Stochastic and Deterministic State-Dependent Social Networks

This article investigates a political party or an association social network where members share a common set of beliefs. In modeling it as a distributed iterative algorithm with network dynamics mimicking the interactions between people, the problem of interest becomes that of determining: 1) the conditions when convergence happens in finite time and 2) the corresponding steady-state opinion. For a traditional model, it is shown that finite-time convergence requires a complete topology and that by removing neighbors with duplicate opinions reduces in half the number of links. Finite-time convergence is proved for two novel models even when nodes contact two other nodes of close opinion. In a deterministic setting, the network connectivity influences the final consensus and changes the relative weight of each node on the final value. In the case of mobile robots, a similar communication constraint is present which makes the analysis of the social network so relevant in the domain of control systems as a guideline to save resources and obtain finite-time consensus. Through simulations, the main results regarding convergence are illustrated paying special attention to the rates at which consensus is achieved.