremain committed to A . I conv is the number of interactions atwhich 95% or more of the network agrees upon opinion A . In boththe original community based network and the rewired network,shown in the insets, activists (bottom) show considerable advan-tage over randomly selected committed minorities (top) in needingfewer committed agents. Figure 5 shows the results of the NG in the secondscenario. All agents begin with knowledge of opinion B except for a small segment of the population in onecommunity who introduce opinion A and remain com-mitted to their cause. This scenario is similar to the ini-tial configuration we previously considered for activistsin activity-driven networks. Again I conv is the number ofinteractions required to get 95% or more of the networkto hold only opinion A . Here activists show the sameadvantage over a random committed minority, needinga smaller minimal committed group in both the origi-nal network ( p ca = 0 .
05 vs p cr = 0 .
15) and the rewirednetwork ( p ca = 9 × − vs p cr = 0 . A and hinder the effectiveness of commit-ted agents. Even activists in the community structurednetwork need a larger minimum a size of the network (atleast ∼ V. CONCLUSION
We investigated the role of committed individuals inthe Naming Game model on activity-driven networks.First, we considered a variant of the Naming Game, inwhich only the listeners update their opinion, consider-ing a homogenous distribution of activity. Interestingly,we found the critical threshold p c , defining the minimalfraction of committed individuals needed for a fast opin-ion flipping of the majority, to be smaller than the samevalue obtained when both agents negotiate their posi-tions. Then we considered the presence of heterogeneousactivity patterns in the propensity to communicate. Sur-prisingly, we found that this characteristic, observed inmany networks, does not alter the critical threshold p c .Furthermore, we considered the effects of activists by cor-relating activity, or propensity to speak, with commit-ment. We found that activists can reduce their numbersby two orders of magnitude compared to random com-mitted agents and still persuade their peers to adopt theiropinion or social convention. Finally, we considered thepresence of communities in a real social network. Weshowed that communities inhibit the ability of commit-ted agents to effectively spread their message and subse-quently require larger committed groups to convince therest of the population to join their position.Taken together our results indicate that strategicallyselecting individuals for a given cause or social conven-tion can greatly reduce the cost of associated campaignsin terms of the sheer number of individuals needed forits success. It is worth noticing that an approach similarto the one presented here was proposed in [25] for therumor spreading model [35], with the difference that theauthors considered a static network mimicking the factthat agents would remember their connections and ex-plore only a fixed subset of the network. Recent resultson the effects of social memory and the heterogeneity ofsocial ties on spreading phenomena suggests that futurework may benefit from including more realistic partnerselection mechanisms also in temporal networks to betterreflect what is observed in real social networks [30, 36–38]. Other interesting points left for future explorationare the influence of fatigue on demobilization and disen-gagement of activists [39, 40] (here modeled as endlesslycommitted), and the role of broadcasting agents able toreach a large part of the population (i.e., mass media)in the spirit of previous studies that addressed this pointfor Axelrod’s model of the dissemination of culture [41]. APPENDIX
To find a critical threshold for the NG in activity drivennetworks we need to find the conditions under which so-lutions for x and y exist. The rate equations for bothtell us that their solutions rely on the existence of validsolutions for e X , e Y , and e Z . First we will consider ourdefinition for e Z = P a z a a . To determine the existence ofvalid solutions for e Z we need an expression for z a . Thedefinition for z gives us a hint of how to define z a : it isthe density of agents with activity class a who are not instate in A or B : z a = N a N − x a − y a − p a (20)The term N a /N is simply the density of all agents withactivity class a and with activity rates fixed a priori this term must also be fixed. On the other hand theprobability of agents being in activity class a is givenby the probability distribution function P ( a ), therefore N a /N = P ( a ). With this z a and e Z become: z a = P ( a ) − x a − y a − pP ( a ) (21) e Z = X a z a a = (cid:10) a (cid:11) − e X − e Y − p (cid:10) a (cid:11) (22)Now we see that the existence of solutions for x and y relies only on the existence of valid solutions for e X and e Y . The steady states for e X and e Y are found througha fixed point stability analysis of their rate equations. These rate equations are obtained by multiplying (11)and (12) by a and then summing over all classes to yield:˙ e X = − e X e Y + 12 e X e Z + 12 e Z + e Zp (cid:10) a (cid:11) (23)˙ e Y = − e X e Y + 12 e Y e Z + 12 e Z − e Y p (cid:10) a (cid:11) (24)The steady state solutions for e X and e Y are then foundto be:( e X, e Y ) = a − ap, a (cid:16) − p − √ δ (cid:17) , a (cid:16) − p + √ δ (cid:17) a (cid:16) − p + √ δ (cid:17) , a (cid:16) − p − √ δ (cid:17) (25) δ = 1 − p + p The first of these solutions represents the trivial casewhere the entire population begins in the state knowingonly opinion A (consensus), whether they are committedor not since e Y = 0 and thus y = 0.The other two solutions lead to stable opinion statesof the population with restrictions on the value of p . Thefirst of these restrictions comes from the fact that opinionstates must remain physical and thus the discriminant δ = 1 − p + p >
0. From this two limiting values of p result: p c = 7 ± √
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