Biased Opinion Dynamics: When the Devil Is in the Details
Aris Anagnostopoulos, Luca Becchetti, Emilio Cruciani, Francesco Pasquale, Sara Rizzo
BBiased Opinion Dynamics:When the Devil Is in the Details
Aris Anagnostopoulos
Sapienza Universit`a di RomaRome, Italy [email protected]
Luca Becchetti
Sapienza Universit`a di RomaRome, Italy [email protected]
Emilio Cruciani
Inria, I3S Lab, UCA, CNRSSophia Antipolis, France [email protected]
Francesco Pasquale
Universit`a di Roma Tor VergataRome, Italy [email protected]
Sara Rizzo
Gran Sasso Science InstituteL’Aquila, Italy [email protected]
Abstract
We investigate opinion dynamics in multi-agent networks when a bias toward one of twopossible opinions exists; for example, reflecting a status quo vs a superior alternative.Starting with all agents sharing an initial opinion representing the status quo, the systemevolves in steps. In each step, one agent selected uniformly at random adopts the superioropinion with some probability α , and with probability 1 − α it follows an underlying updaterule to revise its opinion on the basis of those held by its neighbors. We analyze convergenceof the resulting process under two well-known update rules, namely majority and voter .The framework we propose exhibits a rich structure, with a non-obvious interplay be-tween topology and underlying update rule. For example, for the voter rule we show that thespeed of convergence bears no significant dependence on the underlying topology, whereasthe picture changes completely under the majority rule, where network density negativelyaffects convergence.We believe that the model we propose is at the same time simple, rich, and modular,affording mathematical characterization of the interplay between bias, underlying opiniondynamics, and social structure in a unified setting. a r X i v : . [ c s . M A ] A ug Introduction
Opinion formation in social groups has been the focus of extensive research. While many modelsconsidered in the literature confer the same intrinsic value to all opinions [CHK18], one mightexpect a group to quickly reach consensus on a clearly “superior” alternative, if present. Yet,phenomena such as groupthink may delay or even prevent such an outcome.In this perspective, we investigate models of opinion formation in which a bias towards oneof two possible opinions exists, for instance, reflecting intrinsic superiority of one alternativeover the other. In the remainder, we use labels 0 and 1 for the two opinions and we assume 1is the dominant opinion, that is, the one towards which the agents have a bias. We investigatethis question in a mathematically tractable setting, informally described as follows.Assume some underlying opinion dynamics D . Starting from an initial state in which allagents share opinion 0, the system evolves in rounds. In each round, one agent is selecteduniformly at random. With some probability α , the agent adopts 1, while with probability1 − α , the agent follows D to revise its opinion on the basis of those held by its neighbors in anunderlying network.Although the general model we consider is simple and, under mild conditions on D , thefamily of processes it describes always admits global adoption of opinion 1 as the only absorbingstate, convergence to this absorbing state exhibits a rich variety of behaviors, which in non-obvious ways depends on the interplay between the network structure and the underlying opiniondynamics. The relatively simple, yet general, model we consider allows analytical investigationof the following question:How does a particular combination of network structure and opinion dynamics af-fects convergence to global adoption of the dominant opinion? In particular, howconducive is a particular combination to rapid adoption? Main findings.
In general, the interplay between underlying network structure and opiniondynamics may elicit quite different collective behaviors.In Section 3, we show that the expected time for consensus on the dominant opinion growsexponentially with the minimum degree under the majority update rule, in which agents updatetheir opinion to the majority opinion in their neighborhoods [KR03]. Using asymptotic notationand denoting the number of agents in the network by n , we obtain that convergence time issuper-polynomial in expectation whenever the minimum degree is ω (log n ). One might wonder,if the converse occurs, namely, if a logarithmic maximum degree affords (expected) polynomialconvergence to the absorbing state. Even though we prove that this is indeed the case forspecific topologies as cycles or restricted graph families, this does not seem to hold in general(see discussion in Section 5).The results for majority are at odds with those we obtain in Section 4 for the voter model ,where agents copy the opinion of a randomly selected neighbor [Lig12]. In this case, convergenceto the absorbing state occurs within O (cid:0) α n log n (cid:1) rounds with high probability , regardless of theunderlying network structure. We emphasize that convergence time remains O (cid:0) n s log n (cid:1) when α = Θ (cid:0) n s (cid:1) for any s > Characterizing the notion of “superiority” is typically context-dependent and may be far from obvious. Weremark that this aspect is outside the scope of this paper.
2t a higher level, we provide a simple mathematical framework to investigate the interplaybetween opinion dynamics and underlying network structure in a unified setting, allowing com-parison of different update rules with respect to a common framework. In this respect, we hopethat our work moves in the direction of a shared framework to investigate opinion dynamics, asadvocated in [CHK18].
The problem we consider touches a number of areas where similar settings have been considered,with various motivations. The corresponding literature is vast and providing an exhaustivereview is unfeasible here. In the paragraphs that follow, we discuss contributions that mostclosely relate to the topic of this paper.
Opinion diffusion and consensus.
Opinion dynamics are widely used to investigate howgroup of agents modify their beliefs under the influence of other agents and possibly exogenousfactors. A number of models have been proposed in the more or less recent past, mostlymotivated by phenomena that arise in several areas, ranging from social sciences, to physicsand biology. The reader is referred to [CHK18] and references therein for a recent, generaloverview of opinion dynamics in multi-agent systems. A first distinction is between settings inwhich the set of possible beliefs is continuos, e.g., the interval [0 , majority rule and the voter model . Investigation of the majority update rule originatesfrom the study of agreement phenomena in spin systems [KR03], while the voter model wasmotivated by the study of spatial conflict between species in biology and interacting stochasticprocesses/particle systems in probability theory and statistics [CS73, HL +
75, Lig12]. Thesetwo models received renewed attention in the recent past, the focus mostly being on the timeto achieve consensus and/or conditions under which consensus on one of the initial opinions isachieved with a minimum degree of confidence. The voter model is by now well understood.In particular, increasingly tight bounds on convergence time for general and specific topologieshave been proposed over the recent past [HP99, CEOR13], while it is known that the probabilityof one particular opinion to prevail is proportional to the sum of the degrees of nodes holdingthat opinion at the onset of the process [DW83].
Consensus and network structure.
Network structure has been known to play an impor-tant role in opinion diffusion and influence spreading for quite some time [Mor00], under a varietyof models. For example, consensus under the voter model and dependence of its convergence onthe underlying network topology have been thoroughly investigated [DW83, HP99, CEOR13].For majority dynamics, [ACF +
15] characterized topologies for which an initial majority can besubverted, showing that this is possible for all but a handful of topologies, including cliques andquasi-cliques. On the other hand, regardless of the network, there is always an initial opiniondistribution, such that the final majority will reflect the initial one, while computing an initialopinion configuration that will subvert an initial majority is topology-dependent and NP-hardin general [AFG18].A number of recent contributions investigated (among other aspects) the relationship be-tween network structure and consensus in opinion formation games [FGV16, FV17], while ex-tensions of the Friedkin-Johnsen model to evolving networks were investigated in [AFF19].While expansion of the underlying graph typically accelerates convergence [CEOR12, KMTS19]in many opinion dynamics, some recent work explicitly points to potentially adverse effects ofnetwork structure on the spread of innovation, at least in scenarios where opinion update occurs3n the basis of private utilities that reflect both the degree of local consensus and intrinsic valueof the competing opinions [MS10, You11].While some of our findings are qualitatively consistent with previous work albeit undercompletely different models (in particular, [MS10]), our overall approach is very different, since itcompletely decouples the mechanism of opinion formation from modelling of the bias, affordinga clear-cut mathematical characterization of the interplay between bias, underlying opiniondynamics and network structure.
Different forms of bias.
Bias in opinion dynamics has been considered previously in theliterature. We briefly review contributions that are at least loosely related to our framework. Forthe voter and majority update rules, [MMR16] introduces bias in the form of different, opinion-dependent firing rate frequencies of the Poisson clocks that trigger agents’ opinion updates,implicitly enforcing a bias toward the opinion with lower associated rate. While different, theirmodel is similar to ours in spirit and some of their results for the voter model are consistentwith ours. Yet, these results only apply in expectation and to very dense networks with degreeΩ( n ), whereas our results for the voter model hold for every undirected graph.A somewhat related line of research addresses the presence of stubborn agents or zealots.Loosely speaking, stubborn agents have a bias toward some (initially or currently) held opinion,while zealots are agents that never deflect from some initial opinion. Restricting to the discrete-opinion setting, which is the focus of this paper, the role of zealots and their ability to subvert aninitial majority have been investigated for the voter model (see [Mob03] and follow-up work),while [ACF +
17] investigates majority dynamics in the presence of stubborn agents that arebiased toward the currently held opinion, providing a full characterization of conditions underwhich an initial majority can be subverted.
Let G = ( V, E ) be an undirected graph with | V | = n nodes, each representing an agent. Withoutloss of generality, we assume that V = [ n ] := { , . . . , n } . The system evolves in discrete timesteps and, at any given time t ∈ N , each node v ∈ V holds an opinion x ( t ) v ∈ { , } . Weuse the term opinion liberally here, in the sense that 0 and 1 in general represent competingalternatives, whose meaning is context-dependent and outside the scope of this paper. Wedenote by x ( t ) = (cid:0) x ( t )1 , . . . , x ( t ) n (cid:1) (cid:124) the corresponding state of the system at time t . We assumethat the initial state of the system is x (0) = = (0 , . . . , (cid:124) ; such assumption is discussed inSection 5. For each v ∈ V , we denote the neighborhood of v with N v := { u ∈ V : { u, v } ∈ E } and the degree of v with d v := | N v | . Finally, ∆ := min v ∈ V d v is the minimum degree of thenodes in G .Our framework assumes that agents exhibit a bias toward one of the opinions (e.g., reflectingintrinsic superiority of a technological innovation over the status quo ), without loss of generality1, which we henceforth call the dominant opinion . We model bias as a probability, with aparameter α ∈ (0 , Markovian , that is, given the underlyinggraph G , the distribution of the state x ( t ) at round t only depends on the state x ( t − at theend of the previous round. Moreover, they have x = = (1 , . . . , (cid:124) as the only absorbing state.We use τ to denote the absorption time , which is the number of rounds for the process to reachthe absorbing state . Finally, for a family of events {E n } n ∈ N we say that E n occurs with highprobability ( w.h.p. , in short) if a constant γ > P ( E n ) = 1 − O ( n − γ ), for everysufficiently large n . For the continuous case, there is a vast literature; see the seminal paper [FJ90] and follow-up work. This is equivalent to the asynchronous model in which a node revises its opinion at the arrival of an inde-pendent Poisson clock with rate 1 [BGPS06]. Absorption Time for Majority Dynamics
In this section, we investigate the time to reach consensus on the dominant opinion under themajority update rule. More formally, we study the following random process: Starting from theinitial state x (0) = (0 , . . . , (cid:124) , in each round t a node u ∈ [ n ] is chosen uniformly at randomand u updates its value according to the rule x ( t ) u = (cid:26) α,M G ( u, x ) with probability 1 − α, where α ∈ (0 ,
1] is the bias toward the dominant opinion 1 and M G ( u, x ) is the value held inconfiguration x ( t − = x by the majority of the neighbors of node u in graph G : M G ( u, x ) = (cid:26) (cid:80) v ∈ N u x v < | N u | / , (cid:80) v ∈ N u x v > | N u | / , and ties are broken uniformly at random, that is, if (cid:80) v ∈ N u x v = | N u | / M G ( u, x ) = 0 or1 with probability 1 / α , the above Markov chain has as the onlyabsorbing state. However, the rate of convergence is strongly influenced by the underlying graph G . In Subsection 3.1 we prove a lower bound on the expected absorption time that dependsexponentially on the minimum degree. This result implies super-polynomial expected absorptiontimes for graphs whose minimum degree is ω (log n ). On the other hand, in Subsection 3.2 weprove that the absorption time is O ( n log n ) on cycle graphs, and further graph families with sub-logarithmic maximum degree and polynomial (expected) absorption time are briefly discussedin Subsection 3.3. In this section we prove a general lower bound on the expected absorption time, which onlydepends on the minimum degree ∆. To this purpose, we use the following standard lemma onbirth-and-death chains (see, e.g., [LP17, Section 17.3] for a proof). Lemma 3.1.
Let { X t } t be a birth-and-death chain with state space { , , . . . , n } such that forevery (cid:54) k (cid:54) n − P ( X t +1 = k + 1 | X t = k ) = p, P ( X t +1 = k − | X t = k ) = q, P ( X t +1 = k | X t = k ) = r, with p + q + r = 1 . For every i ∈ { , , . . . , n } let τ i be the first time the chain hits state i , thatis, τ i = inf { t | X t = i } . If < p < / , the probability that starting from state k the chain hitsstate n before state is P k ( τ n < τ ) = ( q/p ) k − q/p ) n − (cid:54) (cid:18) pq (cid:19) n − k . It is not difficult to show that, for α (cid:62) /
2, every graph with minimum degree ∆ = Ω(log n )has O ( n log n ) absorption time, w.h.p. Indeed, since every time a node updates its opinion thenode chooses opinion 1 with probability at least α , as soon as all nodes update their opinionat least once (it happens within O ( n log n ) time steps, w.h.p., by coupon collector argument) if α (cid:62) /
2, every node u will have a majority of 1s in its neighborhood, w.h.p.In the next theorem we prove that, as soon as α is smaller than 1 /
2, the absorption timeinstead becomes exponential in the minimum degree. Birth-and-death chains are Markov processes for which, if in state k , a transition could only go to eitherstate k + 1 or state k − heorem 3.2. Let G = ( V, E ) be an undirected graph with minimum degree ∆ . Assume α (cid:54) (1 − ε )2 , for an arbitrary constant < ε < . The expected absorption time for the biasedopinion dynamics under the majority update rule is E [ τ ] (cid:62) e ε ∆ n . Proof.
Let S ( t ) be the set of nodes with value 1 at time t . For each node u ∈ V , let n ( t ) u be thefraction of its neighbors with value 1 at round t : n ( t ) u = | N u ∩ S ( t ) || N u | . Finally, let ¯ τ be the first round in which n ( t ) u (cid:62) / v ∈ V , namely,¯ τ = inf (cid:110) t ∈ N : n ( t ) u (cid:62) / , for some u ∈ [ n ] (cid:111) . Note that for each round t (cid:54) ¯ τ all nodes have a majority of neighbors sharing opinion 0, thusthe selected agent at time t updates its state to 1 with probability α and to 0 with probability1 − α . Moreover, clearly τ (cid:62) ¯ τ . We next prove that E [¯ τ ] (cid:62) e ε ∆ / (6 n ), which implies our thesis.Observe that, for a node u with degree d u that has k neighbors with value 1 in some roundand for every t (cid:54) ¯ τ , the probabilities p k ( u ) and q k ( u ) of increasing and decreasing, respectively,of one unit the number of its neighbors with value 1 are p k ( u ) = d u − kn α, and q k ( u ) = kn (1 − α ) . Hence, because α (cid:54) (1 − ε ) /
2, for every k (cid:62) d u / (2 + ε ) we have that p k ( u ) q k ( u ) = d u − kk · α − α (cid:54) (1 + ε ) · − ε ε = 1 − ε. Note that d u − d u ε = d u ε ε ) (cid:62) ε d u . From Lemma 3.1 it thus follows that, for each node u , as soon as the number of its neighborswith value 1 enters in the range ( d u / (2 + ε ) , d u / d u / d u / (2 + ε ) is at most(1 − ε ) εd u / (cid:54) e − ε d u / (cid:54) e − ε ∆ , using (1 − x ) x (cid:54) e − x for x ∈ [0 , Y u the random variable indicatingthe number of trials before having at least 1 / u at 1 we have that for every t (cid:62) P ( Y u (cid:62) t ) (cid:62) (cid:18) − e − ε ∆ (cid:19) t (cid:62) e − (3 t/ e − ε
26 ∆ , where in the last inequality we used that 1 − x (cid:62) e − x/ for every x ∈ [0 , ). Thus, P ( Y u < t ) (cid:54) − e − (3 t/ e − ε
26 ∆ (cid:54) t e − ε ∆ , using 1 − e − x (cid:54) x for every x . Finally, by using the union bound over all nodes, we have that P (¯ τ < t ) = P ( ∃ u ∈ [ n ] : Y u < t ) (cid:54) n · t e − ε ∆ . Thus, for ¯ t = e ε ∆ / n we have P (¯ τ (cid:54) ¯ t ) (cid:54) / E [¯ τ ] (cid:62) ¯ t P (¯ τ (cid:62) ¯ t ) (cid:62) ¯ t . .2 Fast Convergence on the Cycle In this section, we prove that the absorption time on an n -node cycle graph is O ( α n log n ),w.h.p. We make use of the following structural lemma. Lemma 3.3 (Structural property of cycles) . Let C n be the cycle on n nodes and let every node v ∈ V have an associated state x v ∈ { , } . Let us call B i and S i the set of nodes in state i such that: every node v ∈ B i has both neighbors in the opposite state and every node v ∈ S i hasone single neighbor in the opposite state. The following holds: | B | + | S | | B | + | S | . Proof.
Given any possible binary coloring of C n each node v belongs to one of the followingcategories: • v ∈ B i : node v is in state i and both its neighbors are in state j (cid:54) = i . • v ∈ R i : node v is in state i , its left neighbor is in state i , and its right neighbor is in state j (cid:54) = i . • v ∈ L i : node v is in state i , its right neighbor is in state i , and its left neighbor is in state j (cid:54) = i . • v ∈ Z i : node v is in state i and zero of its neighbors are in state j (cid:54) = i , i.e., both are instate i .We also call S i = R i ∪ L i . Figure 1 illustrates the eight (counting symmetries) possible categories.Let us consider a clockwise walk through C n that returns to its starting point. Keeping into Figure 1: Categories of a node v in C n ; node v is black while its left and right neighbors are white. account the categories of the nodes previously described it is possible to generate a graph H C that describes all possible binary configurations of a C n graph, for every n ∈ N . We call H C the Cycle Binary Configuration Graph (Figure 2). The nodes of H C represent the possible categoriesof the nodes of C n while the edges the possible neighbors in C n , considering a clockwise walk.For example, there is no edge from B to R since the neighbors of B are both in state 1, whilea node in R is in state 0.Let us pick any node v in C n and let us walk through clockwise until we return to v . Let uspick the node of H C corresponding to the category v belongs to and follow the clockwise walkthat we do on C n also on H C , by moving on the corresponding states. It follows that after n steps the walk on C n will be back to v and the walk on H C will be back to the node representingthe category of v . Note that this implies that the walk on H C is a cycle and, more in general,that every cycle of length n on H C represent a possible binary configuration of the nodes of acorresponding cycle graph C n .Note that every possible cycle in H C is a combination of simple cycles (that go througheach node at most once) on H C . We prove that the structural property of the lemma holds for7 igure 2: The Cycle Binary Configuration Graph H C . every simple cycle on H C . By commutativity and associativity of addition, the property directlytransfer also to composition of simple graphs. In order to reduce the number of simple cycles(which are 17; they are easy to find on a computer given the small size of the graph H C —theproblem is P -hard [AB09]), we avoid cycles that pass through Z i since | Z i | does not appearin the lemma; in fact, every cycle passing through Z i does L i → Z i → R i and the only otheroutgoing edge of L i is L i → R i . In other words, excluding simple cycles passing through node Z i does not have any effect on the following calculations. By also taking advantage of symmetriesin i and j , all the remaining simple cycles are the following four, for which the equality of thelemma is true: • ( B i → B j ): | B i | cancels out with | B j | . • ( B i → L j → R j ): | B i | cancels out with | R j | + | L j | . • ( R i → L j → R j → L i ): | R i | + | L i | cancels out with | R j | + | L j | . • ( B i → B j → L i → R i → L j → R j ): | B i | cancels out with | B j | ; | R i | + | L i | cancels out with | R j | + | L j | . Theorem 3.4 (Cycles) . Let G = C n be the cycle on n nodes. Under the majority update rule,we have τ = O (cid:0) α n log n (cid:1) , with high probability.Proof. Denote by V i the set of nodes with state i . Given a configuration x ∈ { , } n of C n , let B i = { v ∈ V i : ∀ u ∈ N v , x u (cid:54) = i } and S i = { v ∈ V i : ∃ u, w ∈ N v , x u (cid:54) = x w } (see Lemma 3.3). Let X t be the random variable indicating the number of nodes in state 1 at round t and observethat for every k , we have: P ( X t = h | X t − = k ) = q k if h = k − ,r k if h = k,p k if h = k + 1 , where q k = (1 − α ) (cid:16) | B | n + | S | n (cid:17) , p k = α n − kn + (1 − α ) (cid:16) | B | n + | S | n (cid:17) , and r k = 1 − q k − p k .Therefore, the expected value of X t , conditioned to X t − = k , is E [ X t | X t − = k ] = ( k − q k + kr k + ( k + 1) p k = k − q k + p k = k + α n − kn + 1 − αn (cid:18) | B | + | S | − | B | − | S | (cid:19) ( a ) = k + α n − kn , a ) we use Lemma 3.3. We therefore have: E [ X t ] = n (cid:88) k =0 E [ X t | X t − = k ] P ( X t − = k )= n (cid:88) k =0 (cid:16) α + (cid:16) − αn (cid:17) k (cid:17) P ( X t − = k )= α n (cid:88) k =0 P ( X t − = k ) + (cid:16) − αn (cid:17) n (cid:88) k =0 k P ( X t − = k )= α + (cid:16) − αn (cid:17) E [ X t − ] . Solving this recursion with E [ X ] = 0 we get E [ X t ] = α t − (cid:88) i =0 (cid:16) − αn (cid:17) i = α − (1 − α/n ) t α/n . The expected number n − X t of nodes in state 0 at round t is thus E [ n − X t ] = n (cid:16) − αn (cid:17) t , that is smaller than n for t (cid:62) α n log n . Hence, P (cid:18) τ > α n log n (cid:19) = P (cid:16) n − X α n log n (cid:62) (cid:17) ( b ) (cid:54) E (cid:104) n − X α n log n (cid:105) (cid:54) /n, where in ( b ) we use the Markov inequality. It is not difficult to show that convergence times are also polynomial in the cases of trees ofdegree O (log n ) and disconnected cliques of size O (log n ). These results are summarized as thefollowing theorem. Theorem 3.5 (Trees and disconnected cliques) . Assume G = ( V, E ) is a tree of degree O (log n ) (resp. a set of disconnected cliques, each of size O (log n ) ). Then, for every constant α ∈ (0 , ,the expected absorption time is polynomial. As mentioned in the introduction, the voter model has received considerable attention as anopinion dynamics in the more and less recent past [Lig12]. It may be regarded as a “linearized”form of the majority update rule, in the sense that, upon selection, a node pulls each of thetwo available opinions with probability proportional to the opinion’s support within the node’sneighborhood. Despite such apparent similarity, the two update rules result in quite differentbehaviors of the biased opinion dynamics. Namely, for the voter model, absorption times to thedominant opinion are polynomial with high probability as long as 1 /α is polynomial, regardlessof the underlying topology. These results are clearly at odds with those of Section 3.The biased voter model can formally be defined as follows: Starting from some initial state x (0) , at each round t a node u ∈ [ n ] is chosen uniformly at random and its opinion is updatedas x ( t ) u = (cid:26) α,V G ( u, x ) with probability 1 − α, α ∈ (0 ,
1] is a parameter measuring the bias toward the better opinion 1 and V G ( u, x )is the value held in configuration x ( t − = x by a node sampled uniformly at random from theneighborhood of node u . We assume x (0) = for simplicity, though we remark that Theorem4.1 below holds for any x (0) ∈ { , } n .As the proof of Theorem 4.1 highlights, the biased opinion dynamics under the voter updaterule can be succinctly described by a nonhomogeneous Markov chain [Sen06]. Although non-trivial to study in general, we are able to provide tight bounds in probability for the simplifiedsetting we consider.
Theorem 4.1.
Let G = ( V, E ) be an arbitrary graph. The biased opinion dynamics with voteras update rule reaches state within τ = O ( α n log n ) steps, with high probability.Proof. For every node v ∈ V , the expected state of v at time t + 1, conditioned on x ( t ) = x is E (cid:104) x ( t +1) v | x ( t ) = x (cid:105) = 1 n (cid:34) α + (1 − α ) d v (cid:88) u ∈ N v x u (cid:35) + (cid:18) − n (cid:19) x v = αn + 1 n (cid:104) (1 − α )( P x ) v + ( n − I x ) v (cid:105) , where P = D − A is the transition matrix of the simple random walk on G (with D the di-agonal degree matrix and A the adjacency matrix of the graph) and I is the identity matrix.Considering all nodes we can write the vector form of the previous equation as follows: E (cid:104) x ( t ) | x ( t − = x (cid:105) = αn + 1 n (cid:104) (1 − α ) P + ( n − I (cid:105) x . This immediately implies the following equation, relating expected states at times t − t (with E (cid:2) x (0) (cid:3) = x ): E (cid:104) x ( t ) (cid:105) = αn + 1 n (cid:104) (1 − α ) P + ( n − I (cid:105) E (cid:104) x ( t − (cid:105) . Now, consider − x ( t ) , the difference between the absorbing state vector and the statevector at a generic time t . Obviously, ( − x ( t ) ) v (cid:62) v and for every t . As for the expectation of this difference, we have: E (cid:104) − x ( t ) (cid:105) = 1 n (cid:104) (1 − α ) P + ( n − I (cid:105) E (cid:104) − x ( t − (cid:105) , (1) where the equality follows by collecting and rearranging terms, after observing that bothmatrices P and I have eigenvalue 1 with associated eigenvector . Moreover, we have1 n (cid:104) (1 − α ) P + ( n − I (cid:105) = (cid:16) − αn (cid:17) ˆ P , with ˆ P := n − n − α (cid:104)(cid:16) − αn − (cid:17) P + I (cid:105) a stochastic matrix. This follows immediately by observing thatboth P and I are stochastic, so that all rows of (1 − α ) P + ( n − I identically sum to n − α .By solving the recursion in Eq. (1) we obtain E (cid:104) − x ( t ) (cid:105) = (cid:16) − αn (cid:17) t ˆ P t (cid:104) − x (0) (cid:105) ( a ) = (cid:16) − αn (cid:17) t − (cid:16) − αn (cid:17) t ˆ P t x (0) , where in ( a ) we use the fact that ˆ P t is a stochastic matrix, thus with main eigenvalue 1 andassociated eigenvector . Next, observe that for every v , we have (cid:16) ˆ P t x (0) (cid:17) v (cid:62)
0, so we alsohave E (cid:104) − x ( t ) v (cid:105) (cid:54) (cid:0) − αn (cid:1) t . 10herefore, for every time t (cid:62) α n log n we have E (cid:104) − x ( t ) v (cid:105) (cid:54) n for every v ∈ V . Because the x ( t ) v ’s are binary random variables P (cid:16) x ( t ) v = 0 (cid:17) = P (cid:16) − x ( t ) v = 1 (cid:17) (cid:54) P (cid:16) − x ( t ) v (cid:62) (cid:17) (cid:54) E (cid:104) − x ( t ) v (cid:105) (cid:54) n where in the second-to-last inequality we used the Markov inequality. Hence, in O ( α n log n )rounds the process converges to the absorbing state , with high probability.Note that Theorem 4.1 implies that the convergence time is still O (cid:0) n s log n (cid:1) when α =Θ (cid:0) n s (cid:1) for any s >
0, hence polynomial as long as s is constant. In this paper, we considered biased opinion dynamics under two popular update rules, namelymajority [KR03] and the voter model [Lig12]. Although related, these two models exhibitsubstantial differences in our setting. Whereas the voter model enforces a drift toward themajority opinion within a neighborhood, in the sense that this is adopted with probabilityproportional to the size of its support, majority is a nonlinear update rule, a feature that seemsto play a crucial role in the scenario we consider. This is reflected in the absorption time of theresulting biased opinion dynamics, which is O (cid:0) α n log n (cid:1) for the voter model, regardless of theunderlying topology, whereas it exhibits a far richer behavior under the majority rule, beingsuper-polynomial (possibly exponential) in dense graphs. It may be worth mentioning that inthe case of two opinions, the majority rule is actually equivalent to the (unweighted) medianrule, recently proposed as a credible alternative to the weighted averaging of the DeGroot’s andFriedkin-Johnsen’s models [MBCD19]. A modular model.
Both scenarios we studied are instantiations of a general model that iscompletely specified by a triple ( z , α, D ), with z an initial opinion distribution, α ∈ (0 ,
1] aprobability measuring the magnitude of the bias toward the dominant opinion, and D an updaterule that specifies some underlying opinion dynamics. In more detail, a biased opinion dynamicscan be succinctly described as follows.The system starts in some state x (0) = z , corresponding to the initial opinion distribution;for t >
0, let x ( t − = x be the state at the end of step t −
1. In step t , a node v is pickeduniformly at random from V and its state is updated as follows: x ( t ) v = (cid:26) α, D G ( v, x ) with probability 1 − α, where D G : V × { , } n → { , } is the update rule. When the update rule is probabilistic (asin the voter model), D G ( v, x ) is a random variable, conditioned to the value x of the state atthe end of step t − Remark.
It is simple to see that is the only absorbing state of the resulting dynamics, whenever α (cid:54) = 0 and D does not allow update of an agent’s opinion to one that is not held by at least oneof the agent’s neighbors, which is the case for many update rules in the discrete-opinion setting. The subscript G highlights the fact that result of the application of a given update rule D in general dependson both the current state and the underlying graph G . The above definition can be easily adjusted to reflect thepresence of weights on the edges.
11e further remark that the initial condition x (0) = considered in this paper is not intrinsicto the model, it rather reflects scenarios (e.g., technology adoption) where a new, superioralternative to the status quo is introduced, but its adoption is possibly slowed by inertia of thesystem. Although the reasons behind system’s inertia are not the focus of this paper, inertiaitself is expressed here as a social pressure in the form of some update rule D G . Another reasonfor choosing a fixed initial state ( in our case) is being able to compare the behavior of thebiased opinion dynamics under different update rules on a common basis.Finally, it is worth mentioning that Theorem 4.1 and the upper bounds given in Section 3.3hold regardless of the initial opinion distribution. Outlook.
This paper leaves a number of open questions. A first one concerns general upperbounds on convergence times under the majority update rule. Even though the topology-specific upper bounds given in Section 3 might suggest general upper bounds that depend onthe maximum degree, thus mirroring the result of Theorem 3.2, this turns out to not be thecase, with preliminary experimental results suggesting a more complicated dependence on degreedistribution. In particular, convergence time is fast for reasonably large values of α < / α (cid:62) . α becomes sufficiently small (e.g., this happensaround 0 .
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