Online and Offline Dynamic Influence Maximization Games Over Social Networks

Abstract

In this work, we consider dynamic influence maximization games over social networks with multiple players (influencers). At the beginning of each campaign opportunity, individuals' opinion dynamics take independent and identically distributed realizations based on an arbitrary distribution. Upon observing the realizations, influencers allocate some of their budgets to affect their opinion dynamics. Then, individuals' opinion dynamics evolve according to the well-known DeGroot model. In the end, influencers collect their reward based on the final opinion dynamics. Each influencer's goal is to maximize their own reward subject to their limited total budget rate constraints, leading to a dynamic game problem. We first consider the offline and online versions of a single influencer's optimization problem where the opinion dynamics and campaign durations are either known or not known a priori. Then, we consider the game formulation with multiple influencers in offline and online settings. For the offline setting, we show that the dynamic game admits a unique Nash equilibrium policy and provide a method to compute it. For the online setting and with two influencers, we show that if each influencer applies the same no-regret online algorithm proposed for the single-influencer maximization problem, they converge to the set of $\epsilon$-Nash equilibrium policies where $\epsilon =\mathcal {O}(1/\sqrt{K})$ scales in average inversely with the number of campaign times $K$ considering the influencers' average utilities. Moreover, we extend this result to any finite number of influencers under more strict requirements on the information structure.