Abstract
This chapter extends the DeGroot–Friedkin model by introducing dynamic network topology, and in particular, issue-varying relative interaction topology (the term “issue” is used interchangeably with “topic”). The dynamic network topology problem is formally defined, with motivating examples provided as to why issue-varying networks are reflective of real-world social networks. Then, the nonlinear contraction analysis framework introduced in Chap. 5 is used to draw a key conclusion: under mild assumptions on the properties of the issue-varying network, each individual’s initial (perceived) social power is forgotten exponentially fast, and in the limit of the issue sequence, each individual’s social power converges to a “unique limiting trajectory” that depends only on the issue-varying network topology. The results in Sect. 5.3 concerning the upper bound on an individual’s social power at equilibrium and convergence rate for a class of topologies are also extended, with obvious modifications due to the dynamic topology. As a special case, periodically-varying topologies are also considered.
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Notes
- 1.
- 2.
The sets \(\varDelta _n\), \(\widetilde{\varDelta }_n\), and \(\text {int}(\varDelta _n)\) were defined in Sect. 2.1, and are now redefined for the reader’s convenience: The n-simplex is \(\varDelta _n = \{\varvec{x}\in \mathbb {R}^n : 0 \le \varvec{x}, \varvec{1}_n^\top \varvec{x} = 1 \}\), and \(\widetilde{\varDelta }_n = \varDelta _n \backslash \{ \mathbf {e}_1, \ldots , \mathbf {e}_n \}\) and \(\text {int}(\varDelta _n) = \{\varvec{x}\in \mathbb {R}^n : 0 < \varvec{x}, \varvec{1}_n^\top \varvec{x} = 1 \}\).
- 3.
A well-known set is \(\mathcal {K}(\varvec{1}_n/n)\), the set of \(n\times n\) doubly-stochastic \(\varvec{C}\).
- 4.
Any given \(s\in \mathcal {S}\) can be uniquely expressed by a given fixed positive integer P, a nonnegative integer q, and positive \(p\in \mathcal {P}\), as shown.
- 5.
During the thesis examination process, an examiner has identified an alternative proof, which we summarise here for the interested reader. The existence of \(\varvec{x}^*(s)\) was established in Theorem 6.1. Since \(\varvec{x}(2s+2)\) is also a trajectory, \(\Vert \varvec{x}^*(2s+2) - \varvec{x}^*(2s)\Vert \) converges to 0 exponentially fast. From this, one has that for any \(s \ge 0\) and \(k \in \mathbb {N}\), \(\Vert \varvec{x}^*(s+2k) - \varvec{x}(2s) \Vert \) converges to 0 exponentially fast; that is \(\varvec{x}^*(2s)\) is a Cauchy sequence in \(\bar{\mathcal {A}}\), which must converge to a point, denoted as \(\varvec{z}_1^*\). From Eq. (6.14), and by continuity, \(\varvec{z}_1^*\) must be a fixed point of \(\varvec{F}_1(\varvec{F}_2(.))\). A similar approach shows that \(\varvec{x}^*(2s+1)\) must converge to a point, \(\varvec{z}_2^*\), which must be a fixed point of \(\varvec{F}_2(\varvec{F}_1(.))\).
- 6.
The initial condition vector \(\widehat{\varvec{x}}(0)\) also illustrates the relaxed initial condition assumption.
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Ye, M. (2019). Dynamic Social Networks: Exponential Forgetting of Perceived Social Power. In: Opinion Dynamics and the Evolution of Social Power in Social Networks. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-10606-5_6
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