Agents Attraction Competition in an Extended Friedkin-Johnsen Social Network

Abstract

Unlike many complex social networks investigated in the existing literatures concerning about the communication, consensus, or cooperative behaviors, this paper is devoted to an agents attraction competition problem in an extended Friedkin-Johnsen network. In this problem, two strategic agents compete to attract the nonstrategic agents to adopt opinions as closer to theirs as possible, by connecting with these nonstrategic agents and allocating their strengths on these connections. Therefore a 2-person zero-sum competitive game is characterized. We develop mathematical tools to provide some useful properties of the game, including the partial derivatives and positive definiteness of the payoff functions, based on which the Nash equilibrium of the game is analyzed. We rigorously examine that each player will allocate their strengths according to the susceptibilities, the intrinsic opinions and the costs of the nonstrategic agents, and the weighted column-sums of the row-stochastic matrix which describes the interpersonal influences of the network, if the network satisfies a condition. Otherwise, a Nash equilibrium seeking algorithm is proposed with simplex constraints, by which each strategic agent can asymptotically learn its optimal strategy. Finally, numerical examples are presented to illustrate the validity of the obtained results.