ProBLEMAS DE LAS PErSoNAS EN MEDIo orIENtE y PoSIBLES CAMINoS

To begin, the DeGroot–Friedkin model is extended to consider dynamic topology, and an assumption is then placed on the properties of the dynamicC(s).

First, the opinion dynamics for each issue is given by

y(s,t+1) =W(s)y(s,t), (6.2) where

W(s) =X(s) + (In−X(s))C(s). (6.3) This records the fact thatC(s)varies between issues but is constant for alltfor a given issue, in distinction to Eq. (5.2) in the previous chapter, whereCwas constant for each issue, and constant over the issue sequence. Precise details of the adjustments to the model arising from dynamic C are stated below. Here, an assumption is placed on the way in whichC(s)varies∀s ∈ S.

Assumption 6.1. For a given sequence ofC(s),s=0, 1, 2, . . ., the sequence is such that the entries ofC(s1), given as cij(s1), do not depend onx(s0),∀s0 ≤s1.

Assumption 6.1 ensures that the dominant left eigenvectorγ>(s)is independent

of the state x(s), because C(s) is independent of x(s). Notice that both the issue- driven and individual-driven examples in Section 6.1.1 satisfy Assumption 6.1. Al- most all issue-driven dynamicsC(s)satisfy the assumption because the sequence of C(s) depends only on the sequence of issues. That is, for analysis purposes the se- quence ofC(s)is considered to be determined before discussion begins ons=0, but individuali may not necessarily know the sequenceof cij(s)a priori.

However, a situation where individualiadjustscij(s)to be larger becauseiobser- ved that individualjhad large impactζj(s−1), does not satisfy the assumption. For

social network models with state-dependent parameters, limiting behaviour depends critically on the function relating the parameters to the state (as detailed earlier). For social systems (as opposed to e.g. autonomous robotic systems), this functio- nal dependence must necessarily reflect socio-psychological concepts. It is beyond the scope of this chapter to propose such functional dependence. Thus, Assump- tion 6.1 is in place, and investigation of how individuals might determineC(s)based on x(s), . . . ,x(0) may give rise to interesting future work. Furthermore, Assump- tion 5.1 is also assumed to hold throughout this chapter, and it is restated here for the reader’s convenience, with a minor adjustment to reflect the dynamic topology. Assumption 6.2. For all s ∈ S, the graph G[C(s)] = (V,E[C(s)],C(s)), with the same node setV of n≥3nodes, is strongly connected and no node vi has a self-loop. Furthermore,

the relative interaction matrixC(s)row-stochastic for all s∈ S.

The problem treated in this chapter is embedded in the following key objective. Objective 6.1. Consider a network G[C(s)] of n ≥ 3 individuals that discuss a sequence of issues indexed by s ∈ S = {0, 1, 2, . . .}. Suppose that Assumptions 6.1 and 6.2 hold. For each issue s, the DeGroot process Eq. (6.2) is used to describe the opinion dynamics, with the influence matrixW(s)determined according to Eq. (6.3). Suppose that at the end of each issue, the vector of social power x(s) defined above Eq. (5.2) changes according to Eq. (5.3). Then, the objective is to study the dynamical evolution (including convergence) of x(s)over the sequence of issuesS, and determine what (if any) parameters ofG[C(s)]affect the evolution ofx(s)

For a population of n ≥ 3 individuals, consider a social network G[C(s)] = (V,E[C(s)],C(s)), with a finite setC of Ppossible relative interaction matrices defi- ned as C = {Cp ∈ Rn×n : p ∈ P } where P = {1, 2, . . . ,P}. For simplicity, it is also assumed that@psuch that the graphG[Cp]has star topology (see Definition 5.1). Let σ(s) : N → P be a switching signal, and suppose that σ(s) determines the dyna- mic switching as C(s) = C_σ(s). Suppose that Assumption 6.2 holds1 for all s ∈ S.

Suppose further that Assumption 6.1 holds, which implies for any s1 ≥ 0, σ(s1) is

independent of the statex(s), for alls< s1. Then, the DeGroot–Friedkin model with

dynamic relative interaction matrices is

x(s+1) =F_σ(s)(x(s)), (6.4)

where the nonlinear map Fp(x)forp ∈ P, is defined as

Fp(x) =                  ei if x=ei for anyi αp(x)     γp,1 1−x1 .. . γp,n 1−xn     otherwise , (6.5)

1_{Assumption 6.2 is equivalent to requiring that Assumption 5.1 holds separately for allC}

where αp(x) =1/∑n_i=1

γp,i

1−xi and γp,i is thei

th _{entry of the dominant left eigenvector} of Cp, γ>_p = [γp,1,γp,2, . . . ,γp,n]. The derivation for Eq. (6.5) is an extension of the derivation of Eq. (5.5) using [Jia et al., 2015, Lemma 2.2], by noting thatC(s) =C_σ(s).

Remark 6.1. Analysis using the usual techniques for switched systems is difficult for the system Eq. (6.4). For arbitrary switching, one might typically seek to find a common Lya- punov function, i.e., one which would establish convergence for any fixed value of p ∈ P. This, however, appears to be difficult (if not impossible) for Eq. (6.4). In the constant Ccase studied in [Jia et al., 2015], the convergence result relied on 1)a Lyapunov function which was dependent on the unique equilibrium point x∗, and 2) LaSalle’s Invariance Principle. Both1)and2)are invalid when analysing Eq. (6.4). In the case of1), the system Eq. (6.4) does not have a unique equilibrium point x∗ but rather a unique trajectoryx∗(s)(as will be made clear in the sequel). In the case of2), LaSalle’s Invariance Principle is not applicable to general non-autonomous systems.

The following three properties were detailed in Chapter 5, and are detailed here again (with adjusted numbering) for the convenience of the reader.

Property 6.1. The map Fp(x)in Eq. (6.5) is continuous on∆nfor all p∈ P.

Property 6.2. For the system x(s+1) = Fp(x(s)) with map Fp given in Eq. (6.5), and

G[Cp]does not have star topology, there exists a sufficiently small r such that for any r0 ≤r,

xi(s) =1−r0implies xi(s+1)<1−r0, for all i. Thus, for the set2A={x∈∆n: 1−r≥

xi ≥0,∀i∈ {1, . . . ,n}}, there holds Fp(A)⊂ A.

Property 6.3. For the system x(s+1) = Fp(x(s)) with map Fp given in Eq. (6.5), if

x(s1)∈∆ne for some s₁<∞, thenx(s)∈int(∆n)for all s> s₁.

The DeGroot–Friedkin model with dynamic relative interpersonal interactions has now been formally defined, and Objective 6.1 is investigated in the remaining sections of this chapter.

6.3

Exponential Convergence to a Unique Limiting Trajectory