Stochastic and Deterministic State- Dependent Social Networks

Abstract

This work addresses the problem of finite-time convergence of, and the determination of the factors that impact on, the final opinion in a social network for a political party or an association, modeled as a distributed iterative system with graph dynamics chosen to mimic how people interact. It is firstly shown that, in this setting, finite-time convergence is achieved only when nodes form a complete network, and that contacting with agents with distinct opinions reduces to a half the required interconnections. Two novel strategies are presented that enable finite-time convergence, even for the case where each node only contacts the two closest neighbors. It is shown that, in a deterministic setting, the final opinion depends on a so-called connectivity parameter, which influences the relative contribution of each agent's initial belief. In the stochastic case, analogous conclusions are drawn, but in terms of expected values. The proposed strategies and results are relevant also in the context of mobile robot networks where the same assumption of having nodes communicating to their closest neighbors is satisfied. In addition, the results obtained are relevant in terms of saving resources while ensuring finite-time consensus. The performance of the proposed strategies is evaluated through simulation, illustrating, in particular, the key nodes that drive the network, as well as the associated rate of convergence.