The stability analysis of stochastic opinion dynamics systems with multiplicative noise and time delays

This paper studies the stochastic stability of opinion formation systems under both non-time delay and time delay. Individuals whose opinions are affected by multiplicative noise are considered. Subsequently, a stochastic opinion formation model on coopetitive social networks which take both multiplicative noise and time delay into consideration and give the analysis of stochastic stability is introduced. When the continuous-time system of opinion evolution is disturbed by stochastic multiplicative noise, the stability of the stochastic opinion formation system is dependent on the noise intensity and the topology structure. When the continuous-time system of opinion dynamic is disturbed by time delay and multiplicative noise,


INTRODUCTION
Numerous studies on opinion dynamics of social networks have been conducted for multi-agent systems [1] from different perspectives [2][3][4][5][6]. Beginning with DeGroot model [3], opinion models have been investigated to study the evolution of opinions in social networks, including Friedkin-Johnsen model [4], Hegselmann-Krause model [5], Deffuant-Weisbuch model [6] and so on. A common trait of the existing models is that individuals update their opinions (depicted by real numbers) as a convex combination of their neighbours' and own opinions. In the early works, the investigations about opinion formation are under ideal networks where the individuals only cooperate with each other. In other words, the underlying network topology of interactions is characterised by edge weights that are nonnegative. However, recently more and more works have turned their attention to the real coopetitive (cooperative-competitive) networks, where the relationships between the individuals are represented by signed graphs, i.e. positive weights represent cooperative interactions and negative weights represent competitive interactions. The first opinion dynamic model with coopetitive networks was introduced by Altafini [7], who used the property of signed graphs to analyse bipartite consensus.
To study the phenomenon of opinion separation under signed networks which the relationships between individuals are time-varying, Xia et al. [8] study the opinion evolution of interacting agents with coopetitive relationships in either dynamically changing or fixed topologies. Under coopetitive social networks, Liu et al. [9] generalise the concept of consensusability, polarisability and neutralisability of Altafini model. Subsequently, fruitful discussions extend the consensus with unsigned graphs [10] to bipartite consensus with signed graphs [11,12].
The common basis of all the current research on the opinion dynamic problem on social networks is the focus on deterministic systems. However, when we consider opinion evolution in the real world, if the opinion dynamic system is continuously disturbed by small random noise, the change may affect the stability of the entire dynamical system [13]. Therefore, a central challenge that still exists is it is possible to achieve a form of agreement also in presence of noise interference. While a vast number of tentative explanations [14,15] have been raised, and recently, an interesting research is to double check the differences [16] between deterministic opinion dynamics and stochastic opinion dynamics especially when the evolution dynamics are disturbed by stochastic multiplicative noise. For instance, Pineda et al. [17] involve which aspects of the original systematic dynamics are robust against noise through the H-K model and Su et al. [18] provide rigorous theoretical analysis on how stochastic noises influence the consensus behaviour for the confidence-based the H-K opinion model. Hu et al. [19] extend an existing dynamic distributed output-feedback controller and a broader class of stochastic-approximation gain to signed networks, and they achieve bipartite consensus in the end. Wu et al. [20] co-design a adaptive distributed control laws and a stochastic gain while guaranteeing the high-order multi-agent systems to be bipartite consensus in the underlying signed networks. Liang et al. [21] study the stability of stochastic opinion dynamic systems with stochastic multiplicative noises' and the stubborn agent's disturbances. Xue et al. [22] investigate the opinion evolution and social power on coopetitive social networks whose interconnections changes via a reflected appraisal mechanism over a sequence of issues. Yuan et al. [23] try to make a bit of the stochastic stability analysis of opinion evolution with the stochastic multiplicative noises' disturbances. In a sense, the attack signal can also be regarded as the system noise, and in [24] the recent results on secure control are discussed, which provides a new direction for the study of opinion dynamics. Ding et al. [25] propose the defence strategy which is used to identify the occurring attacks as far as possible.
In a realistic social scenarios, it is also natural to include a time delay in the context of opinion dynamics, because the information transmitted among the agents may be subject to communication delays from time to time. Liu et al. [26] investigate the discrete-time Altafini model with timevarying bounded time delays and provide a sufficient condition for the opinion dynamic systems to asymptotically converge an absolute value via a graphical approach. Recently, Choi et al. [27] study the opinion formation of Hegselmann-Krause model in the presence of a time-variable time delay. In [28], a predictive scheme is proposed to compensate for the influenced induced by the communication delays. Motivated by the above observations, this paper will focus on the stability of a stochastic opinion formation system on coopetitive social networks which take both multiplicative noise and time delay into consideration.
The main contribution of this paper is twofold. First, we investigate the continuous-time opinion dynamic model with stochastic multiplicative noise and provide a sufficient condition for the model to be almost sure exponential stability for the trivial solution of the whole system. The second and more important contribution is that we analyse the stochastic Altafini model with communication delays and provide conservative bounds on the strength of noise and the time delay for the almost sure exponential stability of stochastic systems. The conditions are compliant to both indirected networks and directed networks. The main computational tool that we exploit is the stochastic Lyapunov's framework of the stochastic differential equation, which is used to verify the stochastic stability associated with opinion dynamic systems.
The structure of this paper is as follows. Section 2 presents the model description, preliminaries and problem setup. Section 3 provides the main theoretical results for the stochastic opinion dynamic systems. Section 4 introduces the main theoretical results for the stochastic delay systems. Simulation results on examples are given in Section 5. Conclusion remarks are provided in Section 6.

Basic Notations
Consider a weighted graph G = {V , E, C }, where V = {1, 2, … , n} denotes the vertex set of G , E ⊆ V × V is the edges set of G , and C = [c i j ] ∈ ℝ n×n represents the weighted adjacency matrix. We say that c i j > 0, if the ith agent is a cooperative partner of the jth agent. Hence c i j < 0, if the ith agent is a competitor of the j th agent. And c i j = 0 means that the there is no relationship between the ith agent and j th agent. The adj(i ) denotes the neighbour of the ith agent, and j ∈ adj(i ) if c i j ≠ 0. For the subsequent analysis, we introduce the Laplacian matrix |x| is the Euclidean norm of a vector x for x ∈ ℝ n×1 . We say that a graph is structural balanced [29], if there exists a partition such that the vertex set where the vertexes in V + or V − are connected by +, V + and V − are connected by −.

MODEL DESCRIPTION AND PRELIMINARIES
In this section, we formalise the framework of our problem. We introduce the model of opinion dynamic underlying stochastic multiplicative noise. Moreover, we introduce stochastic opinion dynamic model with time delay. Let us revise a deterministic continuous-time Altafini model [7] described by the differential equation under the cooperative-competitive (coopetitive) network.̇y where the terms of y i (t ) denotes the opinion values of the ith agent at time t , i = 1, 2, … , n, and c i j is the weighting coefficient from the j th agent to the ith agent among the graph.
Specifically, c i j > 0, if the ith agent is a cooperative partner of the j th agent. c i j < 0, if the ith agent is a competitor of the j th agent, otherwise c i j = 0. Moreover, C = [c i j ] ∈ ℝ n×n is the weighted adjacency matrix. We investigate the stochastic continuous-time Altafini opinion evolution system with multiplicative noise' disturbances.
where the noise W is the standard Wiener process with independent increments and zero mean, and i ≥ 0 is the strength of the standard Wiener process. Then we study stochastic opinion dynamic model with time delay.
where a positive number is denoted as time delay.
The matrix form of the opinion evolution system describing a network of agents using Equation (2) are updated as follows where is the opinion vector of all individuals at time t , L is the Laplacian matrix of the corresponding matrix C , the strength of noise vector = [ 1 , 2 , … , n ], and Rewriting the dynamical system (3) in a matrix form by a similar way to Equation (4), The purpose of this article is to firstly consider the stochastic stability of the opinion dynamics (4) underlying stochastic multiplicative noise and Equation (5) for stochastic opinion dynamics with time delay respectively and then illustrate how the stochastic noise and time delay can affect the evolution of opinions in comparison to the corresponding deterministic system. Therefore, we need to firstly clarify the concept of stochastic stability for opinion dynamic systems.

STOCHASTIC OPINION DYNAMIC SYSTEMS
In this section, we investigate the stochastic stability of the opinion dynamic systems with the multiplicative noises' disturbances. As is known to all, we use the Lyapunov direct method to study the asymptotic stability of the deterministic system. Explicitly, in a deterministic case, the definition of stability is that the system state is not sensitive to the initial state or small disturbances for the system parameters. Compared to the deterministic counterparts, scholars have believed that the notion of stochastic stability can give at least three different explanations for a long time: stability in probability, stability in moment and stability in an almost sure sense [30,31]. Because the stochastic system has an important element in the evolution process which is the random characteristic of noise [32]. Compared to existing work, the objective of this paper is to investigate the almost sure exponential stability [33]. The almost sure exponential stability means that almost all sample paths of the solution will tend to the equilibrium position x = 0 exponentially fast. To address the aforementioned issues, the main mathematical theory that we develop is the stochastic Lyapunov frameworks [34], which is described in detail in the following. The stochastic differential equation is on t ≥ t 0 with the original value y(t 0 ) = y 0 .
Before proceeding with the main results, we first assume the following local Lipschitz continuous condition for the functions f (⋅) and g(⋅): Assumption 1. There exist positive constants Γ 1 and Γ 2 such that for all t ≥ 0, ∀y, z ∈ ℝ, and f (y, t ), g(y, t ) are Borel measurable.
In addition, the following facts are true according to Assumption 1. When y(0) = 0, y(t ) = 0 for all t > 0. In these instances y(t ) ≡ 0 is a trivial solution of the stochastic differential equation (6).
Choose the nonnegative functions V(y, t ), and the infinitesimal operator  associated with Equation (6) which acts on the function V(y, t ) is defined as Definition 1. [34] The trivial solution of the stochastic differential equation (6) is called the almost sure exponential stable if the following inequality is satisfied for all y 0 ∈ ℝ.
Roughly speaking, the almost sure exponential convergence is that almost all of the sample paths for the solution will tend to the trivial solution y = 0 exponentially fast. The detail technical discussion procedure as [34] and its following works explaining that such a function V(y, t ) in general exists. Besides its insight into the almost sure exponential stability of stochastic differential equation, the function V(y, t ) can be found for most cases. The computation relies on the following Lemma. Lemma 1. [34] Suppose that Assumption 1 holds. If there exists a function V(y, t ) and constants q > 0, 1 ≥  1 > 0,  2 ∈ ℝ,  3 ≥ 0, such that for all y ≠ 0, and T ≥ t 0 , then for all y 0 ∈ ℝ. In particular, if  3 > 2 2 , the trivial solution y = 0 is almost sure exponential stable. For the stochastic equation (4) that we want to study in this paper, the stochastic stability is stated in the following theorem.

Theorem 1. Given a connected signed graph G = {V , E, C } and the opinion dynamic system with stochastic multiplicative noise is adopted by Equation (2). The trivial solution Y = 0 of Equation (4) is the almost sure exponential stable if
where min = min{ 1 , 2 , … , n } and max = max{ 1 , 2 , … , n }.
Proof. Notice that the stochastic differential equation (4) is the matrix form of Equation (2). The stochastic stability of Equation (2) is converted into the stochastic stability of the Equation (4). In the following, we will study the almost sure exponential stability of the stochastic system (4) based on the Lemma 1.
Recalling that the Equations (4) and (6), it can be obtained that It follows that Clearly, For simplicity of discussion, it is straightforward to verify for the function g(⋅) by a similar way.
For the next step of the proof, we need to calculate the value of constants q,  1 ,  2 , and  3 based on the Lemma 1 and the system (4).
In order to facilitate the analysis of the system, let Since (8), it follows from the concept of the Lyapunov-Krasovskii function V that 1 =  1 = 1 and q = 2. Now, we aim to exploit  2 .
where max = max{ 1 , 2 , … , n }. So by comparing the above inequality and the inequality (9), we obtain where min = min{ 1 , 2 , … , n }. The above inequality means  3 = 4 2 min . According to Lemma 1 and Definition 1, calculate Then we can obtain that the trivial solution Y = 0 of Equation (4) is the almost sure exponential stable if The theorem is one of our main results and it points out the almost sure exponential convergence of stochastic opinion dynamic systems with multiplicative noises. In the next section, we demonstrate the almost sure exponential convergence of stochastic opinion dynamic model with time delay.

STOCHASTIC OPINION DYNAMIC SYSTEMS WITH TIME DELAY
Following the discussion in the previous section, we investigate the stochastic stability in signed networks with multiplicative noise and time delay. In particular, we are interested in how the time delay can affect the evolution of opinions in comparison to the stochastic opinion dynamic systems. To make the corresponding theoretical results more accessible, we also focus on the almost sure exponential convergence of stochastic systems in this section. The stochastic differential delay equation is on t ≥ t 0 with the initial value y(t 0 ) = y 0 and time delay > 0.
Lemma 2. [35] Suppose that Assumption 1 holds. Moreover, suppose that the trivial solution Y = 0 of Equation (4) is the almost sure exponential stable. If there exists a positive number * such that for any initial value ∈ ℝ, then the following inequality for the solution of Equation (11) hold: provided < * . Actually, give a constant ∈ (0, 1) for q < 1 and 0.5(1 − ) 3 >  2 below to hold and let p = q; next choose another constant ∈ (0, 1) and let and finally assume * > 0 be the unique solution to the equation (in ) e (Γ 1 +0.5Γ 2 2 )p + 2 p Ω 1 ( , p, + T ) where M , will be defined by M = ( 1 ∕ 1 ) and = (0 ] . Regarding the almost sure exponential stability of the stochastic differential delay equation (11), we accomplish this goal by investigating the almost sure exponential stability of the corresponding non-delay stochastic differential equation (6) and appropriately choosing (the time delay) < * (the lower bound) based on the theory provided in Lemma 2. Now, we are ready to give our the second main result. Theorem 2. Given a connected signed graph G = {V , E, C } and the stochastic opinion dynamic system with time delay is adopted by Equation (3). If the multiplicative noise and time delay satisfies the following two properties: then the trivial solution Y = 0 of Equation (5) is the almost sure exponential stable. Furthermore, it is straightforward to obtain an implicit lower bound for * and * is the unique root to Equation (13).
From Lemma 2 and Definition 1, it is clear that the trivial solution Y = 0 of Equation (5) is the almost sure exponential stable According to the previous Theorem, we give a method to determine the value of * by incoming two random variables and . Unfortunately, the calculation of the unique solution * for Equation (13) is a bit more involved and we cannot work out a more precise solution in this paper. Therefore the boundary on * is conservatively constrained, and one central challenging question that remains open is to obtain the optimal bound.

SIMULATION RESULTS
In this section, numerical experiments are shown to demonstrate the proposed stochastic opinion dynamic models and theoretical analysis.

Stochastic opinion dynamic systems
Consider a opinion dynamic system consisting of six agents. We adapt the continuous-time Altafini model as the stochastic model with multiplicative noise. The communication network topologies are shown in the Figure 1 and the corresponding matrices L are given by From the structurally unbalanced topology Figure 1(a) and structurally balanced topology Figure 1(b), we can get the corresponding matrix L a and L b . We can easily get {1, 5} ∈ V + and {2, 3, 4, 6} ∈ V − from the structurally balanced topology Figure 1 Figures 1(a,b) with different noise intensities respectively where the line colours correspond to the node colours in Figure 1. In Figures 2(a) and 3(a), we show the evolutions of the systems without the disturbance of stochastic multiplicative noise, and it is worth noting that the opinion  From Figures 2(b) and 3(b), it is simple to notice that the opinion values fluctuate around 0, when the noise intensities increase to [1, 1, 3, 0, 5, 7] T which do not meet the condition in Theorem 1. When the stochastic noise intensity satisfy the inequality constraint 2 2 min − 2 max > 2Γ 1 , the opinion values almost converge to 0 which verifies the stochastic stability of the equilibrium position Y = 0 where min = min{ 1 , 2 , … , n } and max = max{ 1 , 2 , … , n }, which verify the Theorem 1 above. According to the above simulation results, the structural balance, as a topological structure with special properties under determination systems, will no longer impact on the evolution of opinion values under the effect of multiplicative noise.

Stochastic opinion dynamic systems with time delay
We now take the impact of both multiplicative noise and time delay into consideration. Considered the two topologies in Figure 1, where the stochastic opinion dynamics systems are affected by multiplicative noise and time delay. The topology of signed graphs in this section is the same as the Section 5.1.  Figures 1(a) and 1(b). According to Theorem 2, we can calculate Γ 1 ≈ 21.45 under both topologies Figures 1(a) and 1(b). In this case, we will show the effective of time delay . In Figures 2(d) and 3(d), we set the both of the time delay to a large value = 0.05. It is clearly Y (t ) fluctuate around 0 sharply even if the noise intensity meet the requirement. When reduces to 0.016, the path of opinion value will almost tend to zero both in Figures 1(a) and 1(b). We can see the evolutions in Figure 2(e) and 3(e), where both noise intensity and time delay meet the conditions.

CONCLUSION
This paper addresses a stochastic stability problem of continuous-time opinion dynamic with multiplicative noise and time delay on coopetitive social networks. We have introduced a stochastic Lyapunov method to investigate the stochastic stability. In the case that the continuous-time system of opinion evolution is disturbed by stochastic multiplicative noise, the sufficient condition is established to make sure the almost surely stability of the stable equilibrium point of the system when the strength of stochastic noise is below the threshold. In the case that the continuous-time system of opinion dynamic is disturbed by multiplicative noise and time delay, we were able to provide conservative bounds on the strength of noise and the time delay for the almost sure exponential stability of stochas-tic systems. Finally, numerical simulation examples are given to show the theoretical results.