Quasi-synchronization of bounded confidence opinion dynamics with stochastic asynchronous rule
aa r X i v : . [ c s . M A ] S e p Quasi-synchronization of bounded confidence opiniondynamics with stochastic asynchronous rule*
Wei Su Xueqiao Wang Ge Chen Kai Shen
Abstract
Recently the theory of noise-induced synchronization of Hegselmann-Krause (HK) dy-namics has been well developed. As a typical opinion dynamics of bounded confidence,the HK model obeys a synchronous updating rule, i.e., all agents check and update theiropinions at each time point. However, whether asynchronous bounded confidence models,including the famous Deffuant-Weisbuch (DW) model, can be synchronized by noise havenot been theoretically proved. In this paper, we propose a generalized bounded confidencemodel which possesses a stochastic asynchronous rule. The model takes the DW model andthe HK model as special cases and can significantly generalize the bounded confidence mod-els to practical application. We discover that the asynchronous model possesses a differentnoise-based synchronization behavior compared to the synchronous HK model. Generally,the HK dynamics can achieve quasi-synchronization almost surely under the drive of noise.For the asynchronous dynamics, we prove that the model can achieve quasi-synchronization in mean , which is a new type of quasi-synchronization weaker than the “almost surely” sense.The results unify the theory of noise-induced synchronization of bounded confidence opiniondynamics and hence proves the noise-induced synchronization of DW model theoretically forthe first time. Moreover, the results provide a theoretical foundation for developing noise-based control strategy of more complex social opinion systems with stochastic asynchronousrules.
Keywords :Quasi-synchronization in mean, noise, asynchronous, bounded confidence, stochas-tic opinion dynamics
In recent years, opinion dynamics are attracting increasing attention of researchers in variousareas [1–4]. One of the interesting topics in opinion dynamics is the noise-induced synchro-nization of opinion systems. During the study of opinion dynamics in a noisy environment, *This research was supported by the National Key Research and Development Program of Ministry of Sci-ence and Technology of China under Grant No. 2018AAA0101002, the National Natural Science Foundationof China under grants No. 61803024, 11688101,61906016, the General Project of Scientific Research Project ofthe Beijing Education Committee under Grant No. KM201811417002, the Foundation of Beijing Union Univer-sity under Grant No. BPHR2019DZ08, the Young Elite Scientists Sponsorship Program by CAST under Grant2018QNRC001, and Beijing Institute of Technology Research Fund Program for Young Scholars.Wei Su is with School of Automation and Electrical Engineering, University of Science and Technology Beijing& Key Laboratory of Knowledge Automation for Industrial Processes, Ministry of Education, Beijing 100083,China, [email protected] . Xueqiao Wang is with Beijing Key Laboratory of Information Service Engineering,Beijing Union University, Beijing 100101, China, [email protected] . Ge Chen is with National Center forMathematics and Interdisciplinary Sciences & Key Laboratory of Systems and Control, Academy of Mathematicsand Systems Science, Chinese Academy of Sciences, Beijing 100190, China, [email protected] . Kai Shen is withSchool of Automation, Beijing Institute of Technology, Beijing, 100081, China, [email protected] .
1t has been found in a mass of simulations that the bounded confidence opinion models dis-play a very positive tendency to be synchronized under the disturbance of noise [10–13]. Usingthe widely known Hegselmann-Krause (HK) model [15], Su et al made a rigorous theoreticalanalysis of this phenomenon [5]. They proved that the noisy HK model can achieve a type ofquasi-synchronization (a definition of synchronization with noise) in finite time almost surelyfor any initial state. After that a series of conclusions and applications of noise-based synchro-nization of opinion systems were obtained, including the noise-induced truth seeking [6], andthe noise-induced synchronization of HK model in full space [7] and also the heterogeneous HKmodel [8], etc. These simulation and theoretical studies effectively reveal the opinion evolutionin noisy circumstances. More crucially, due to the advantages of random information working asan effective social opinion control method [9], the well establishment of noise-induced synchro-nization of opinion dynamics could provide a theoretical foundation for developing noise-basedcontrol strategies of social opinion system.The analysis of noise-induced synchronization mentioned above are mainly based on the HKmodel, which is a typical bounded confidence opinion model. Notably, the HK model obeysa synchronous opinion updating rule, i.e., all agents continuously check and decide whether ornot they will update their opinions at each time. However, a more practical case in reality isthat people are more likely to communicate with each other in an asynchronous way, where onlya random fraction of agents are communicating with each other at each time. Hence a moregeneral asynchronous model is needed. An extreme example of asynchronous opinion dynamics isthe famous Deffuant-Weisbuch (DW) model [14], where only two agents are randomly selectedto communicate and therefore an asynchronous updating rule is adopted. For asynchronousmodels of bounded confidence, only a fraction of agents (we may call them as communicatingagents) update their opinions according to the bounded confidence mechanism at each timepoint. Obviously, the asynchronous rule presents a more practical process of opinion evolution,especially for a large-scale social system. Currently, there are several literatures that studiedthe opinion formation with asynchronous rule. Touri et al [16] and Etesami et al [17] bothconsidered an asynchronous HK model, in which only one agent is communicating at each time.Rossi et al studied an opinion dynamics on the k -nearest-neighbors graph, which is essentiallyan asynchronous model [19]. Ding et al considered an asynchronous opinion scenario with onlineand offline interactions [18].At the same time, there are some theoretical studies that focus on the noise-driven proper-ties of asynchronous opinion models. Just as the HK model, simulation studies revealed thatnoise-driven synchronization also arises in the DW model [11]. However, using the method of an-alyzing noisy HK model, we can show that the DW model cannot achieve quasi-synchronizationalmost surely as the HK model does. Baccelli et al studied a noisy DW model on graph andproved some sufficient conditions for the system to converge. The sufficient conditions given bythe authors are essentially related to the state connectivity of the system, which is difficult tobe predetermined in bounded confidence models. Zhang et al carried out a analysis of consensusproperty for a noisy DW model in [21], where the definition of T -robust consensus was intro-duced. The authors proved that the noisy DW model can achieve T -robust consensus a.s. Theresults imply that the noisy DW model can achieve a synchronization state at some moment T ∗ ≥ T ∗ , T ∗ + T ]. In addition, the authorsshowed that there is a positive probability with which robust consensus happens. With thesepioneering studies, however, it is still unclear whether the DW model as well as the more generalasynchronous models of bounded confidence possess a property of noise-induced synchronization,and a complete theoretical conclusion is far to be obtained.In this paper, we intend to make a theoretical analysis of noise-induced synchronization in2epth for general asynchronous opinion dynamics. Firstly, we propose a generalized stochasticasynchronous model, for which the HK model and DW model may be considered as its specialexamples. In the stochastic asynchronous model, communicating agents are randomly chosen ateach time, and then it is possible to update their opinions according to the bounded confidencemechanism. Interestingly, we prove that the stochastic asynchronous model cannot achievethe noise-driven synchronization almost surely as the HK model does. Meanwhile, simulationstudies suggest that synchronization in some sense emerges in the presence of noise. Finally,by introducing the definition of quasi-synchronization in mean , we prove that the stochasticasynchronous model will achieve quasi-synchronization in mean under the drive of noise. Thisresult naturally implies the noise-induced synchronization of DW model, i.e., for any initial state,the DW model will achieve quasi-synchronization in mean under the drive of noise. We alsoshow that the quasi-synchronization in mean is weaker than the quasi-synchronization almostsurely. The results hence reveal that the quasi-synchronization in mean we propose here isan essentially new concept which is necessary to study the noise-induced synchronization ofasynchronous models. This discovery also accords with the fact that the asynchronous updatingrule possesses a lower interaction frequency than that of the synchronous model. At last, asimulation result is demonstrated to understand the quasi-synchronization in mean. Comparedto the existing studies on the noise-induced synchronization of synchronous HK model, theresults in this paper provide essentially new definitions and discoveries, which completes thetheory of noise-induced synchronization of bounded confidence opinion dynamics and for thefirst time solves the noise-induced synchronization problem of DW model. More importantly,the results in this paper lay a theoretical foundation for developing noise-based control strategyof more complex social systems with stochastic asynchronous rules.The rest of the paper is organized as follows: Section 2 presents some preliminaries ofmodels and definition. Section 3 gives the main theoretical results of the paper; Section 4 showsa simulation result to illustrate the quasi-synchronization in mean. Finally some concludingremarks are given in Section 5. Denote V = { , , . . . , n } , n ≥ n agents, x i ( t ) ∈ [0 , , i ∈ V , t ≥ i at time t , and let U ( t ) ⊂ V be the set of communicating agents at time t , and { ξ i ( t ) , i ∈ V , t ≥ } be the noise. Usually, noise is used to model the external and internalfactors that randomly affect people’s opinions, such as the random information flow on socialmedia each day and the intrinsic opinion uncertainty resulting from one’s free will. Moreover,let Ω be the sample space of { ξ i ( t ) , i ∈ S , U ( t ) , t ≥ } , F be the generated σ − algebra, and P be the probability measure on F , so the underlying probability space is written as (Ω , F , P ). Inaddition, denote E {·} as the expectation of a random variable.To proceed, we first propose our general bounded confidence model with stochastic asyn-chronous rule. Denote y [0 , = , y < y, ≤ y ≤ , y > x i ( t + 1) = (cid:16) α i ( t ) x i ( t ) + (1 − α i ( t )) P j ∈N i ( t ) x j ( t ) |N i ( t ) | + ξ i ( t + 1) (cid:17) [0 , , if i ∈ U ( t ) , N i ( t ) = ∅ ;( x i ( t ) + ξ i ( t + 1)) [0 , , otherwise. (2.1)where α i ( t ) ∈ [ α, − α ] (2.2)with α ∈ (0 , n ] is the inertial coefficient of agent i at t , N i ( t ) = { j ∈ U ( t ) − { i } (cid:12)(cid:12) | x j ( t ) − x i ( t ) | ≤ ǫ } (2.3)is the neighbor set of i at t and ǫ ∈ (0 ,
1] represents the confidence threshold of the agents. Here, | · | is the cardinal number of a set or the absolute value of a real number accordingly.From (2.1), we can see that when an agent i ∈ U ( t ), i.e., i is a communicating agent at t , i will communicate with other communicating agents at t , and then updates its opinions accordingto the bounded confidence mechanism. When i is not a communicating agent at t , it will notcommunicate with other agents, while its opinion can only be affected by environment noise.Now we discuss the communicating set U ( t ). In reality, people seldom update opinion valuessimultaneously at each time. Hence the elements of U ( t ) are randomly selected from V . Formally,we suppose that for t ≥ U ( t ) satisfies( a ) P {|U ( t ) | = k } = p k , where 0 ≤ k ≤ n, ≤ p k ≤ X k p k = 1 , ≤ p + p < b ) for any i , i ∈ V , i = i , P { i , i ∈ U ( t ) } = P { i ∈ U ( t ) } P { i ∈ U ( t ) } , P { i ∈ U ( t ) } = P { i ∈ U ( t ) } ;( c ) {U ( t ) , ξ i ( t ) , i ∈ V , t ≥ } are mutually independent . (2.4)(2.4) provides the assumptions of communicating agents. The assumptions are natural andwe will further provide a methodological explanation of them. (2.4)(a) assumes that the numberof communicating agents is randomly selected at each time, and also the number of communi-cating agents is not always 0 or 1, since we suppose that there do exist communication amongagents with a positive probability. (2.4)(b) assumes that each agent is independently selected tobe the communicating agent with an equal probability at each time. With (2.4)(b) we can have P {U ( t ) = { i , . . . , i k }} = 1 C kn p k , (2.5)where { i , . . . , i k } ⊂ V is an arbitrary choice of k agents and C kn is the combinatorial symbol thatabbreviates n ! k !( n − k )! for 0 ≤ k ≤ n . (2.5) indicates that given the communicating agents number is k , then any choice of a combination of k agents possesses an equal probability. (2.4)(c) assumesthat the event of which agents are communicating and the noises are mutually independentat each time, and also the updating process of communicating agents and noise are mutuallyindependent for different time point.The model represented by equations (2.1)-(2.4) can be considered as a generalized boundedconfidence opinion dynamics with a stochastic asynchronous rule, of which the HK model andthe DW model are two special examples under some conditions. For example, consider thedeterministic case of model (2.1)-(2.4)( ξ i ( t ) ≡ , i ∈ V , t ≥ U ( t ) = V , α i ( t ) = |N i ( t ) | +1 , the model degenerates into the standard HK model; While, if we let U ( t ) = { i , i } { i , i } is an arbitrary choice of two agents, and α i ( t ) ≡ β ∈ (0 , quasi-synchronization almost surely and quasi-synchronization in mean . Definition 2.1.
Denote d V ( t ) = max i,j ∈V | x i ( t ) − x j ( t ) | .1. If P n lim sup t →∞ d V ( t ) ≤ ǫ o = 1, we say the system (2.1)-(2.4) achieves quasi-synchronizationalmost surely (a.s.);2. If lim sup t →∞ E d V ( t ) ≤ ǫ , we say the system (2.1)-(2.4) achieves quasi-synchronization in mean(i.m.).Different from the synchronous bounded confidence model, we can show that the asyn-chronous model (2.1)-(2.4) cannot achieve quasi-synchronization a.s. (Theorem 3.9). However,simulation studies suggest that an approximate “synchronization” state truly occurs for theasynchronous model. In the following section, we will reveal that this new kind of synchroniza-tion state is quasi-synchronization i.m. Remark . The definition of quasi-synchronization provides a normative description of anapproximate synchronization in the sense of confidence threshold ǫ . When a system achievesquasi-synchronization (a.s. or i.m.), all agents become neighbor to each other (a.s. or i.m.),and a cluster of social significance forms. Meanwhile, quasi-synchronization does not implya precise error of synchronization. Actually, as it is revealed by both the previous studiesabout quasi-synchronization a.s. (Lemma 7 [5], Lemma 3.2 [7]) and the present study on quasi-synchronization i.m. (Lemma 3.6 and (3.21)), a more accurate error of synchronization is heavilydependent on the noise amplitude δ . It shows that the accurate error of synchronization can bearbitrarily small as long as δ is small enough. In this section, we intend to introduce the main results of noise-induced synchronization of thesystem represented by equations (2.1)-(2.4). Firstly, we prove that the system (2.1)-(2.4) canachieve quasi-synchronization i.m. from any initial state, when there are noises with properstrength. Secondly, we testify that the system (2.1)-(2.4) cannot achieve quasi-synchronizationa.s. Finally, we verify that a system achieves quasi-synchronization i.m. when it reaches quasi-synchronization a.s.
When there is no noise in the system (2.1)-(2.4) (i.e., ξ i ( t ) ≡ , i ∈ V , t ≥ t →∞ d V ( t ) = 0) under some initial state x (0) ∈ [0 , n and ǫ ∈ (0 , Theorem 3.1.
Suppose that the noises { ξ i ( t ) } i ∈V ,t ≥ are zero-mean random variables with in-dependent and identical distribution (i.i.d.), and E ξ (1) > , | ξ (1) | ≤ δ a.s. for δ > . Thengiven any x (0) ∈ [0 , n and ǫ ∈ (0 , , there exists ¯ δ = ¯ δ ( n, ǫ, α, p , p ) > such that the system(2.1)-(2.4) achieves quasi-synchronization i.m. for all δ ∈ (0 , ¯ δ ] .
5y Theorem 3.1, the quasi-synchronization i.m. of DW model can be obtained directly. Let U ( t ) = { i , i } in the system (2.1)-(2.4), where { i , i } ⊂ V is an arbitrary choice of two agentsand β ∈ (0 , x i r ( t + 1) = (cid:26) βx i r ( t ) + (1 − β ) x i − r ( t ) , if | x i ( t ) − x i ( t ) | ≤ ǫ ; x i r ( t ) , otherwise. x k ( t + 1) = x k ( t ) , k / ∈ U ( t ) . (3.1)where r = 1 , x i ( t + 1) =(˜ x i ( t ) + ξ i ( t + 1)) [0 , , (3.2)where ˜ x i ( t ) is the right side of (3.1). Corollary 3.2. (Quasi-synchronization i.m. of DW model) Given any x (0) ∈ [0 , n , ǫ ∈ (0 , for the system (3.2), there exists ¯ δ = ¯ δ ( n, ǫ, β ) > , such that lim sup t →∞ E d V ( t ) ≤ ǫ for all δ ∈ (0 , ¯ δ ] . To prove Theorem 3.1, some lemmas are needed. The following lemma provides a basictool for analyzing the noise-induced properties of bounded confidence opinion dynamics, whichroughly states that when a random walk has a uniform positive probability entering a regionwithin a finite time, it will almost surely enter that region in finite time.
Lemma 3.3.
Let { w t , t ≥ } be a random walk on R n , { T i : Ω → N + , i ≥ } be a sequence ofincreasing random variables. For D ⊂ R n , denote T = inf t ≥ { t : w t ∈ D } and ¯ D = R n − D . If forany T i , i ≥ , there is a constant < p ≤ such that P n w T i +1 ∈ D (cid:12)(cid:12)(cid:12) T k ≤ i { w T k ∈ ¯ D } o ≥ p , then P { T < ∞} = 1 .Proof. The proof of Lemma 3.3 was given in the last part of Proposition 3.1 in [7].
Lemma 3.4.
Given any initial state x (0) ∈ [0 , n , ǫ ∈ (0 , of the system (2.1)-(2.4), and any λ ∈ (0 , , define T = inf t ≥ { t : d V ( t ) ≤ λǫ } , then P { T < ∞} = 1 for all < δ ≤ λǫ . The proof of Lemma 3.4 designs a noise protocol which drives the system to reach the goal,as in [5]. We give the proof in Appendix.In the following, we intend to analyze the system properties of the asynchronous model(2.1)-(2.4). Specially, the analysis methodology of quasi-synchronization i.m. of the proposedasynchronous model is quite different compared to the previous studies of quasi-synchronizationa.s. of synchronous models. For t ≥
0, we take m t ∈ n i ∈ V : x i ( t ) = min j ∈V x j ( t ) o ,M t ∈ n i ∈ V : x i ( t ) = max j ∈V x j ( t ) o , and define the event A ( t ) = { m t ∈ U ( t ) , M t ∈ U ( t ) } . (3.3) A ( t ) is the set of two agents with maximum and minimum opinion values who are also commu-nicating agents at t . Then we have the following lemma.6 emma 3.5. Consider the system (2.1)-(2.4) and denote L = n ( n − . Given λ ∈ (0 , , if thereexists T < ∞ a.s. such that d V ( T ) ≤ λǫ , then P n d V ( T + t ) ≤ d V ( T ) + 2 tδ ≤ λǫ o = 1 (3.4) for all ≤ t ≤ L , δ ∈ (cid:16) , αλǫ n ( n − i , and P (cid:26) d V (cid:16) T + L (cid:17) ≤ λǫ − αλǫ n − (cid:12)(cid:12)(cid:12)(cid:12) T + L − \ r = T A ( r ) (cid:27) = 1 . (3.5) Proof.
For convenience of notation, suppose T = 0 a.s. To prove (3.4), we only need to prove d V ( t ) ≤ d V (0) + 2 tδ ≤ λǫ, a.s. (3.6)for 1 ≤ t ≤ L .Since | ξ i ( t ) | ≤ δ a.s., from (2.1), d V ( t ) ≤ d V ( t −
1) + 2 δ ≤ . . . ≤ d V (0) + 2 tδ , implying thefirst part of (3.6). The second part of (3.6) can be directly obtained by 1 ≤ t ≤ n ( n − , α ≤ n and δ ≤ αλǫ n ( n − .Now we proceed to prove (3.5). By (2.4)(a) and (2.5), for t ≥ P { A ( t ) } = n X k =2 C kn p k C k − n − = n X k =2 k ( k − n ( n − p k ≥ n ( n −
1) ( p + . . . + p n )= 2 n ( n −
1) (1 − p − p ) > . (3.7)By (2.4)(c), { A ( t ) , t ≥ } are independent, then we can gain P (cid:26) L − \ r =0 A ( r ) (cid:27) = L − Y r =0 P { A ( r ) } ≥ (cid:18) − p − p ) n ( n − (cid:19) L > . (3.8)To prove (3.5), we assume P (cid:26) L − T r =0 A ( r ) (cid:27) = 1 without loss of generality, and we then need toprove d V ( L ) ≤ λǫ − αλǫ n − , a.s. (3.9)By the above assumption, we know P { A (0) } = 1, which implies that m and M are commu-nicating agents at t = 0. And, d V (0) ≤ λǫ ≤ ǫ implies that m and M are neighbors to eachother. From (2.1), we know x m (1) = α m (0) x m (0) + (1 − α m (0)) P j ∈N m (0) x j (0) |N m (0) | + ξ m (1) (3.10)then, by α ≤ α m (0) ≤ − α in (2.2) and | ξ m (1) | ≤ δ a.s., it follows a.s. x m (1) − x m (0) =(1 − α m (0) P j ∈N m (0) ( x j (0) − x m (0)) |N m (0) | + ξ m (1) ≥ (1 − α m (0)) x M (0) − x m (0) n − ξ m (1) ≥ αn − d V (0) − δ. (3.11)7imilarly, we can get x M (1) = α M (0) x M (0) + (1 − α M (0)) P j ∈N M (0) x j (0) |N M (0) | + ξ M (1) (3.12)and a.s. x M (1) − x M (0) =(1 − α M (0) P j ∈N M (0) ( x j (0) − x M (0)) |N M (0) | + ξ M (1) ≤ (1 − α M (0)) x m (0) − x M (0) n − ξ M (1) ≤ − αn − d V (0) + δ. (3.13)Equations (3.11) and (3.13) yield a.s. | x M (1) − x m (1) | ≤ d V (0) − αn − d V (0) + 2 δ ≤ λǫ − (cid:16) αλǫn − − δ (cid:17) (3.14)For any i ∈ U (0), we know x m (0) ≤ x i (0) ≤ x M (0). Hence, following a similar argument asillustrated above, (3.14) implies a.s. | x i (1) − x j (1) | ≤ λǫ − (cid:16) αλǫn − − δ (cid:17) (3.15)for any i, j ∈ U (0).Equation (3.15) yields that once two agents are communicating at t = 0, their distance at t = 1 has an upper bound which is represented by the right side of (3.15). During the followingmovement, their distance can exceed the upper bound only when they are not communicatingagents simultaneously. In this case, their distance may increase by no more than 2 δ after eachtime step. Since δ ≤ αλǫ n ( n − , then a.s. | x i ( t ) − x j ( t ) | ≤ λǫ − (cid:16) αλǫn − − tδ (cid:17) ≤ λǫ − αλǫ n −
1) (3.16)for all i, j ∈ U (0) and 1 ≤ t ≤ n ( n − .Given any 2 ≤ t ≤ L −
1, by (3.6), we can get d V ( t ) ≤ d V (0) + 2 t δ . Similar to the processof obtaining (3.14), we have a.s. | x M t ( t + 1) − x m t ( t + 1) | ≤ d V ( t ) − αn − d V ( t ) + 2 δ ≤ λǫ − (cid:16) αλǫn − − t δ (cid:17) + 2 δ (3.17)Then just as the process of obtaining the equation (3.16), | x i ( t ) − x j ( t ) | ≤ λǫ − (cid:16) αλǫn − − t δ (cid:17) + (cid:16) L − t (cid:17) (2 δ ) ≤ λǫ − αλǫ n − , a.s. (3.18)8or all i, j ∈ U ( t ) and t + 1 ≤ t ≤ L .For any i ∈ V , since P (cid:26) L − T r =0 A ( r ) (cid:27) = 1 by assumption, it is certain that i ∈ A ( t ) for some0 ≤ t ≤ L , or x m t ( t ) ≤ x i ( t ) ≤ x M t ( t ) for all 0 ≤ t ≤ L . For both cases, by (3.16) and (3.18),we have (cid:12)(cid:12)(cid:12) x i ( L ) − x j ( L ) (cid:12)(cid:12)(cid:12) ≤ λǫ − αλǫ n − , a.s. (3.19)for all i, j ∈ V . Hence (3.9) can be obtained. This complete the proof.Lemma 3.5 indicates that, once the system enters a region which is narrow enough, it willnot get away from the region, provided some special agents with extreme opinion values arealways communicating.Furthermore, in order to achieve the final result of quasi-synchronization i.m. of the asyn-chronous model, we would like to introduce the following lemma. In the proof of the lemma, weintroduce a stopping time as a bridge to obtain the moment property of d V ( t ). Lemma 3.6.
Suppose the noises { ξ i ( t ) } i ∈V ,t ≥ are given in Theorem 3.1. Let x (0) ∈ [0 , n , ǫ ∈ (0 , be arbitrarily given, then for any µ ∈ (0 , , there is ¯ δ = ¯ δ ( µ, n, ǫ, α, p , p ) > , such that lim t →∞ E d V ( t ) ≤ µǫ for all δ ∈ (0 , ¯ δ ] .Proof. Denote ˜ p = n ( n − (1 − p − p ), L = n ( n − , L = min n l > − ˜ p L ) l ≤ µǫ o and T = inf t ≥ n t : d V ( t ) ≤ µǫ L L ) o , then d V ( T ) ≤ µǫ L L ) , a.s. (3.20)Denote ¯ δ = min n αµǫ n ( n − , µǫ L L ) o , (3.21)and we next prove that E d V ( T + k ) ≤ µǫ (3.22)for all k ≥ δ ∈ (0 , ¯ δ ].In order to prove (3.22), we first consider P n d V ( T + k ) > µǫ o for any given k >
0. Take λ = µ in Lemma 3.5, and notice that d V ( T ) ≤ µǫ L L ) ≤ µǫ a.s., then by (3.4) in Lemma 3.5and (3.20), we can get d V ( T + k ) ≤ d V ( T ) + 2 kδ ≤ µǫ L L ) + 2 L L ¯ δ ≤ µǫ , a.s. (3.23)for 1 ≤ k ≤ L L , implying P n d V ( T + k ) > µǫ o = 0 , ≤ k ≤ L L. (3.24)Next we consider P n d V ( T + k ) > µǫ o for k > L L . Since d V ( T ) ≤ µǫ L L ) ≤ µǫ a.s., by Lemma3.5, we know that once A ( t ) occurs for T ≤ t ≤ T + L −
1, then d V ( T + L ) ≤ µǫ − αµǫ n − a.s.By δ ≤ αµǫ n − L L , we can get d V ( T + L + t ) ≤ d V ( T + L ) + 2 δt ≤ µǫ , a.s. (3.25)9or 1 ≤ t ≤ L L .Denote B ( s, r ) = T T + s + rL − t = T + s +( r − L A ( t ) , s ≥ , r ≥
1. (3.25) implies that when there is amoment T such that d V ( T ) ≤ µǫ , and A ( t ) occurs in the following L times, then d V ( T + t ) ≤ µǫ for all L ≤ t ≤ L + L L . In other words, once B (0 ,
1) occurs, d V ( t ) can not exceed µǫ duringthe next L L steps. By (3.23), d V ( T + t ) ≤ µǫ a.s. for all 1 ≤ t ≤ L L . Hence, if there is k > L L such that d V ( T + k ) > µǫ a.s., there must exist a period of length L L and someinteger s ≥ { B ( s, , . . . , B ( s, L ) } cannot happen, i.e., n d V ( T + k ) > µǫ o ⊂ (cid:26) L \ r =1 { Ω − B ( s, r ) } (cid:27) . (3.26)By (2.4) and (3.7), { A ( t ) , t ≥ } are i.i.d., and so are { B ( s, r ) , s ≥ , r ≥ } by strong Markovproperty. As a result, for any given k > L L , we can get by (3.26) P n d V ( T + k ) > µǫ o ≤ P (cid:26) L \ r =1 { Ω − B ( s, r ) } (cid:27) =(1 − P { B (0 , } ) L (3.27)where Ω is the sample space.By (3.8) and (3.27), we have P n d V ( T + k ) > µǫ o ≤ (1 − P { B (0 , } ) L = (cid:18) − L − Y r =0 P { A ( T + r ) } (cid:19) L ≤ (cid:16) − ˜ p L (cid:17) L (3.28)for any given k > L L .Since d V ( t ) ≤ t ≥
0, by (3.24), (3.28) and the definition of L , it follows E d V ( T + k ) = E (cid:16) d V ( T + k ) I { d V ( T + k ) ≤ µǫ } + d V ( T + k ) I { d V ( T + k ) > µǫ } (cid:17) ≤ µǫ P n d V ( T + k ) > µǫ o ≤ µǫ − ˜ p L (cid:17) L ≤ µǫ. (3.29)for all k ≥ t ≥
0, we gain by (3.29) E d V ( t ) = E (cid:16) d V ( t ) I { T ≤ t } + d V ( t ) I { T >t } (cid:17) = E (cid:18) t X k =0 d V ( T + k ) I { T = t − k } (cid:19) + E d V ( t ) I { T >t } = t X k =0 E d V ( T + k ) I { T = t − k } + E d V ( t ) I { T >t } ≤ µǫ t X k =0 E I { T = t − k } + E d V ( t ) I { T >t } ≤ µǫ P { T ≤ t } + P { T > t } (3.30)10y Lemma 3.4, P { T < ∞} = 1, we can thus obtainlim sup t →∞ E d V ( t ) ≤ lim sup t →∞ (cid:16) µǫ P { T ≤ t } + P { T > t } (cid:17) ≤ µǫ. (3.31)This completes the proof. Remark . Lemma 3.6 provides a general conclusion of noise-induced quasi-synchronizationi.m. of the asynchronous system and also an estimation of the synchronization error. It showsthat the synchronization error is heavily dependent on noise amplitude δ . Deduced from (3.21),one can see that the synchronization error can be arbitrarily small (taking any 0 < µ ≤
1) when δ is small enough. Remark . Lemma 3.6 only shows that the system can achieve quasi-synchronization i.m. whennoise amplitude is suitably small, while how large noise affects the synchronization is absent.Intuitively and as we can show readily, large noise could destroy the quasi-synchronization i.m.of the system. In fact, given any 0 < p ≤
1, we consider the systems with 0 < ǫ ≤ p . When P { ξ (1) > ǫ p } ≥ p, P { ξ (1) < − ǫ p } ≥ p , then for all t ≥ P { d V ( t + 1) > ǫp } ≥ P { ξ m ( t ) ( t + 1) < − ǫ p , ξ M ( t ) ( t + 1) > ǫ p } ≥ p , implying E d V ( t + 1) > ǫp p = ǫ for all t ≥ Proof of Theorem 3.1:
Take µ = 1 in Lemma 3.6, and we obtain the conclusion. ✷ Lemmas 3.4 and 3.5 show that when the system (2.1)-(2.4) is synchronous (i.e., |U ( t ) | ≡ n, t ≥ p n =1), the system can achieve quasi-synchronization a.s. for all δ ∈ (0 , αǫn ( n − ) (this isactually the case of HK model [5]). But when the model is asynchronous, i.e., p n <
1, we havethe following conclusion.
Theorem 3.9.
Suppose that the noises { ξ i ( t ) } i ∈V ,t ≥ are zero-mean i.i.d. random variables,and E ξ (1) > , | ξ i ( t ) | ≤ δ a.s. for δ > . If p n < , then the system (2.1)-(2.4) cannot achievequasi-synchronization a.s for any x (0) ∈ [0 , n , ǫ ∈ (0 , and δ > .Proof. We only need to prove that for any x (0) ∈ [0 , n , ǫ ∈ (0 ,
1) and δ >
0, lim sup t →∞ d V ( t ) = 1a.s., i.e. P (cid:26) ∞ [ g =0 { d V ( t ) < , t ≥ g } (cid:27) =1 − P (cid:26) ∞ \ g =0 ∞ [ t = g { d V ( t ) = 1 } (cid:27) =1 − P n lim sup t →∞ d V ( t ) = 1 o = 0 . (3.32)Given any g ≥
0, denote T = inf t ≥ g { t : d V ( t ) = 1 } , then by Lemma 3.3, we need to prove that, forany initial state x (0) ∈ [0 , n , there are t L > g, < p < P { d V ( t L ) = 1 } ≥ p .Since E ξ (1) = 0 , E ξ (1) > | ξ i (1) | ≤ δ a.s., there exist constants 0 < a ≤ δ, < ¯ p < P { a < ξ (1) ≤ δ } ≥ ¯ p, P {− δ ≤ ξ (1) < − a } ≥ ¯ p. (3.33)For t ≥ g , consider the following noise protocol ξ i ( t + 1) ∈ [ − δ, − a ] , if min j ∈V x j ( t ) ≤ x i ( t ) ≤ min j ∈V x j ( t ) + d V ( t )2 ; ξ i ( t + 1) ∈ [ a, δ ] , if min j ∈V x j ( t ) + d V ( t )2 < x i ( t ) ≤ max j ∈V x j ( t ) . (3.34)11enote A ( t ) C = Ω − A ( t ) , t ≥
0, where A ( t ) is defined in (3.3), then by p n < P {{ A ( t ) C } =1 − P { A ( t ) } = 1 − n X k =2 C kn p k C k − n − =1 − n − X k =2 k ( k − n ( n − p k − p n ≥ − p n − n − n ( p + . . . + p n − ) ≥ − p n − n − n (1 − p n )= 2(1 − p n ) n > . (3.35)Hence by the independence of { ξ i ( t ) , U ( t ) , i ∈ V , t ≥ } and (3.33), we can get P { d V ( t + 1) ≥ d V ( t ) + a } ≥ P { A ( t ) C , protocol (3 .
34) occures at t } = P { A ( t ) C } · P { protocol (3 .
34) occures at t }≥ ˜ p ¯ p n > , (3.36)where ˜ p = − p n ) n .Denote t L = ⌈ a ⌉ , then under the protocol (3.34)max i ∈V x i ( g + t L ) = 1 , min i ∈V x i ( g + t L ) = 0 , yielding d V ( g + t L ) = 1. By (3.36) and the independence of { ξ i ( t ) , U ( t ) , i ∈ V , t ≥ } , we gain P { d V ( g + t L ) = 1 } ≥ g + t L Y t = g +1 P { A ( t ) C , protocol (3.34) occurs at t }≥ ˜ p t L ¯ p nt L > . (3.37)Let p = ˜ p t L ¯ p nt L , and this completes the proof.Theorem 3.9 shows that quite different from the synchronous model, the asynchronous modelcannot achieve quasi-synchronization a.s., no matter how small the non-zero noise is. Theorems 3.1 and 3.9 indicate that, for the system (2.1)-(2.4), quasi-synchronization i.m. doesnot necessarily imply quasi-synchronization a.s. The following theorem reveals that the converseconclusion is true, i.e., quasi-synchronization a.s. leads to quasi-synchronization i.m.
Theorem 3.10. If P n lim sup t →∞ d V ( t ) ≤ ǫ o = 1 , then lim sup t →∞ E d V ( t ) ≤ ǫ .Proof: Since P n lim sup t →∞ d V ( t ) > ǫ o = 0, P (cid:26) lim s →∞ [ t ≥ s { d V ( t ) > ǫ } (cid:27) = lim s →∞ P (cid:26) [ t ≥ s { d V ( t ) > ǫ } (cid:27) = 0 , d V ( t ) ≤ E d V ( t ) = E (cid:16) d V ( t ) I { d V ( t ) ≤ ǫ } + d V ( t ) I { d V ( t ) >ǫ } (cid:17) ≤ ǫ P { d V ( t ) ≤ ǫ } + P { d V ( t ) > ǫ } . (3.38)Consequently, we can getlim sup t →∞ E d V ( t ) ≤ lim sup t →∞ (cid:16) ǫ P { d V ( t ) ≤ ǫ } + P { d V ( t ) > ǫ } (cid:17) ≤ ǫ. This completes the proof. ✷ In this section, we introduce our simulation result to help understand the meaning of quasi-synchronization i.m. Let n = 40 , ǫ = 0 . , α i ( t ) = |N i ( t ) | +1 , and |U ( t ) | is randomly selectedfrom 0 , , . . . , n with equal probability at each time. The initial opinion values are randomlygenerated on [0 , − δ, δ ] to thesystem (2.1)-(2.4). Take δ = 0 .
01, then Fig. 1 shows that the system achieves a synchronizationat about t = 10000. However, it is not the almost sure quasi-synchronization (refer to [5]for the simulation study of almost sure quasi-synchronization), since it can be calculated thatmax t d V ( t ) = 0 . > ǫ = 0 . t = 15000 to 40000. Such an approximate synchronizationcan be measured by quasi-synchronization i.m.Figure 1: Opinion evolution of system (2.1)-(2.4) of 40 agents. The initial system states arerandomly generated on [0 , ǫ = 0 .
1, noise strength δ = 0 . In this paper, we studied the noise-induced synchronization of an asynchronous opinion dynamicsof bounded confidence. A noisy stochastic asynchronous model was proposed, and we proved13hat, though the proposed model can not achieve quasi-synchronization a.s. as the synchronousHK model does, it can achieve quasi-synchronization i.m. The results for the first time provetheoretically the noise-driven synchronization of the DW model, which has been observed byprevious simulation studies. Moreover, quasi-synchronization i.m. was verified to be weaker thanquasi-synchronization a.s. The results in this paper help complete the theory of noise-inducedsynchronization of bounded confidence dynamics. Moreover, due to the limitations of applyingtraditional control method, which relies heavily on the accurate information of systems states,to the control of complex systems, the present results lay a theoretical foundation for developingnoise-based control strategy of more general complex social opinion systems.Recently, Li et al designed a particle robotics system which used stochastic movement ofloosely coupled robotic systems to realize a global coordination [22]. For a further interpretationof this study, we want to mention that the results in this paper also highlight the noise-drivenproperties of loosely coupled self-organizing particle robotic systems.
Appendix
Proof of Lemma 3.3:
Notice (3.33) and consider the following noise protocol: for all i ∈ V , t ≥ ξ i ( t + 1) ∈ [ a, δ ] , if min j ∈V x j ( t ) ≤ e x i ( t ) ≤ min j ∈V x j ( t ) + d V ( t )2 ; ξ i ( t + 1) ∈ [ − δ, − a ] , if min j ∈V x j ( t ) + d V ( t )2 < e x i ( t ) ≤ max j ∈V x j ( t ) , (5.1)where e x i ( t ) = ( α i ( t ) x i ( t ) + (1 − α i ( t )) P j ∈N i ( t ) x j ( t ) |N i ( t ) | , if i ∈ U ( t ) and N i ( t ) = ∅ ; x i ( t ) , otherwise . (5.2)Since δ ≤ λǫ , following a similar argument of the proof of Theorem 5 in [5], there exists L > P (cid:26) d V ( mL ) ≤ λǫ (cid:12)(cid:12)(cid:12) \ j
1. Let D = { x ∈ R n : max i,j | x i − x j | ≤ λǫ } , T k = kL, p = ¯ p L , then by (5.3) and Lemma3.3, P { T < ∞} = 1. ✷ References [1] A. Proskurnikov, R. Tempo, “A tutorial on modeling and analysis of dynamic social net-works. Part I”,
Annual Reviews in Control , vol. 43, pp. 65-79, 2017.[2] A. Proskurnikov, R. Tempo, “A tutorial on modeling and analysis of dynamic social net-works. Part II”,
Annual Reviews in Control , vol. 45, pp. 166-190, 2018.[3] M. Ye, J. Liu, B. Anderson, C. Yu, and T. Ba¸sar, “Evolution of social power in socialnetworks with dynamic topology”,
IEEE Trans. Autom. Control , vol. 63, no. 11, pp. 3793-3808, 2018. 144] G. Chen, X. Duan, N. Friedkin, and F. Bullo, “Social power dynamics over switching andstochastic influence networks”,
IEEE Trans. Autom. Control , vol. 64, no. 2, pp. 582C597,2019.[5] W. Su, G. Chen, and Y. Hong, “Noise leads to quasi-consensus of Hegselmann-Krauseopinion dynamics”,
Automatica , vol. 85, pp. 448-454, 2017.[6] W. Su, Y. Yu, “Free information flow benefits truth seeking”,
J. Sys. Sci. Complex , vol. 31,pp.964-974, 2018.[7] W. Su, J. Guo, X. Chen, G. Chen, “Noise-induced synchronization of Hegselmann-Krausedynamics in full space”,
IEEE Trans. Auto. Control , vol. 64, no. 9, pp.3804-3808, 2019.[8] G. Chen, W. Su, S. Ding, Y. Hong, “Heterogeneous Hegselmann-Krause Dynamics withEnvironment and Communication Noise”,
IEEE Trans. Auto. Control , 2019.[9] W. Su, X. Chen, Y. Yu, G. Chen, “Noise-based control of opinion dynamics”,arXiv:1806.03781v3, 2018.[10] M. M¨as, A. Flache, and D. Helbing, “Individualization as driving force of clustering phe-nomena in humans”,
PLoS Computational Biology , vol.6, pp. e1000959, 2010.[11] A. Carro, R. Toral, and M. San Miguel, “The role of noise and initial conditions in theasymptotic solution of a bounded confidence, continuous-opinion model”,
J. Statis. Phys. ,vol.151, pp. 131-149, 2013.[12] M. Pineda, R. Toral, and E. Hernandez-Garcia, “The noisy Hegselmann-Krause model foropinion dynamics”,
Eur. Phys. J. B , vol.86, pp. 1-10, 2013.[13] T. Hadzibeganovic, D. Stauffer, and X. P. Han, “Randomness in the evolution of coopera-tion”,
Behav. Process. , vol. 113, pp. 86-93, 2015.[14] G. Deffuant, D. Neau, F. Amblard, and G. Weisbuch, “Mixing beliefs among interactingagents”,
Adv. Compl. Syst. , vol.3, no.01n04, pp.87-98, 2000.[15] R. Hegselmann, U. Krause, “Opinion dynamics and bounded confidence models, analysis,and simulation”,
J. Artificial Societies and Social Simulation , vol.5, pp. 1-33, 2002.[16] B. Touri, C. Langbort, “On endogenous random consensus and averaging dynamics”,
IEEETrans. control of network systems , vol. 1, no. 3, pp. 241-248, 2014.[17] S. Etesami, T. Ba¸sar, “Game-theoretic analysis of the Hegselmann-Krause model for opiniondynamics in finite dimensions”,
IEEE Trans. Auto. Control , vol. 60, no. 1, pp. 1886-1897,2015.[18] Z. Ding, Y. Dong, H. Liang, and F. Chiclana, “Asynchronous opinion dynamics with onlineand offline interactions in bounded confidence model”,
J. Artificial Societies and SocialSimulation , vol. 20, no. 4, 6, 2017.[19] W. Rossi, P. Frasca, “Asynchronous opinion dynamics on the k -nearest-neighbors graph”, , CDC 2018, Miami, FL, USA, December17-19, pp. 3648-3653, 2018. 1520] F. Baccelli, A. Chatterjee , and S. Vishwanath, “Pairwise stochastic bounded confidenceopinion dynamics: heavy tails and stability”, IEEE Trans. Autom. Control , vol. 62, no. 11,pp. 5678-5693, 2017.[21] J. Zhang, Y. Zhao, “The robust consensus of a noisy Deffuant-Weisbuch model”,
Mathe-matical Problems in Engineering , Article ID 1065451, 10 pages, 2018.[22] S. Li, R. Batra, D. Brown, H. Chang, N. Ranganathan, C. Hoberman, D. Rus, and H.Lipson, “Particle robotics based on statistical mechanics of loosely coupled components”,
Nature , vol. 567, pp. 361-365, 2019.[23] Y. Chow, H. Teicher,