E R A D C M M U N I C A TNC Stochastic Hybrid Dynamic Multicultural Social Networks

In this work, we investigate the cohesive properties of a stochastic hybrid dynamic multi-cultural network under random environmental perturbations. By considering a multi-agent dynamic network, we model a social structure and find conditions under which cohesion and coexistence is maintained using Lyapunov’s Second Method and the comparison method. In this paper, we present a prototype illustration that exhibits the significance of the framework and approach. Moreover, the explicit sufficient conditions in terms of system parameters are given to exhibit when the network is cohesive both locally and globally. The sufficient conditions are algebraically simple, easy to verify, and robust. Further, we decompose the cultural state domain into invariant sets and consider the behavior of members within each set. We also analyze the degree of conservativeness of the estimates using Euler-Maruyama type numerical approximation schemes based on the given illustration.


Introduction
The aim of this work is to explore and extend the cohesive properties of a dynamic network of multiagents/members with a desired minimum safe distance between the members of the network [1][2][3] under the influence of both continuous and discrete-time stochastic perturbations. Dynamic network models play an important role in a variety of modeling applications. For example, economics, finance, engineering, management sciences, and biological networks have considered such large-scale dynamic models to investigate connectivity, stability, dynamic reliability, and convergence [4][5][6][7].
One of the concepts studied using a dynamic social network is that of consensus [8][9][10][11]. In such models, the conditions under which a group collectively comes to an agreement on an issue under consideration are studied. Another question of interest for such a network is when the group may divide into subgroups with an agreement reached within the subgroup but never reaching a consensus at an overall group level. Most of the work done in these areas look to develop consensus seeking algorithms and consider long term stability of the network in consideration [12][13][14][15].
The concepts of cohesion, coordination, and cooperation within a group are often multi-faceted, dynamic and complex, but are important concepts when trying to better understand how nations or communities function [16]. We seek to better understand the group dynamics of such a society in order to create policies and practices that encourage a sense of community among individuals from a variety of cultural backgrounds.
In fact, we systematically initiated the study of this issue [2,3] to better understand the social dynamics of a group seeking to find such a balance under the influence of both continuous and discrete-time deterministic and stochastic perturbations. In doing so, we are interested in better understanding the cohesive properties of a multi-cultural social network. In this work, we further extend the developed results in the framework of hybrid stochastic dynamic model for which we explore the features of the network. By considering a hybrid dynamic [17], we are able to consider the impact that events both from external and internal stochastic fluctuations coupled with an intervention process on the network have on the cultural dynamics. The presented work is used to exhibit the quantitative and qualitative properties of the network. Further, the techniques used are computationally attractive and algebraically simple relating with the underlying network parameters. This feature plays an important role for planning and decision processes.
In Section 2, we present a general problem under consideration and the underlining assumptions. We then present an illustration of such a network in Section 3 to exhibit the role and scope of the underlying complexity with the simplicity without loss of generality. Using an appropriate energy function and the comparison method, upper and lower estimates on cultural states are established in Section 4. In Section 5, the long-term behavior of the solutions to the comparison equations are examined and we explore the study of the cultural state invariant sets in the context of the illustration presented in Section 3. In Section 6, we use numerical simulations to model the network and to better understand to what extent the analytically developed estimates in Section 5 are feasible. Overall, the presented results are conservative but are reliable and robust.

Problem Formulation
The network consists of m agents whose position at time t is represented by , with . In our model, this vector does not represent a geographical location but rather a cultural position of the ith member. That is to say, the vector xi is a numerical representation of the ith member's beliefs or background on certain cultural or ethnic practices relevant to the network under study. Further, we assume that is a normalized Wiener process such that and for and are independent. We then consider a system of Itô-Doob type stochastic system of differential equations that describes the cultural state dynamic process: , where I in (1) stands for a discrete time intervention dynamic process. We will also make the following assumptions: is an n-dimensional initial cultural state random vector defined on the complete It is assumed that the initial value problem (1) for the system of stochastic differential equations has a solution process.
We wish to investigate the stochastic cohesive property of such a network. Further, we will explore the behavior of a member of the network based on the cultural state distance between a network member cultural state and the cultural state center of the network.
Below, we state a few definitions with regard the quantitative and qualitative behavior of the cultural network.
Let and be cultural state random vectors for and . For , we define the relative cultural state affinity with probability 1 sense by (9) i, j ÎI(1,m) We note that the relative cultural state affinity in the a.s. sense exists as is Borel measurable.

Prototype Dynamic Model
Let us define a prototype multicultural network dynamic model under the stochastic environmental perturbations described by the Itô-Doob type stochastic system of differential equations (10) for , and where and are positive real numbers, and We note that the solution process of (10) is defined by (12) Here, is the center of the multicultural dynamic system (1) defined by: (13) and note that by substituting for by into (10), we have The center of the multicultural dynamic model (10)   repulsive forces over are described by and the magnitude of the long range deterministic attractive forces are characterized by (16) Further, is the sine-cyclical influence of the ith member's relative distance to the center of the network. The stochastic term represents the environmental influence due to long-range attractive forces. In particular, in the case of a multi-cultural network, the noise captures the uncertainty generated due to the membership interactions and deliberations under the influence of the long-range cultural forces.
We remark that the solution process of (15) can be re-casted as (12). In order to study the multicultural dynamics (15), we use Lyapunov's Second Method in conjunction with the comparison method [18]. These methods are computationally attractive and provide a means of better understanding the movement and behavior of the state memberships of the network. By utilizing these methods, we are able to establish conditions for which we have both upper and lower estimates on the members' cultural state positions on the interval for . In this work, we assume that all inequalities are with probability 1.

Upper and Lower Comparison Equations
Using Lyapunov's Second Method and differential inequalities, we first seek a function such that (17) From Definition 1, relation (17) generates a concept of a locally upper-cohesive cultural network in the almost sure sense on the k-1th interval for .
To this end, for let us choose an energy function as: We have previously shown [2,3] that the differential of V k-1 in the direction of the vector field represented by (15) is and In the following, we present a result that will be used subsequently.
Lemma 1: Let V k-1 be the energy function defined in (18) and z i k-1 be a solution of the initial value problem defined in (15). Then, for each i ÎI(1,m),k ÎI(1,¥), and t Î[t k-1 ,t k ), where E stands for the expected value.
) be the solution process of (15). Let F t be an increasing family of sub -s algebras as previously defined and set m t where the last equality holds as z for all Dt > 0 sufficiently small such This together with (19), yields We note that for small Dt , we have

Upper Estimate of
We seek constraints on the parameters a k-1 ,b k-1 ,c k-1 ,q where h k-1 is defined by From the inequality (28) utilizing the comparison method [18] and Lemma 1, we establish the following lemma. For each interval [t k-1 ,t k ) and k ÎI(1,¥), the presented result establishes not only an upper bound but also the locally upper cohesive property almost surely. Hereafter, all inequalities and equalities are assumed to be valid with probability one.

Lemma 2:
Let V k-1 be the energy function defined in (18), k ÎI(1,¥),t Î[t k-1 ,t k ), and z i k-1 be a solution of the initial value problem defined in (15). Let r k-1 (t ) be the maximal solution [18] of a random initial value where h k-1 is defined as in (29). For each V k-1 (z i k-1 ) , i ÎI(1,m) , and k ÎI(1,¥) satisfying the differential inequality (28) and V k-1 (z i k-1 (t k-1 )) £ u k-1 , it follows that the multicultural dynamic network (10) is upper cohesive on [t k-1 ,t k ) with probability 1 and Proof: From Lemma 1, (28), and the application of stochastic comparison theorem [18], with probability 1, it follows that

Lower Estimate of
Next we consider the lower comparison equation. Using Lyapunov's Second Method and differential inequalities, we next seek a function r Again, from Definition 1, relation (33) initiates a notion of a locally lower cohesive cultural dynamic network in the almost sure sense.
Using the energy function defined in (18) and relation (21), it follows that By inequality (37) and the comparison method [18] and Lemma 1, we establish the following lemma. The presented result provides the lower estimate that in turn establishes the locally lower cohesive property of (15).
where a k-1 is as defined in (35). For each V k-1 (z i k-1 ) , i ÎI(1,m) , and k ÎI(1,¥) satisfying the differential inequality (37) and V(z i k-1 (t k-1 )) ³ u k-1 , it follows that the multicultural dynamic network (10) is lower cohesive on [t k-1 ,t k ) with probability 1 and Proof: From inequality (37) and Lemma 1 and the imitating the outline of the proof of Lemma 2, it follows that As the minimal solution of (38) is a lower bound, the network is lower cohesive almost surely. Moreover, a remark similar to Remark 1 establishes the locally stochastic mean and probability of (15).
We note that comparison differential equations (30) and (38) each have a unique solution process. Therefore the maximal and minimal solutions of (30) and (38) are the unique solutions of the respective random initial value problems.

Long-term Behavior of Comparison Differential Equations and Invariant Sets
To appreciate the role and scope of Lemmas 2 and 3, we seek to better understand both the behavior of the network on each interval [t k-1 ,t k ) and the long-term behavior of the network. For this purpose, for (38). Moreover, we analyze the qualitative properties of the solutions to the comparison equations. Using the comparison method [18], we are able to establish, quantitatively, the behavior of the individual member cultural dynamic states on the interval [t k-1 ,t k ). Using this, we also establish the overall longterm behavior of both individual member cultural dynamic states in the network as well as multicultural network state as a whole.
Following the method of finding the closed form solution process of the initial value problem [19], the solution of (38) is represented by By squaring both sides and rearranging the terms, we can write the above as u k-1 We now set where y(t k-1 ) = y k-1 on the interval [t k-1 ,t k ). Next, we take the derivative of both sides Therefore, on the interval [t k-1 ,t k ), the solution of Error! Reference source not found. is We assume that for k -1 ÎI(1,¥), the solution of (44) Then for k ÎI(1,m), the solution of (44) on [t and Therefore, using mathematical induction, it follows that for any k ÎI (1,¥),   T r a n s a c t i o n s o n N e t w o r k s a n d C o m m u n i c a t i o n s ; V o l u m e 5 , N o . 5 , O c t o b e r 2 0 1 7   C o p y r i g h t © S o c i e t y f o r S c i e n c e a n d E d u c a t i o n U n i t e d K i n g d o m 15 From the definition of y k and (50), for k ÎI(1,¥) and t Moreover, the solution process of (15) is globally lower cohesive a.s. on [t 0 ,¥).
Using the long term behavior of the comparison equations in conjunction with Lemmas 2 and 3, we establish the following theorem.
for t ³ t 0 . As the solutions r and r are bounded, the network is globally cohesive with probability 1.

Invariant Sets
In the case of the hybrid stochastic dynamical network, we can first consider the behavior of the solution process on the interval [t k-1 ,t k ). For k ÎI(1,¥), let us denote Further, let us define the following sets: From the analysis developed in that section, we establish the following theorem for the solution on the interval [t k-1 ,t k ). iii.
the set B k-1 ÈC k-1 is conditionally invariant relative to C k-1 . Proof: Following the proof outlined in [3], the result follows directly.    By considering the limit as k ® ¥, we also establish the following result for the long-range invariant sets of (15).
for both the upper and lower comparison equations, then as k ® ¥ and for sufficiently large k Î(1,¥). Thus, (15) exhibits long-range self-invariance for every member of the network.
In Section 6, we use numerical simulations to better understand the estimates and network behavior on the intervals [t k-1 ,t k ) for a finite number k .

Numerical Simulations
In this section, we consider numerical simulations for the multicultural dynamic network governed by the stochastic differential equation (15). We use a Euler-Maruyama [20][21][22] type numerical approximation scheme. We consider a network of six members, using the same initial position and varying the parameters a k-1 ,b k-1 , and b k-1 , k ÎI(1,¥) . Further, we consider the case such that x Often in a cultural network, events such as natural disasters, sudden political or economic changes, etc., can cause rippling effects in the cultural network. These changes can be characterized by the parametric changes in the stochastic differential equation (15). Therefore, we choose to simulate such a situation in the models in this section. Here, we choose 5 arbitrary times t k on the interval (0,1) for which the model experiences an intervention on the dynamic. Further, for each t k , k ÎI(1,5) , we set In order to consider the effects of changing the parametric quantity a k-1 , we consider various models for which b k-1 = 2,b k-1 = 1,c k-1 = 2 , and q k-1 = 1/7 are held constant for k ÎI(1,5) and a k = a k-1 +1 , a 0 = 2. The plot of the position z i (t ) for t Î[0,1] is given in Figure 1.
In order to consider the effects of changing the parametric quantity b k-1 , we consider the model for which a k-1 = 2,b k-1 = 2,c k-1 = 2, and q k-1 = 1/7 are held constant for k ÎI(1,5) and b k = b k-1 +1 , b 0 = 1.   In order to consider the effects of a change in the parametric quantity a k-1 and b k-1 , we consider the model for which b k = 1,c k-1 = 2, and q k-1 = 1/7 are held constant for k ÎI(1,5) and a k = a k-1 +1, and The plot of the member's positions of the simulated network is given in Figure 4.

Conclusion
Maintaining diversity while simultaneously fostering a sense of community membership, individual cultural identity, and cohesion is currently a goal among communities worldwide. It is important for members in a society to both feel as a part of the community in which they live and interact as well as feel free to embrace a strong sense of self and individuality. We seek to better understand the factors that play a role in obtaining such a balance by considering the impact of the repulsive and attractive forces influencing the multicultural network as in the previous work [2,3]. Attractive influences can be thought of as attributes that bring people to active membership within the group. Social acceptance, gaining social status, economic opportunity, career growth, common purpose and membership, personal development, and a sense of mutual respect, trust and understanding are examples of attractive influences within a social cultural network. Repelling forces are attributes that create some desire for individuals to leave or be less involved in the group or to preserve some personal identity from one other with their individual magnitude of inner repulsive force. A desire to retain a sense of individuality, economic or emotional cost, interpersonal conflict within the group, or disagreement with parts of the overall philosophies of the group are forces that may be considered as repulsive forces. The goal of the presented multicultural dynamic network is model the balance sought by members of the network in achieving these types of objectives. By doing so, we can consider the impact that policies and environmental factors may have on such a network.
By considering a hybrid dynamic model, we are able to better understand the impacts of outside influences that occur within community members and the cultural impacts such events have on the modeled cultural network. We have considered change based on the parameters that allow the perturbed multicultural dynamic network to remain cohesive while retaining a cultural state that is distinctive from the cultural state center of the network. We established qualitative and quantitative conditions that are computationally attractive and verifiable. We also conducted simulations of the multicultural network that exhibit the influence of the random perturbations and intervention processes as well as demonstrate the long-term behavior of the multicultural network. The presented results provide a tool for planning, performance, and implementations of policies and procedures within a social network.
We are interested in further exploring similar multicultural networks in the context of better understanding the relative cultural affinity x ij between members within the network and not just the cultural affinity between the cultural state of a member relative to the center of the network. The goal is to better understand the environmental factors that help foster a sense of individuality and diversity between all members within the network while maintaining a cohesive structure.