7.6 Appendix: Proofs
7.6.5 Proof for Theorem 7.6
The relative interaction matrix for Topology Variation 7.5 is given by
C(β1,β2) = 0 c12 c13 . . . β1 0 . . . 0 1 0 0 . . . 0 0 . . . 0 .. . ... ... . .. ... ... . .. ... 1 0 0 . . . 0 0 . . . 0 β2 0 0 . . . 0 cn+1,n+2 . . . cn+1,n 0 0 0 . . . 1 0 . . . 0 .. . ... ... . .. ... ... . .. ... 0 0 0 . . . 1 0 . . . 0 (7.15)
And the expressionγ>C =γ> yields the following equalities
γ1=
∑
1<i≤n γi+β2γn+1 (7.16a) γi =c1,iγ1, ∀i∈ {2, ...,n} (7.16b) γn+1=∑
n+1<i≤n+m γi+β1γ1 (7.16c) γi =cn+1,iγn+1, ∀i∈ {n+2, ...,n+m} (7.16d)Statement (i) is obtained trivially from Eq. (7.16b) and Eq. (7.16d). In regards to Statement (ii), first substitute Eq. (7.16b) into Eq. (7.16a) to obtain γ1 = β2γn+1+ ∑1<i≤nc1,iγ1which is rearranged to yieldγ1(1−∑1<i≤nc1,i) =β2γn+1, and which is
equivalent to β1γ1 = β2γn+1 because 1−∑1<i≤nc1,i = β1. Statement (iii) is obtained
Nonlinear Mapping Convergence
and Application to Social Power
Analysis
Chapter Summary
This chapter considers the original DeGroot–Friedkin model. Recently, a nonlinear mapping convergence result was developed using Lefschetz fixed point theory; a map can be shown to have a unique fixed point that is locally exponentially stable if the Jacobian evaluated at any fixed point satisfies a certain property. This result is used to analyse the DeGroot–Friedkin model, allowing an exponential convergence result to be established using a tool different to the nonlinear contraction analysis employed in Chapters 5 and 6.
It must be noted that both the general result and the application to the DeGroot– Friedkin model will appear in the same conference paper at the 2018 European Con- trol Conference (see page vii, paper [10]). The general result was developed by B. D. O. Anderson, and will be presented in Section 8.2. The application to the DeGroot– Friedkin model was the work of the author, and will thus be presented separately.
8.1
Introduction
Recursive equations of the form
x(k+1) =F(x(k)), (8.1)
are fundamental to control and signal processing. Very often F is linear or affine, but this is not always so and as a result, established linear system theory techniques cannot be applied. However, and as is the case in this chapter, F is often suitably smooth. Usually also,x(k)resides in a Euclidean space of known dimension, though this is not always the case, and indeed will not always be the case in the problem considered in this chapter.
In many situations, it is possible to examine local behaviour of the nonlinear map 137
Fin Eq. (8.1) around an equilibrium point, through a linearisation process. If ¯xis an equilibrium point, i.e. a fixed point of the mapping F satisfying ¯x = F(x¯), then the Jacobian J(x¯) = ∂F
∂x|x¯ can often provide guidance as to behaviour in the vicinity of ¯x. Ifkx(k)−x¯kis small, for some normk · k, then approximately
x(k+1)−x¯ = J(x¯)[x(k)−x¯] (8.2) If the eigenvalues of J(x¯)do not lie on the unit circle, then the asymptotic stability or instability of the linear equation Eq. (8.2) implies the same property for the nonlinear equation Eq. (8.1), albeit locally.
The DeGroot–Friedkin model, introduced in [Jia et al., 2015] and studied in Chap- ters 5–7 thus far, is what amounts to a particular version of Eq. (8.1). [Jia et al., 2015] established by two different sets of rather specialised calculations, tailored to the specific algebraic form of F, that under normal circumstances, the system Eq. (8.1) (i) possesses a unique fixed point for the map F, and (ii) the equilibrium is globally asymptotically stable. Chapter 5 used nonlinear contraction theory to obtain an ex- ponential convergence result, again exploiting the specific algebraic form of F. In that instance, the specific algebraic form ofF was used to find the expression of J(x)
over the entire state-space, and from this expression of the Jacobian, a differential coordinate transform was introduced to establish a contractive property.
It is natural to speculate whether the conclusion that there is a single attractive equilibrium is indeed intrinsic to the algebraic form of F, or whether rather, it is a consequence of some more general property, and consequently also one that will follow for a whole class of F of which that in the DeGroot–Friedkin model is just a special case. It turns out that this conclusion is indeed a consequence of a more general property. The recent paper [Anderson and Ye, 2018] showed, using Lefschetz fixed point theory, that if a system of the form Eq. (8.1) acting on a positive invariant set (with the set satisfying some further topological conditions) has at least one fixed point and the Jacobian at every fixed point has eigenvalues inside the unit circle, then the mapFhas a unique fixed point and Eq. (8.1) is locally exponentially stable about that point. This chapter shows that the DeGroot–Friedkin model has a mapF which is in the larger class of maps for which the result in [Anderson and Ye, 2018] holds.
By way of brief background, Lefschetz fixed point theory (of which more de- tails are summarised subsequently) is a tool for relating the local behaviour of maps to some global properties, taking into account the underlying topological space in which the maps act. The local properties are associated with the linearised equations Eq. (8.2), potentially studied at multiple equilibrium points (and with in general a different J(x¯) associated with each equilibrium point). Such local properties were flagged in Chapters 5 and 6 as of central concern in, respectively, a time-invariant and a time-varying version of the DeGroot–Friedkin model.
To sum up the contribution of this chapter, an exponential convergence result is established for the DeGroot–Friedkin model using a recently developed nonlinear mapping convergence result. The analysis method avoids the detailed calculations required in [Jia et al., 2015] and Chapter 5 to establish uniqueness and attractiveness
of the equilibrium, by only requiring evaluation of the Jacobian at each fixed point of the map (assuming at least one fixed point exists); uniqueness and local exponential stability are guaranteed if the Jacobian at each fixed point satisfies a certain property. 8.1.1 Chapter Organization
In the remaining part of this chapter, Section 8.2 reviews a recently developed non- linear mapping result, and Section 8.3 shows how this result may be applied to the DeGroot–Friedkin model. Conclusions are drawn in Section 8.4.