Asynchronous opinion dynamics on the k -nearest-neighbors graph
AAsynchronous opinion dynamics on the k -nearest-neighbors graph Wilbert Samuel Rossi and Paolo Frasca ∗† July 26, 2018
Abstract
This paper is about a new model of opinion dynamics with opinion-dependent connectivity. We assume that agents update their opinionsasynchronously and that each agent’s new opinion depends on the opinionsof the k agents that are closest to it. We show that the resulting dynamicsis substantially different from comparable models in the literature, suchas bounded-confidence models. We study the equilibria of the dynamics,observing that they are robust to perturbations caused by the introductionof new agents. We also prove that if the number of agents n is smaller than2 k , the dynamics converge to consensus. This condition is only sufficient. Driven by the evolution of digital communication, there is an increasing interestfor mathematical models of opinion dynamics in social networks. A few suchmodels have become popular in the control community, see the surveys [1, 2]. Inthe perspective of the control community, opinion dynamics distinguish them-selves from consensus dynamics because consensus is prevented by some otherdynamical feature. In many popular models, this feature is an opinion depen-dent limitation of the connectivity. This is the case of bounded confidence(BC) models [3, 4], where social agents influence each other iff their opinionsare closer than a threshold. This way of defining connectivity, however, seemsat odds with several social situations, since it may require an agent to be in-fluenced by an unbounded number of fellow agents. Instead, the number ofpossible interactions is capped in practice by the limited capability of attentionby the individuals. For instance, online social network services are based on ∗ W.S. Rossi and P. Frasca are with Department of Applied Mathematics, University ofTwente, 7500 AE Enschede, The Netherlands [email protected] . P. Frasca is withUniv. Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA-lab, F-38000 Grenoble, France [email protected] † This work has been partly supported by IDEX Universit´e Grenoble Alpes under C2S2“Strategic Research Initiative” grant. The authors also acknowledge the inspiring conversa-tions with J.M. Hendrickx and S. Martin. a r X i v : . [ c s . M A ] J u l ecommender systems that select a certain number of news items, those whichare closer to the user’s presumed tastes. However, to the best of our knowledge,this important observation has not been incorporated in any suitable model ofopinion dynamics, with the partial exception of [5]. The latter paper comparesdifferent models of interaction, including one in which each agent is influencedby a fixed number of neighbors.In a striking contrast, this observation has been made in the field of biologyby a number of quantitative studies about flocking in animal groups (theseinclude both theoretical and experimental works) [6, 7, 8, 9]. The importance ofthis way of defining connectivity has been also captured by graph theorists, whohave studied a the properties of what they call k -nearest-neighbors graph. Forinstance, it is known that k must be logarithmic in n to ensure connectivity [10]and flocking behavior [11].In this paper, we provide the first analysis of the k -nearest-neighbor opiniondynamics. In this analysis, our contribution is threefold: (1) We describe theequilibria of the dynamics, distinguishing a special type of clustered equilibria that are constituted of separate clusters; (2) We discuss the robustness of clus-tered equilibria to perturbations consisting in the addition of new agents; (3)We provide a proof of convergence for small groups, that is, groups such that n < k .Our work differs from [5] in several aspects. As per the model, the dynamicalmodel in [5] is synchronous and continuous-time, whereas ours is asynchronousand discrete-time. As per the analysis, [5] focuses on the equilibria and theirproperties (for instance, the distribution of their clusters’ sizes) are studied byextensive simulations, whereas we study the dynamical properties (robustnessto perturbations, convergence) by a mix of simulations and analytical results.Our robustness analysis is based on the approach taken by Blondel, Hendrickxand Tsitsiklis for BC models [12]. Our convergence result is inspired by classicalproofs of convergence for randomized consensus dynamics [13, Chapter 3], butits interest and difficulty originate from the lack of reciprocity in the interactions:this feature clearly distinguishes our model from bounded confidence models,where interactions are reciprocal as long as the interaction thresholds are equalfor all agents [3, 5, 14, 15, 16, 17]. Let n and k be two integers with 1 ≤ k ≤ n, and let V = { , . . . , n } be the set of agents. Each agent is endowed with a scalaropinion x i ∈ R , to be updated asynchronously. The update law x + = f ( x , i ) (1)goes as follows. An agent i is selected from V ; the elements of V are ordered byincreasing values of | x j − x i | ; then, the first k elements of the list (i.e. those withmallest distance from i ) form the set N i of current neighbors of i . Should a tiebetween two or more agents arise, priority is given to agents with lower index.Agent i may but not necessarily does belong to N i . Once N i is determined,agent i updates his opinion x i to x + i = 1 k (cid:88) j ∈ N i x j , while the remaining agents do not change their opinions x + j = x j for every j (cid:54) = i . We show a couple of simulations to illustrate the possible behaviours of themodel, see Figure 1 and 2. For these simulations we set n = 20, k = 5 and choosethe initial opinion of every agent uniformly at random in [0 , V the node that updates opinion, independently and uniformly atrandom. The simulation of Figure 1 shows a typical outcome: the agents formtwo distinct groups (of 10 agents each) with homogeneous opinions; for everyagent, his neighbors at time t = 1000 have almost the same opinion. This lastobservation does not hold in the simulation of Figure 2: the two pairs of agentsthat at time t = 1000 have opinion about 0.6 and about 0.7, respectively, haveneighbors with different opinions. These distinct behaviors lead us to distinguishdifferent kinds of equilibria: this will be the topic of the next section. In this section we discuss some properties of the equilibria of system (1). Mo-tivated by the simulations, we introduce the following terminology. Given aconfiguration x ∈ R n , the directed graph that represents the possible interac-tions (i.e. the opinion dependancies for any possible selection of the node to beupdated) is G ( x ) = ( V, E ( x )) with E ( x ) = (cid:91) i ∈ V { ( i, j ) , j ∈ N i } , where N i is the set of neighbors of i , should i be selected to update his opinion.Clearly, if k = n the graph G ( x ) = ( V, V × V ) is complete. A configuration x ∈ R n is an equilibrium for the asynchronous dynamics if x = f ( x , i ) for every i . If k = 1, then G ( x ) contains only links between nodes with the same opinion:in this trivial case, every configuration is an equilibrium because agents cannotchange opinion.A configuration x is called clustered if x N i = x i N i for every i , Figure 1: Simulation of the model (1) with n = 20, k = 50, initial opinionschosen uniformly at random in [0 ,
1] and update sequence chosen uniformly atrandom. The plot contains a typical trajectory that converges to a clusteredequilibrium. Figure 2: Simulation of the model (1) with n = 20, k = 50, initial opinionschosen uniformly at random in [0 ,
1] and update sequence chosen uniformlyat random. The plot contains a less common trajectory that converges to anon-clustered equilibrium.hat is, if for every node all of his neighbors have the same opinion. Furthermore,a clustered configuration x = c for some c ∈ R is called consensus .It is immediate to see that clustered configurations are equilibria. However,there exist equilibria that are not clustered. It is possible to obtain a simplecounterexample with n = 7 and k = 3 and exploiting the tie break rule. Considerany configuration x ∈ R of the form x { , , } = α { , , } , x { , , } = β { , , } , x = α + β , where α, β ∈ R and α < β . The above is an equilibrium even if x N = x { , , } (cid:54) = ( α + β ) { , , } .The tie breaking rule is not central for the existence of non-clustered equi-libria, as one can see in the following example inspired by Figure 2. Example 1.
Consider x ∈ R with x { , ,..., } = α { , ,..., } ,x = x = α +2 β ,x = x = α +3 β , x { , ,..., } = β { , ,..., } , where α, β ∈ R and α < β . For instance, the neighbors of agent 12 are N = { , , , , } because | x − x | = | x − x | = 0 , | x − x | = | x − x | = ( β − α ) , | x − x | = ( β − α ) , while the remaining agents are at distance ( β − α ) or larger. Such configurationis an equilibrium with x N (cid:54) = x N . A simple analysis shows that clustered configuration are those in which theagents form clusters of at least k participants with the same opinion. To makethis claim formal, let V i = { j : x j = x i } be the set of nodes that share the sameopinion of i . Lemma 1.
A configuration is clustered if and only if | V i | ≥ k for every i .Proof. By definition, in a clustered configuration N i ⊆ V i for every i . Assume | V i | ≥ k for every i . For any i there are at least k nodes j (including i ) with x j = x i : such nodes have zero distance from i and hence N i ⊆ V i . This holdsfor every i so the configuration is clustered. On the other hand, assume thatexists i with | V i | ≤ k −
1. The set N i must contain a node j with x j (cid:54) = x i sonot in V i , violating the definition of clustered configuration.From this result, it follows that a clustered configuration allows up to (cid:106) nk (cid:107) istinct sets V i (and this bound is tight). For the special case of consensus, thisclaim becomes the following corollary. Corollary 2.
Consensus is the only possible clustered configuration if and onlyif n < k . The clustered equilibria of the dynamics described above have interesting ro-bustness properties regarding the addition of new nodes or the removal of nodes.The model shows different behavior with respect to a standard AsynchronousBounded Confidence (ABC) model. In this section, we briefly introduce forcomparison the ABC model; then we provide a few simulations to motivate thefollowing discussion of the robustness properties.
Given a fixed range of confidence d >
0, we introduce the Asynchronous BoundedConfidence (ABC) update law x + = f ABC ( x , i ) . (2)where i is the agent that updates his opinion. The neighborhood of i is N ABCi = { j : | x j − x i | ≤ d } and always contains i itself. The new opinion of agent i is x + i = 1 | N ABCi | (cid:88) j ∈ N ABCi x j , while the remaining agents do do not change opinion x + j = x j for every j (cid:54) = i . We present a simulation to show the difference between model (1) and model (2)when a few agents are added to a consensus configuration (which is an equi-librium for both models). We set k = 5 for model (1) and d = 0 .
25 for model(2). We start with 10 agents sharing opinion 0 .
4; at steps t = 2 , , , , Figure 3: The addition of four new nodes to a consensus configuration withten nodes. Upper plot: the trajectory of the model (1) with k = 5. Lower plot:the trajectory of the model (2) with d = 0 .
25. The same initial conditions andupdate order are used.he agent added at step t = 3 remains isolated during the dynamics and keepshis opinion. The other three new agents join the original ten; this group of 13agents converge to the same opinion which however is different from the originalconsensus value. We now provide a general discussion that explains the observations from Fig-ure 3. Let n, k with 1 ≤ k ≤ n be given and consider a clustered equilibria x ∈ R n of the model (1). We first discuss the addition of a new agent with opin-ion x n +1 = α to the configuration x , that becomes [ x ; α ] ∈ R n (cid:48) with n (cid:48) = n + 1.Before the addition of the new node, clusters have to contain at least k agents.This fact remains true after the addition and we have that f ([ x ; α ] , i ) = [ x ; α (cid:48) ]for every i , meaning that the original (clustered) portion of the configuration[ x ; α ] remains unperturbed. For a generic value of α the limit of the dynamicshas the same cluster locations of x , with one of the clusters getting a newmember. For some specific values, it may happen that the configuration [ x ; α ]is a non-clustered equilibrium. In any case, none of the original agents changesopinion. Instead, in the metric ABC model (2) with uniform visibility radius d , either the new agent is further apart from the original agents and nothinghappens or he falls within the visibility radius of a cluster of agents. In thelatter case both the new agents and the agents in the cluster change opinions,converging to an intermediate value.Assuming n sufficiently large, the removal of an agent from a clustered equi-librium presents interesting differences too. In the metric ABC model (2) theremoval of an agent does not trigger any dynamics in the remaining agents. Inmodel (1), if the agent is removed from a cluster with k + 1 agents or more,nothing happens. But if the agent is removed from a cluster with k agents, thenew configuration is not an equilibrium anymore and the remaining nodes fromthat group will evolve towards some new equilibrium. In this section we show that process (1) converges to a consensus, provided n < k and the choice of the agent that updates his opinion at time t is an i.i.d.uniform random variable over V . We recall from Section 3 that the consensusis the unique clustered equilibrium for n < k .For t ≥
0, let x ( t ) ∈ R n be the sequence of opinion vectors and I ( t ) ∈ V asequence of agents. Given an initial configuration x (0) = x , we consider thedynamics x ( t + 1) = f ( x ( t ) , I ( t )) for every t ≥ , (3)where I ( t ) is the agent that updates his opinion at time t .e introduce two functions µ, M : R n → V that, given an opinion vector x ,return respectively the index of the smallest and largest components, with tiessorted µ ( x ) = min(arg min i x i ) , M ( x ) = min(arg max i x i ) . The outer min sorts possible ties; note that M ( x ) = µ ( − x ).In the following two lemmas we prove the properties of the dynamics inwhich the agent with smallest opinion is the one that updates his opinion. Lemma 3.
Given n, k with ≤ k ≤ n and an initial configuration x ∈ R n consider dynamics (3) with I ( t ) = µ ( x ( t )) and the scalar sequence y ( t ) :=max i ∈ N µ ( x ( t )) x i ( t ) . Then: • the set sequence N µ ( x ( t )) and the scalar sequence y ( t ) are constant; • for every i ∈ N µ ( x (0)) the sequences x i ( t ) are non-decreasing and satisfy x i ( t ) ≤ y (0) ; • for every i / ∈ N µ ( x (0)) the sequences x i ( t ) are constant.Proof. The proof goes by induction. First, consider the trivial case with x µ ( x ( t )) ( t ) = y ( t ). This condition means x i ( t ) = y ( t ) for every i ∈ N µ ( x ( t )) and thus x µ ( x ( t )) ( t +1) = x µ ( x ( t )) ( t ) so everything remains unchanged.Next, consider the case with x µ ( x ( t )) ( t ) < y ( t ). We have x µ ( x ( t )) ( t + 1) = 1 k (cid:88) j ∈ N µ ( x ( t )) x j ( t ) ∈ (cid:0) x µ ( x ( t )) ( t ) , y ( t ) (cid:1) . Therefore, { i : x i ( t ) < y ( t ) } = { i : x i ( t + 1) < y ( t ) } and { i : x i ( t ) = y ( t ) } = { i : x i ( t + 1) = y ( t ) } . Moreover, the cardinality of the set { i : x i ( t ) < y ( t ) } is strictly smaller than k . This implies that N µ ( x ( t +1)) = N µ ( x ( t )) and also y ( t + 1) = y ( t ). Theclaims follow by induction and by observing that only the agents i ∈ N µ ( x (0)) can update their opinions at some time t ≥ x i ( t + 1)belongs to [ x i ( t ) , y ( t )]. emma 4. Given n, k with ≤ k ≤ n and an initial configuration x ∈ R n consider the dynamics (3) with I ( t ) = µ ( x ( t )) and the scalar sequence y ( t ) =max i ∈ N µ ( x ( t )) x i ( t ) . Then y ( k − − min i x i ( k − ≤ (cid:0) − k (cid:1)(cid:0) y (0) − min i x i (0) (cid:1) Proof.
First, compute x µ ( x ( t )) ( t + 1) for a generic t ≥
0. We have x µ ( x ( t )) ( t + 1) = k (cid:80) j ∈ N µ ( x ( t )) x j ( t )= k (cid:80) j ∈ N µ ( x (0)) x j ( t ) ≥ k (cid:80) j ∈ N µ ( x (0)) x j (0)thanks to Lemma 3. Then, x µ ( x ( t )) ( t + 1) ≥ k − k x µ ( x (0)) (0) + k y (0)= x µ ( x (0)) (0) + k (cid:0) y (0) − x µ ( x (0)) (0) (cid:1) . Next, consider the set S ( t ) = (cid:8) i : x i ( t ) < x µ ( x (0)) (0) + k (cid:0) y (0) − x µ ( x (0)) (0) (cid:1)(cid:9) , and observe that either S ( t ) = ∅ or | S ( t + 1) | = | S ( t ) | − µ ( x ( t )) / ∈ S ( t + 1). Since the set S (0) contains at most k − S ( k −
1) isempty. Hence, x i ( k − ≥ x µ ( x (0)) (0) + k (cid:0) y (0) − x µ ( x (0)) (0) (cid:1) for every i , a fact that implies x µ ( x ( k − ( k − ≥ x µ ( x (0)) (0) + k (cid:0) y (0) − x µ ( x (0)) (0) (cid:1) . Using Lemma 3 we know that N µ ( x ( t )) = N µ ( x (0)) for every t ≥ i therein, x i ( t ) ≤ y ( t ) = y (0). Therefore y ( k − − x µ ( x ( k − ( k − ≤ y (0) − x µ ( x (0)) (0) − k (cid:0) y (0) − x µ ( x (0)) (0) (cid:1) and the thesis follows because x µ ( x ( t )) = min i x i ( t ).The following lemma follows from Lemma 3 and 4 using the property M ( x ) = µ ( − x ). emma 5. Given n, k with ≤ k ≤ n and an initial configuration x ∈ R n consider the dynamics (3) with I ( t ) = M ( x ( t )) and the scalar sequence z ( t ) :=min i ∈ N M ( x ( t )) x i ( t ) . Then: • the set sequence N M ( x ( t )) and the scalar sequence z ( t ) are constant; • for every i ∈ N M ( x (0)) the sequences x i ( t ) are non-increasing and satisfy x i ( t ) ≥ z (0) ; • for every i / ∈ N M ( x (0)) the sequences x i ( t ) are constant.Moreover, max i x ( k − − z ( k − ≤ (cid:0) − k (cid:1)(cid:0) max i x i (0) − z (0) (cid:1) . The next equivalence will be crucial in the following.
Lemma 6.
Given n, k with ≤ k ≤ n , consider x ∈ R n and define the quanti-ties y := max i ∈ N µ ( x ) x i and z := min i ∈ N M ( x ) x i . Then, z ≤ y for every x ∈ R n if and only if n < k .Proof. We prove the equivalent claim that x ∈ R n with z > y exists if and onlyif n ≥ k . Indeed, if n ≥ k consider the vector x ∈ R n such that x ≤ x ≤ . . . ≤ x k < x k +1 ≤ . . . ≤ x n − k +1 ≤ . . . ≤ x n where n − k + 1 > k . The set N µ ( x ) contains the k smallest elements of x so y = x k , while the set N M ( x ) contains the k largest elements of x , so z = x n − k +1 > x k = y . For the converse, assume that x with z > y exists, meaning (cid:0) max i ∈ N µ ( x ) x i (cid:1) < (cid:0) min i ∈ N M ( x ) x i (cid:1) . Both sets N µ ( x ) and N M ( x ) contain k elements, so the sets { j : x j ≤ max i ∈ N µ ( x ) x i } and { j : x j ≥ min i ∈ N M ( x ) x i } contain at least k elements each. These two sets are disjoint, thus the vector x ∈ R n has at least n ≥ k components.The next lemma describes a “shrinking sequence”. emma 7. Given n, k with ≤ k ≤ n and an initial configuration x ∈ R n consider the dynamics (3) with I ( t ) = (cid:40) µ ( x ( t )) for t ∈ { , . . . , k − } M ( x ( t )) for t ∈ { k − , . . . , k − } If n < k then max i x i ( T ) − min i x i ( T ) ≤ (cid:0) − k (cid:1)(cid:0) max i x i (0) − min i x i (0) (cid:1) where T = 2 k − .Proof. For the sake of compactness, we set α ( t ) := min i x i ( t ) , β ( t ) := max i x i ( t ) , γ := (cid:0) − k (cid:1) , introduce the two sequences y ( t ) := max i ∈ N µ ( x ( t )) x i ( t ) and z ( t ) := min i ∈ N M ( x ( t )) x i ( t ) , and set R = k −
1. We have β ( T ) − α ( T ) = β ( T ) − z ( T ) + z ( T ) − α ( T ) ≤ γ (cid:0) β ( R ) − z ( R ) (cid:1) + z ( R ) − α ( R )using Lemma 5 with initial configuration x ( R ). Then= γ (cid:0) β ( R ) − y ( R ) (cid:1) + γ (cid:0) y ( R ) − z ( R ) (cid:1) + z ( R ) − α ( R ) ≤ γ (cid:0) β ( R ) − y ( R ) (cid:1) + (cid:0) y ( R ) − z ( R ) (cid:1) + z ( R ) − α ( R )since γ < y ( R ) − z ( R ) ≥ n < k by Lemma 6. Then= γ (cid:0) β ( R ) − y ( R ) (cid:1) + y ( R ) − α ( R ) ≤ γ (cid:0) β (0) − y (0) (cid:1) + γ (cid:0) y (0) − α (0) (cid:1) = γ (cid:0) β (0) − α (0) (cid:1) using Lemma 3 and 4 with initial configuration x (0). We have finally obtained β ( T ) − α ( T ) ≤ γ (cid:0) β (0) − α (0) (cid:1) .f n < k and the agent I ( t ) that updates his opinion at time t is chosenindependently and uniformly at random over V , then process (3) convergesalmost surely to a consensus, from any initial configuration. The almost sureconvergence is guaranteed because the finite sequence of updates introduced inthe Lemma 7 appears infinitely often with probability one. This fact is provedin the following theorem, which provides the desired converge result. Theorem 8.
Let n, k with ≤ k ≤ n be given. Let { I ( t ) , t ≥ } be a sequenceof independent and uniformly distributed random variables over { , . . . , n } andconsider dynamics (3) . If n < k , then lim t →∞ x ( t ) = c almost surelyfor any x ∈ R n , with c ∈ [min i ( x i ) , max i ( x i )] .Proof. Let δ ( t ) = max i x i ( t ) − min i x i ( t ) and observe that, for any x (0) = x and { I ( t ) , t ≥ } , δ (0) ≥ ≤ δ ( t + 1) ≤ δ ( t ) for every t ≥ , because the updates in the dynamics (3), based on model (1), involve convexcombinations: the element with highest opinion cannot increase it and the el-ement with lowest opinion cannot decrease it. We introduce the sequence ofevents { A t , t ≥ k − } with A t = (cid:8) I ( s ) = µ ( x ( s )) for s ∈ { t − k +3 , . . . , t − k +1 } and I ( s ) = M ( x ( s )) for s ∈ { t − k +2 , . . . , t } (cid:9) , i.e. the event A t is the occurrence of the finite sequence introduced in Lemma 7in the time window { t − (2 k − , . . . , t } . In the same lemma we proved that,given the occurrence of A t , we have δ ( t +1) ≤ (1 − k ) δ ( t − k +3). Observe that0 ≤ lim t →∞ δ ( t ) ≤ lim t →∞ (cid:0) − k (cid:1) n t δ (0)where n t is the number of times A t occurred up to time t . If P ( A t infinitely often) =1 then n t → ∞ for t → ∞ and the rightmost limit above is zero almost surely.Hence, lim t →∞ δ ( t ) almost surely, which implies the convergence to consensus.Moreover, c ∈ [min i ( x i ) , max i ( x i )] because every update in (3) is a convexcombination of a subset of the current opinions.t remains to prove P ( A t infinitely often) = 1. The events of the sequence { A t , t ≥ k − } are not independent but the events in the subsequence { A t h , h ≥ } where t h = h (2 k − − P ( A t h ) = (cid:18) n (cid:19) k − , thus (cid:80) ∞ h =1 P ( A t h ) = ∞ . Hence, { A t i.o. } ⊃ { A t h i.o. } . From the second Borel-Cantelli lemma [18, Ch. 2, Thm 18.2] P ( A t infinitely often) ≥ P ( A t h infinitely often) =1 . The result continues to hold for dynamics where I ( t ) is not uniformly dis-tributed over { , . . . , n } , as long as the probability to sample each agent isconstant and positive. The proof has been based on exhibiting one suitable“shrinking sequence”: however, it is clear that plenty of other sequences coulddo the job and actually play a role in inducing convergence of the dynamics.Therefore, the proof does not imply any good estimate of the convergence time. In this paper we have introduced a new model of opinion dynamics with opinion-dependent connectivity following the k -nearest-neighbors graph. The model ismotivated by the rise of online social network services, where recommendersystems select a certain number of news items to present to users, reducing thenumber of possible interactions to those which are closer to the user’s presumedtastes. The resulting dynamics is substantially different from comparable modelsin the literature, such as bounded-confidence models. One key difference is theinherent lack of reciprocity of the interactions, which makes all convergenceanalysis challenging. Another key difference is the robustness of the formedclusters, whose opinions are hard to sway by external leader nodes. This featuremakes control approaches based on leadership, like [19], unsuitable to k -nearest-neighbors dynamics. References [1] A. V. Proskurnikov and R. Tempo, “A tutorial on modeling and analysisof dynamic social networks. Part I,”
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