Consensus on Social Graphs under Increasing Peer Pressure

In this paper, we present a novel generalized framework for expressing peer influence dynamics over time in a set of connected individuals, or agents. The proposed framework supports the representation of individual variability through parametrization accounting for differences in susceptibility to peer influence and pairwise relationship strengths. Modeling agents’ opinions and behaviors as strategies changing discretely and simultaneously, we formally describe the evolution of strategies in a social network as the composition of contraction maps. We identify points of convergence and analyze these points under various conditions. DISCUSSION In a connected social network, we show that agents’ strategies converge to the optimal solution, namely consensus, if and only if peer pressure increases without bound. If the peer pressure is bounded and individual agents fail to change their opinions in the presence of dissonance with their peers, then these agents will not converge to an efficient distribution. Our notion of persuasion and peer-pressure is related to the psychology literature on belief formation and social influence. We follow Friedkin’s foundational theory [1] that strong ties are more likely to affect users’ opinions and result in persuasion or social influence. Underpinning our model is also the notion of mimicking. Brewer and more recently Van Bareen [2, 3] suggest that mimicking is used when individuals feel out of a group and therefore will alter their behavior (to a point [3]) to be more socially accepted. Assume that the agents’ network is represented by a simple graph G = (V,E) where vertexes V are users (or agents), and edges E are the social (communications) connections between them. It is clear that disconnected sections of the graph are independent, so we assume that G is connected. For the remainder of the paper, let V = {1, 2, . . . n}, so E is a subset of the two-element subsets of V . Assume we are given a set of non-negative vertex weights si and positive edge weights wij respectively for i, j ∈ V . In addition, we assume at least one vertex weight is positive. Without loss of generality, assume the range of opinions to be the inAppears in: Proceedings of the 16th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2017), S. Das, E. Durfee, K. Larson, M. Winikoff (eds.), May 8–12, 2017, São Paulo, Brazil. Copyright c © 2017, International Foundation for Autonomous Agents and Multiagent Systems (www.ifaamas.org). All rights reserved. terval [0, 1] and each user has a private constant preference x+i ∈ [0, 1]. The user’s opinion value at time k is x (k) i ∈ [0, 1]. The set of all such values are denoted by the vector x while the set of constant private preferences is x. Each agent selects its next choice by minimizing an objective function representing social stress: Ji(x (k) i ,x , k) = si ( x (k) i − x + i )2