Theory of Cooperation in Complex Social Networks

This paper presents a theoretical as well as empirical study on the evolution of cooperation on complex social networks, following the continuous action iterated prisoner’s dilemma (CAIPD) model. In particular, convergence to network-wide agreement is proven for both evolutionary networks with fixed interaction dynamics, as well as for coevolutionary networks where these dynamics change over time. Moreover, an extension to the CAIPD model is proposed that allows to model influence on the evolution of cooperation in social networks. As such, this work contributes to a better understanding of behavioral change on social networks, and provides a first step towards their active control. Introduction Modelling the evolution of cooperation in social networks has recently attracted much attention, aiming to understand how individuals work together and influence each other, and how society as a whole evolves over time (Nowak and May 1992; Santos and Pacheco 2005; Ohtsuki et al. 2006; Lazer et al. 2009; Hofmann, Chakraborty, and Sycara 2011). Progress made towards understanding how this evolution comes about has been mostly empirical in nature. Though compelling, deeper insights are better gained from an analytical analysis of the problem. Meanwhile, the control theory community has developed strong theories for dealing with various types of multiagent systems. Although many of these theories were initially developed for the analysis of artificial networks such as series of chemical reactors, electrical circuits or robotic swarms (Rosenbrock 1963; Jadbabaie, Lin, and Morse 2003; Ren, Beard, and McLain 2005), they can be extended to the analysis of social networks as well. For instance, OlfatiSaber (2005) uses properties of the Laplacian matrix, a typical tool in the control community for analysis of multi-agent systems, to analyze the information flow in small world networks. Additionally, Liu, Slotine, and Barabási (2011) study controllability, focussing on so-called driving nodes that need to be controlled in various complex networks. Summers and Shames (2013) use theory of nonlinear dynamical systems for modeling and influencing behaviors in specific social networks. Copyright © 2014, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. Recently a formal model for understanding the evolution of cooperation in arbitrary social networks has been proposed by Ranjbar-Sahraei et al. (2014). The authors introduce the continuous action iterated prisoner’s dilemma (CAIPD) as a general framework for modelling the behavior of individuals in complex networks. Moreover, they show that CAIPD is better able to capture the nature of cooperation in arbitrary networks than existing models. Due to its generalization capabilities and broad spectrum of applications, this paper adopts CAIPD as the formal framework on which an in-depth analytical study is conducted. Aiming at a broader theoretical understanding of the evolution of cooperation, this paper provides a formal analysis of agreement among individuals in complex social networks, following the CAIPD model. A set of theorems are presented and empirically validated, proving convergence to a final agreement in (co)evolutionary social networks. Additionally, the CAIPD model is extended to allow for the influence on final agreements in such type of networks. In particular, multi-rate adaptation is discussed. It is shown both theoretically and empirically that individuals with the slowest adaptation rates ultimately determine the final agreement. Finally, state-reference tracking is discussed as a special case of the proposed control extension, showing how a varying reference signal can be incorporated to guide the network to any level of agreement. Background Continuous-Action Iterated Prisoner’s Dilemma CAIPD (Ranjbar-Sahraei et al. 2014) is adopted for describing the evolution of cooperation in arbitrary complex networks. In CAIPD, N individuals are positioned on a vertices vi ∈ V for i = 1, 2, . . . , N of a weighted graph G = (V,W). The symmetrically weighted N × N adjacency matrix W = [wij ], with wij ∈ {0, 1}, describes the i to j individual connections with all wii = 0. One of the advantages of CAIPD over other models is the continuous nature in which cooperation and defection levels are modelled. To this end, Ranjbar-Sahraei et al. introduce xi ∈ [0, 1] to represent the cooperation level of each individual i, where xi = 0 corresponds to pure defection while xi = 1 represents pure cooperation; an individual pays a cost cxi while the opponent receives a benefit bxi, with b > c. This way a defector (i.e., xi = 0) pays no cost and distributes no benefits. Accordingly, the fitness of individual Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence