About the power to enforce and prevent consensus

Transcription

1 About the power to enforce and prevent consensus Jan Lorenz Department of Mathematics and Computer Science, Universität Bremen, Bibliothekstraße Bremen, Germany Diemo Urbig Department of Computer Science and School of Business and Economics, Humboldt-Universität zu Berlin, Under den Linden 6 99 Berlin, Germany urbig@diemo.de Received (received date) Revised (revised date) We explore the possibilities of enforcing and preventing consensus in continuous opinion dynamics due to modifying the communication rules. We regard the model of Weisbuch- Deffuant (WD) [, 2], where n agents adjust their continuous opinions as a result of random pairwise encounters whenever their opinions differ not more than a given bound of confidence ε. High ε leads to consensus, while lower ε leads to fragmentation into several opinion clusters. We drop the random encounter assumption and ask: How low may ε be such that consensus is still possible with a certain communication plan for the whole group? Mathematical analysis shows that ε may be significantly lower than in the random pairwise case. On the other hand we ask: How high may epsilon be such that preventing consensus is still possible? In answering this question we prove Fortunato s simulation result [3] that consensus cannot be prevented for ε >.5. Further on we simulate opinion dynamics under different individual strategies and check their power to raise the chances for consensus. One result is that balancing agents raise chances for consensus, especially if the agents are cautious in adapting their opinions. But curious agents raise chances for consensus only if agents are not cautious in adapting. Keywords: Continuous opinion dynamics; communication structure; balancing agents; curious agents.. Introduction What happens if people meet and talk about their opinions toward a political party, toward a brand or a new product? Usually they influence one another and change their opinions. Such opinion formation processes are at the heart of models explaining voting behavior as well as of models of innovation diffusion. If people who are assumed to have continuous opinions (e.g. real numbers) toward something meet and discuss, they may get closer in their opinions and thereby

2 2 Jan Lorenz, Diemo Urbig compromise somehow or they may also distance themselves when they are far away from each other. We will only consider compromising under bounded confidence here, which means that individuals who differ too much in their opinions do not affect each other. This assumption mirrors the psychological concept of selective attention, where people tend to perceive their environment in favor of their own opinions and thereby avoid communication with people with conflicting opinions. Dynamical systems of agents who make their new opinions as averages of other opinions which are not too far away from their own are referred as systems of continuous opinion dynamics under bounded confidence. They have been studied by Hegselmann and Krause [4, 5] and Weisbuch, Deffuant and others [, 2, 6]. In the Hegselmann-Krause model every agent perceives the opinions of every other agent and builds his new opinion as an average of close opinions. They added the bounded confidence assumption to an older linear opinion dynamics model of DeGroot [7, 8]. His main question was under what conditions consensus can be reached. The agents in the Weisbuch-Deffuant model sit in a completely connected society, too. Thus, everybody communicates with everybody else by chance. But they are involved in random pairwise encounters. Both models do not concentrate anymore on the question of conditions for consensus but also on the fragmentation of the agents into opinion clusters which is dependent on the bound of confidence. Both models have been studied with more sophisticated techniques of density based state spaces instead of agents [9,, ], but we will stick to the agent based approach in this paper, because it is closer to reality and also open for analytical results as we will see. Several extensions (e.g. [2]) and combinations [3] of both models have been made, e.g. to a model which includes the centrifugal forces of rejecting agents [4]. There are also results for the models placed in incompletely linked networks, as for instance scale-free networks [3, 5]. But even if we regard a completely linked society as given, the Weisbuch- Deffuant model has an unexplored free parameter in the order of who communicates with whom at what time. We will call rules that modify this order the communication regime. Two main questions have remained open for the Weisbuch-Deffuant model: To what extend does the communication regime matter? And how do individual communication strategies matter? To focus our analysis we concentrate on possibilities and strategies to enforce, foster or prevent consensus. After a short introduction of the Weisbuch-Deffuant model in section 2 we give answers to the first question and prove in section 3 that consensus is possible for a huge parameter space. On the other hand we prove that preventing consensus is also possible for a huge parameter space. This result will be in line with Fortunato [3], who gives numerical evidence by simulation that consensus is reached for ε >.5 under every underlying connected social network. Our result will explicitly define a

3 About the power to enforce and prevent consensus 3 threshold such that for higher bounds of confidence consensus can not be prevented. In that way the simulation result of Fortunato is formally proved in the limit of large numbers of agents and uniformly distributed initial opinions. But the communication regimes we use to reach these results rely on communication regimes, which can only be constructed by the full knowledge of the opinions of all agents. But their properties open the door for developing and analyzing mechanisms that enforce consensus by individual communication strategies. An individual who talks with somebody who has a higher opinion may actually actively seek afterwards somebody with a lower (even higher) opinion and thus show balancing (curious) communication behavior. We check the impact of a communication chain, balancing and curious agents in section 4. We will see that these very simple communication strategies, that could reasonably be applied by individuals, can raise the chances for consensus to a surprisingly large extend, what gives answers to our second leading question. We will conclude with a summary of the results and a discussion of them in the light of opportunities to enforce consensus. 2. Dynamics of continuous opinions We analyze the model of continuous opinion dynamics that was introduced by Weisbuch, Deffuant and others [, 2]. The dynamics are driven by random encounters of two agents, which compromise if their distance in opinion is below a bound of confidence ε. The model always converges to a stabilized opinion formation, where agents in the same cluster have the same opinion in the long term limit [6]. We treat n N agents each having an opinion which is a real number. The opinion of agent i n := {,..., n} at time step t N is represented by x i (t) R. We call the vector x(t) R n the opinion profile at time step t. Definition. (Weisbuch-Deffuant Model) Given a starting opinion profile x() R n, a bound of confidence ε R > and a cautiousness parameter a µ ],.5] we define the Weisbuch-Deffuant process of opinion dynamics as the random process (x(t)) t N that chooses in each time step t N two random agents i, j n which perform the action if x i (t) x j (t) < ε else x i (t + ) = ( µ)x i (t) + µx j (t), x j (t + ) = µx i (t) + ( µ)x j (t), x i (t + ) = x i (t), x j (t + ) = x j (t). With random we mean random and equally distributed in the respective space. a This parameter is called convergence parameter in [].

4 4 Jan Lorenz, Diemo Urbig The bound of confidence ε has shown to be the most significant parameter to control the number of emerging clusters. For randomly distributed initial profiles with opinions between zero and one x [, ] n and n = simulations have shown that consensus is reached nearly in every case for ε >.3 []. For lower ε the usual outcome is polarization into a certain number of opinion clusters. Weisbuch, Deffuant et al. derived by computer simulation the /2ε-rule, which states that the number of surviving clusters is roughly the integer part of /2ε. b The cautiousness parameter µ has been considered to have no effect on clustering in the basic model (only on convergence time) [, 2]. However, there is already some evidence that µ can effect the clustering as well as that the effect of µ interacts with other parameters, e.g. number of agents that participate in an interaction [3]. Further on, different random initial profiles may lead to different numbers of clusters, and even the same initial profile may lead to different numbers of clusters for different random choices of communicating pairs. Furthermore, agents may have communication strategies that do not match the random encounter assumption. We want to shed light on the possible effects that can be reached through different choices in the communication order. 3. Enforcing and Preventing Consensus In this section we give mathematical answers to the questions: How low may ε be such that enforcing consensus is still possible? How high may ε be such that preventing consensus is still possible? Let us regard our initial opinion profile x() and the parameter µ as fixed. We define ε low as the lowest value of epsilon for which there is a communication regime which leads to a consensus. Obviously ε low depends on the initial opinion profile and on µ. We will give a lower and an upper limit for ε low based on a communication regime that looks like a phone chain of closest. For our approximation we must have a detailed look at the initial opinion profile. For that reason we will regard our initial opinion profile x() as ordered in the way x ()... x n (), without loss of generality. For our considerations it is useful to look at the gaps between the opinions. We define for i n the gap to the next neighbor as x i (t) := x i+ (t) x i (t). If we regard an opinion profile as a function x ( ) (t) : n R than we can consider x(t) R n as the discrete derivative with respect to the agent index i. is thus not a differential but a difference operator. For further abbreviation we define the maximal gap max x := max i n x i. In our setting with ordered initial opinions, the function x ( ) () is monotonously increasing. Thus its difference function x ( ) () is non-negative. We now define the phone chain communication regime of closest as the commub Very small surviving clusters are neglected by this rule, but their existence is systematic as shown by the analysis of a rate equation for the density of opinions [9].

5 About the power to enforce and prevent consensus 5 phone chain of closest, n = 5, ε =., 7 time steps phone chain of closest, n = 5, ε =., 2 time steps Fig.. The phone chain communication regime of closest. nication regime that will guide us to fair approximation to ε low. Definition 2. (phone chain communication regime of closest) Let n N be the number of agents. A Weisbuch-Deffuant process of opinion dynamics is ruled by a phone chain communication regime of closest if the communicating agents at time step t N are (t mod (n )) + and (t mod (n )) + 2. The phone chain communication of closest is (, 2), (2, 3), (3, 4),..., (n, n), (, 2), and so forth. This sequencing communication strategy gives a nice proof for the following proposition. Proposition. holds that Let x() R n be an ordered initial profile and let µ ],.5]. It i max x() ε low max µ j x i j (). () i n j= For a proof see appendix A.. Figure shows how the phone chain of closest works.

6 6 Jan Lorenz, Diemo Urbig If we define range(x) = i n x i = x n x and as rounding to the upper integer then we can derive a corollary with a simpler but not as sharp bound. Corollary. Further on it holds ε low range(x()) µ max x() max x() (2) µ For a proof see appendix A.2. Thus ε low is determined mostly by the maximal gap, µ and the ratio of the maximal gap and the difference between the two most extreme opinions in the initial profile. Simulators often use initial profiles x() [, ] n with random and uniformly distributed opinions. The length of the maximal gap in such a profile can be estimated by Whitworth s formula (3) and is thereby dependent on the number of agents. ε ( ) n P (max x > ε) = ( ) k+ ( kε) n. (3) k k= In terms of statistical theory the formula is about the spacings in an order statistics of n independent uniformly distributed random variables (see [7]). Figure 2 shows the probability of the maximal gap to be bigger then ε [, ]. P(max x > ε).5 n= n=2 n=5..2 ε Fig. 2. P (max x > ε) for random equally distributed x R n for some n. Based on this distribution it is possible to derive an estimate for the expected size of the maximal gap in an initial opinion profile. But it gives an additional insight: the larger the population the smaller the expected size of the maximal gap. This leads to the conclusion that for very large numbers of agents that are equally distributed consensus is possible for extremely low values of ɛ. If we assume that the maximal gap converges to zero with rising number of agents, then it is possible to reach consensus for every ε with a big enough number of agents. We now ask the other way round: How high may ε be such that preventing consensus is still possible? At first we see that this question is not detailed enough to be interesting. Preventing consensus is obviously possible if we forbid one agent

7 About the power to enforce and prevent consensus 7 to communicate with all others, e.g. by underlying a disconnected social network. The right question is: How high may ε be such that preventing consensus is still possible, even if we switch at some time step to an arbitrary communication regime? The biggest possible ε is called ε high. Proposition 2. Let x() R n be an ordered initial profile and let < µ <.5. ε high = max n x i () k x j () (4) k n n k k i=k+ For a proof see appendix A.3. A very helpful lemma which helps to prove the proposition is that the mean opinion is conserved by the process of opinion formation (see appendix A.3). If we regard random and uniformly distributed x i () [, ] in the limit for large n, then ε high is computed as the distance of the central points of arbitrary two disjoint intervals which union is [, ]. Thus, ε high.5 with n. This proves the result of Fortunato [3] by showing that preventing consensus is impossible for a large enough number of somehow connected agents for ε >.5. The interval [ε low, ε high ] is the range where both enforcing and preventing consensus is possible with an appropriate communication regime. This interval depends on the initial profile x(). We show figure 3 to give some numerical evidence about the possibilities that can be reached with manipulation of the communication regime. The data in line comes from figure 4 in [] (visually extracted) with 25 simulation runs with random and equally distributed initial profiles with n =. For line 2 we took 25 random equally distributed profiles with n = and give the maximal ε low and minimal ε high that occurred in all 25 profiles (all computed with propositions and 2). Thus, enforcing and preventing consensus was possible in the displayed interval for all 25 selected profiles. The same in line 3 for n = 2. j= ε phases pluralism 2 clust. cluster = consensus 2 3 and 2 clusters possible, n = and 2 clusters possible, n = Fig. 3. Numerical [ε low, ε high ] (second darkest gray) where consensus and polarization happens () and is possible (2 and 3). From this figure we can see that: For finite populations polarization could be possible even for ε >.5.

8 8 Jan Lorenz, Diemo Urbig The bigger the population the smaller ε can be for guaranteeing the possibility of consensus. The bigger the population the smaller is the upper limit of ε for which polarization can be enforced. 4. Individual strategies that raise chances for consensus The phone chain of closest is constructed with the knowledge of all opinions. The chain goes to the extremes of the opinions space and everybody knows for sure who has the closest opinion into both possible direction. Further more, consensus building with this regime lasts much longer than stabilizing with random communication. We now want to skip the idea of global knowledge and the great master plan for communication and go to agent based strategies that may raise the chances for consensus, too. Our agents should not know the opinions of all others and thus do not know if they are in the center or at the extremes of the opinion space. They should only stick to a rule for which they need to know only their own communication history. We check a communication chain and balancing and curious agents. For all these strategies we introduce the new parameter f max which is the maximal frustration. It is the number of unsuccessful times one agent sticks to the rule before neglecting it. In other words: An agent may be balancing in the way that he actively seeks opinions that contradict the opinion of a previous communication partner, but the search only lasts for some trials. Thus, agents are not forced to follow the strategies for ever. 4.. Communication chain The idea of this communication regime is as follows: One agent starts to communicate. He tries to find an agent to compromise with. After a success his communication partner goes on and tries to find a new agent from the opposite side as the agent who contacts him. After a success he passes the initiative to his partner. He should again find an opposite opinion. This is in fact a less demanding version of the phone chain of closest. Here agents do not need to find the closest but only somebody in that direction. We implement the frustration level f max, which is the maximal number of unsuccessful attempts that an agent accepts until he switches to an undirected search. The frustration level should not rise if the compromising fails due to the bound of confidence but only if it fails due to the wrong direction within the bound of confidence. The introduction of frustration level is on the one hand for realism and on the other hand to prevent the process from quitting at one extreme. A pseudo-code for the strategy is given in the appendix B..

9 About the power to enforce and prevent consensus Direction vectors with balancing and curious agents The idea of the communication chain could be criticized, because all agents do not discuss until they are phoned by the one with the initiative. To circumvent this assumption we go to an approach where we store direction and frustration for each agent in a direction vector d Z n. If d i is negative agent i wants to compromise with an agent with lower opinion. If d i is positive agent i wants to compromise with an agent with higher opinion. If d i is zero agent i has no preferred direction. The absolute values of the directions stand for the frustration level. Additional to the bounded confidence criteria, agents i, j only compromise if i s direction is to j (or zero) and j s direction is to i (or zero). If they fail (but are closer then ɛ to each other) they both reduce the absolute value of their frustration levels one point towards zero. Within this setting we distinguish two oppositional agent strategies: After a successful compromise balancing agents set their direction to the direction they came from. Curious agents set their direction after a successful compromise to the direction they moved to. They both set the absolute value of their direction to f max. With this setting we switch back to random pairwise communication. The corresponding pseudo-code can be found in the appendix B.2. Balancing and curious are essentially identical in the dynamics if µ =.5. This is because after a compromise of two balancing agents we have two agents with the same opinion searching in opposite directions. This is exactly the same with curious agents. (Only the indexes of the agents switch.) Thus, clustering outcomes are identical for balancing and curious agents with µ = Simulation Setup We run for the three settings (communication chain, balancing and curious agents) simulations for the values µ =.2,.5, ε =, +....,.35, n = 2 and f max =,, 2, 4, 8, 6, 32. For each point in this parameter space we have 3 independent simulation runs with random initial profiles and random selection of communication partners. The stopping criteria in each process is when we reach a configuration where all indirectly connected c subgroups of agents have a maximal opinion difference lower than ε and thus cannot split anymore into two clusters. The mean preserving property (see lemma in A.3) of the dynamics allows to calculate the long term limit of the convergence process. We regard the average size of the biggest cluster after stabilization (over all 3 runs) as the measure for the possibility of consensus. Simulations have been implemented in ANSI-C and the program code is available on request from the second author. c Two agents are connected if their opinions differ not more than ε.

10 Jan Lorenz, Diemo Urbig 4.4. Simulation Results Figure 4 shows the results. The fat line shows the average size of the biggest cluster when f max =. For balancing and curious agents this is exactly the same as in the basic model. The plateau at ε =.2 shows the characteristic polarization phase where agents form mainly two big clusters (see for instance []). For the communication chain we observe a slight difference from the basic model: We reach lower average biggest clusters for ε <.25. This is due to the fact that the initiative is passed from one communicating agent to another. The other lines show the effects of rising maximal frustration f max under the given setting of the strategy and µ. We omit the figure for the communication chain with µ =.2 because the differences to the µ =.5 case are too small. In figure 5 we show some example runs to give an impression how dynamics really look like because this is not visible in the aggregated data of figure 4. To give an impression we draw one ε-range in each part of the figure. We tried to choose interesting parameter settings for the example runs. To compare the example runs with the general picture of figure 4 we marked the parameter settings there with small circles. In the two plots at the top of figure 5 we see two runs without one of our strategies but with µ =.2 (which is very similar to µ =.5 disregarding convergence time). On the left hand side in the center we have a communication chain finding a consensus for a low ε. At the right hand side in the center we have curious agents which fail to find consensus even for a high ε due to a low frustration level and thus preventing consensus. On the bottom we see processes where agents reach consensus for low ε due to balancing (left hand side) and curious (right hand side). The curious agents need a high maximal frustration. We see that nearly each curious agent has crossed the central opinion before reaching the final consensual value. Notice the different time scales in each part of the figure. We summarize the main effects of the agent-based strategies in the interplay with eps and µ. Evidence can be extracted from figures 4 and 5. The communication chain enforces consensus even for relatively low f max, but a communication chain does not support consensus per se. It works only in conjunction with balancing strategies. Balancing agents have a positive effect on the chances for consensus. For µ =.5 this holds for all maximal frustrations f max >. The same holds trivially for curious agents under µ =.5. A lower µ (so more careful agents) supports the positive effects of balancing agents. While f max = does not enforce consensus under µ =.5 it does under µ =.2. A lower µ does not support the positive effects of curious agents. For µ =.2, f max =, 2, 4, 8 it even prevents consensus. Positive effects appear not until f max = 6. Better chances for consensus are paid by a longer convergence times. But

11 About the power to enforce and prevent consensus average size of biggest cluster chain, µ = average size of biggest cluster balancing/curious, µ = ε..2.3 ε balancing, µ =.2 curious, µ =.2 average size of biggest cluster average size of biggest cluster ε..2.3 ε Fig. 4. The average size of the biggest cluster for initial profiles with 2 agents for the communication chain and µ =.5, balancing agents with µ =.5 (which is the same for curious agents), balancing agents with µ =.2 and curious agents with µ =.2. good improvements can be made by only fairly rising convergence times. An interesting but small effect for balancing agents is that the general tendency that rising f max raises the average size of the biggest cluster is sometimes slightly undermined. E.g. for µ =.2 and f max = 6 we have a slightly higher average size of the biggest cluster as for f max = 32. This effect must be structurally and not caused by randomness due to our huge number of runs. But the effect is so low that we have not studied the reasons.

12 2 Jan Lorenz, Diemo Urbig ε =.25, µ =.2, f max = ε =.3, µ =.2, f max = ε.4 +ε.4 ε.2 ε.2 5 ε =.22, µ =.5, chain, f = 4 max 25 5 ε =.3, µ =.2, curious, f = 4 max.8.8 +ε.6.4 +ε ε.6.4 ε ε =.25, µ =.2, balancing, f = 4 max 5 ε =.25, µ =.2, curious, f = 6 max ε.6 +ε.4.2 ε.4.2 ε Fig. 5. Six example processes with n = Discussion and Conclusion Generally, continuous opinion dynamics under bounded confidence is driven by the opposition of the contractive force of averaging and the dividing force of bounded confidence. Dynamics start at the extremes of the opinion space. The whole opinion range contracts due to averaging. But higher densities of opinions evolves at both extremes. These two high density regions attract agents from the center and may lead to a split in the opinion range. Our analytical results shows the surprisingly huge ε-space where consensus is possible, which means that there is at least one specific communication regime that

13 About the power to enforce and prevent consensus 3 leads to consensus. This also shows the large impact that the control of communication has on consensus formation in the Weisbuch-Deffuant model. Based on our analytical results we derived some agent-based strategies that may reasonably be applied by individuals. All our proposed agent based strategies try to preventing a fast clustering. They all work with a slower is more consensual -effect. The communication chain does it by controlling the discussion by only one agent in each time step who has the initiative, which they try to pass through the whole opinion space. Balancing agents try to get input from both sides and thus prevent getting absorbed by a cluster to fast. Curious agents tend to break out of a cluster they recently reached to explore more of the opinion space. But the prevention of early clustering does not stop the overall contraction of the opinion profile and thus leads to better chances for consensus. The balancing and curious strategies have an interesting interplay with the cautiousness parameter µ. This is somewhat surprising because the parameter has been considered in [, 2] to have no effect on clustering in the basic model (only on convergence time). Our additional results now could be phrased: If you want your agents to foster consensus by balancing, you should appeal to them to be cautious. If you want them to foster consensus by being curious you should appeal to them to be not cautious, otherwise you may even get a negative effect when agents have low frustration maximum. So in general the impact of balancing is higher than for curious. But both individual strategies can work to foster consensus. Appendix A. Appendix for Proofs A.. Proof of Proposition Proof. The left hand side inequality is clear due to the fact that an ε < max i n x i () can obviously not bridge this maximal gap, thus the opinion profile will be divided into the two groups above and below this gap for ever regardless of any communication structure. i To show the right hand side let ε > max i n j= µj x i j (). We will show that the phone chain communication regime drives the dynamic to a consensus. The first thing to mention is that the phone chain communication regime can not change the order of the opinion profile. Thus it holds for all t N that x (t)... x n (t). In a first step we will look at the n first time steps, thus at the first phone chain round. After one round we will see that the maximal gap in x(n ) has shrunk substantially and we can conclude with an inductive argument. Let us consider that there is no bounded confidence restriction by ε, thus in every time step two opinions really change (if they are not already equal). We will derive equations for x in the time steps,..., n under this assumption. After that we will see that ε does not restrict this dynamic.

14 4 Jan Lorenz, Diemo Urbig Let i n be an arbitrary agent. We focus on x i the gap between i and i+ for all time steps, and will deduce formulae only containing values of the initial profile. Agent i at time step i has communicated recently with agent i and will communicate with agent i +. Thus x i (i 2) =... = x i () = x i (). (A.) Due to the communication with agent i, agent i moves towards i thus x i gets bigger. x i (i ) = x i (i 2) + µ x i (i 2) By recursion of (A.) and (A.2) it follows x i (i ) = x i () + µ x i () +... (A.2)... + µ i 2 x 2 () + µ i x () i = µ j x i j () (A.3) j= Going one step further to the communication of i and i + where their opinion gets closer. x i (i) = x i (i ) 2µ x i (i ) (A.4) (We use x i (i ) as an abbreviation for the right hand side of (A.3) which only contains expressions at time step.) The next step further x i is getting bigger again, as i + moves towards i + 2. x i (i + ) = x i (i) + µ x i+ (i) (A.4)(A.2) = ( 2µ) x i (i ) µ( x i+ (i ) + µ x i (i )) (A.) = µ x i+ () + ( 2µ + µ 2 ) x i (i ) To complete all time steps until t = n we have to mention x i (i + ) = x i (i + 2) =... = x i (n ). (A.5) (A.6) (For x n there is no equation (A.5) the last value after the phone chain round is computed by equation (A.4).) To make all these equations valid, thus to ensure that no opinion change is prevented by ε, it must hold for all i n that X i (i ) < ε. Looking at (A.3) makes clear that this is the case by construction of the lower bound of ε. From Equation (A.3), (A.5) and (A.6) we get x i (n ) = µ x i+ ()+ +( 2µ + µ 2 ) i ( j= µj x i j () µ + ( 2µ + µ 2 ) ) i j= µj max x() ( ) = µ + ( µ) 2 ( µ i ) µ max x() = ( µ i + µ i+ ) max x() (A.7)

15 About the power to enforce and prevent consensus 5 Thus it holds max x(n ) ( µ i + µ i+ ) max x(). It is easy to see k := µ i + µ i+ < for < µ <. For the next phone chain rounds we can conclude with the same procedure and it will hold max x(t(n )) k t max x(). Thus max x(t) converges to zero what implies that the process converges to a consensus. A.2. Proof of Corollary Proof. With abbreviation x := x() we use the equations range(x) = n i= x i to derive range(x) range(x) max (x) max x. This gives and, thus, we can derive n x i i= i ε low max µ j x i j i n j= rangex max (x) i= range(x) max (x) i= max (x) µ i max (x) = µ range(x) max x µ (A.8) max x (A.9) A.3. Proof of Proposition 2 Lemma. Let x() R n be an initial profile and (x(t)) t N be a WD process of opinion dynamics with arbitrary ε, µ. For every time step t N it holds n x i (t) = n x i (). (A.) n n i= Proof. Obvious by definition. Proof of propostion 2. Let ε ε high. Let us divide the set of agents according to the maximal k in equation (4) into two subsets I = {,..., k}, I 2 = {k +,..., n}. We choose a communication regime, where both subgroups find their consensuses x =... = x k = c, x k+ =... = x n = c 2. This should be possible, otherwise we are ready. Due to lemma and equation 4 it holds c c 2 ε and no communication is possible between the subgroups any more. i= Appendix B. Appendix Pseudo-Code for Strategies B.. Communication chain with balancing agents : initialize X[] 2: initialize D = 3: choose agent i 4: WHILE not clustered(x) AND changes possible

16 6 Jan Lorenz, Diemo Urbig 5: choose agent j 6: IF X[i]-X[j] <= epsilon 7: IF (X[j]-X[i])*D[i] >= 8: X[i]=X[i] - mu*(x[i]-x[j]) 9: X[j]=X[j] + mu*(x[i]-x[j]) : D = - sign(x[i]-x[j])*fmax : i = j 2: ELSE 3: IF D!= 4: D = D - sign(d) 5: ENDIF 6: ENDIF 7: ENDIF 8: ENDWHILE B.2. Direction vector and balancing/curious agents : initialize X[] 2: initialize D[] = (,,...,) 3: WHILE not clustered(x) AND changes possible 4: choose agent i,j 5: IF X[i]-X[j] <= epsilon 6: IF (X[j]-X[i])*D[i] >= AND (X[j]-X[i])*D[i] >= 7: X[i]=X[i] - mu*(x[i]-x[j]) 8: X[j]=X[j] + mu*(x[i]-x[j]) 9: D[i] = + sign(x[i]-x[j]) * fmax : D[j] = - sign(x[i]-x[j]) * fmax : ELSE 2: IF D[i]!= 3: D[i]= D[i] - sign(d[i]) 4: ENDIF 5: IF D[j]!= 6: D[j]= D[j] - sign(d[j]) 7: ENDIF 8: ENDIF 9: ENDIF 2: ENDWHILE For curious agents use same code with signs switched in lines nine and ten. References [] G. Deffuant, J. P. Nadal, F. Amblard, and G. Weisbuch. Mixing beliefs among interacting agents. Advances in Complex Systems, 3:87 98, 2.

17 About the power to enforce and prevent consensus 7 [2] Gérard Weisbuch, Guillaume Deffuant, Frédéric Amblard, and Jean-Pierre Nadal. Meet, Discuss and Segregate! Complexity, 7(3):55 63, 22. [3] Santo Fortunato. Universality of the threshold for complete consensus for the opinion dynamics of deffuant et al. International Journal of Modern Physics C, 5(9):3 37, 24. [4] Ulrich Krause. A discrete nonlinear and non-autonomous model of consensus formation. In S. Elyadi, G. Ladas, J. Popenda, and J. Rakowski, editors, Communications in Difference Equations, pages Gordon and Breach Pub., Amsterdam, 2. [5] Rainer Hegselmann and Ulrich Krause. Opinion dynamics and bounded confidence, Models, Analysis and Simulation. Journal of Artificial Societies and Social Simulation, 5(3), [6] G. Deffuant, D. Neau, F. Amblard, and G. Weisbuch. How can extremism prevail? A study based on the relative agreement interaction model. Journal of Artificial Societies and Social Simulation, 5(4), [7] M. H. DeGroot. Reaching a consensus. Journal of American Statistical Association, 69(345):8 2, 974. [8] Samprit Chatterjee and Eugene Seneta. Towards consensus: Some convergence theorems on repeated averaging. J. Appl. Prob., 4:59 64, 977. [9] E. Ben-Naim, S. Redner, and P.L. Krapivsky. Bifurcation and patterns in compromise processes. Physica D, 83:9 24, 23. [] Santo Fortunato, Vito Latora, Alessandro Pluchino, and Andrea Rapisarda. Vector opinion dynamics in a bounded confidence consensus model. To appear on International Journal of Modern Physics C, 25. [] Jan Lorenz. Consensus strikes back in the Hegselmann-Krause model of continuous opinion dynamics under bounded confidence. Journal of Artificial Societies and Social Simulation, 9, jlorenz/jasss/. [2] R. Hegselmann. Laws and Models in Science, chapter Opinion dynamics: Insights by radically simplifying models, page [3] Diemo Urbig and Jan Lorenz. Communication regimes in opinion dynamics: Changing the number of communicating agents. In Proceedings of the Second Conference of the European Social Simulation Association (ESSA), September 24. (availabe at and [4] Wander Jager and Frédéric Amblard. Uniformity, bipolarization and pluriformity captured as generic stylized behavior with an agent-based simulation model of attitude change. Computational & Mathematical Organization Theory, (4):295 33, 25. [5] F. Amblard and G. Deffuant. The role of network topology on extremism propagation with the relative agreement opinion dynamics. Physica A, 343: , 24. [6] Jan Lorenz. A stabilization theorem for dynamics of continuous opinions. Physica A, 355():27 223, 25. [7] Luc Devroye. Laws of the iterated logarithm for order statistics of uniform spacings. The Annals of Probability, 9(5):86 867, 98.