Limit cycles in replicator-mutator network dynamics

Abstract

Abstract-The replicator-mutator equations from evolutionary dynamics serve as a model for the evolution of language, behavioral dynamics in social networks, and decision-making dynamics in networked multi-agent systems. Analysis of the stable equilibria of these dynamics has been a focus in the literature, where symmetry in fitness functions is typically assumed. We explore asymmetry in fitness and show that the replicator-mutator equations exhibit Hopf bifurcations and limit cycles. We prove conditions for the existence of stable limit cycles for the dynamics in the case of circulant fitness matrices, and illustrate their existence in the noncirculant case. For decision-making networks, these limit cycles correspond to sustained oscillations in decisions across the group. I. INTRODUCTIONIn this paper we study the replicator-mutator equations from evolutionary dynamics; these serve as a model for the evolution of language [1], for behavior selection in social networks [2], [3], and also for decision-making dynamics in networked multi-agent systems [4]. Our main result is to prove a Hopf bifurcation for these dynamics with certain network interconnection topologies; this implies the existence of limit cycle behavior. For decision-making networks, the limit cycles correspond to sustained oscillations in decisions across the group.Evolutionary dynamics [5], [6], [7] are, broadly speaking, an effort to cast the basic tenets of Darwinian natural selection (replication, competition, strategy dependent fitness, mutation) in a mathematical framework. The set of replicator equations [8] are the simplest model of evolutionary dynamics for a population divided among a finite set of N competing strategies (N ≥ 2). The differential equations model the game theoretic interactions among the sub-populations, each subscribed to a different competing strategy, and determine how the different sub-populations change in size as a consequence of these interactions.Although the replicator dynamics have proved to be a powerful tool in analyzing a variety of classical games from an evolutionary perspective, they do not model mutation, a key ingredient of selection theory. Mutation can be included by adding the possibility that individuals spontaneously change from one strategy to another. This yields the replicatormutator dynamics [9], which have played a prominent role in evolutionary theory and have recently been employed to model social and multi-agent network dynamics [2]. The replicator-mutator dynamics can be interpreted from a

Hopf Bifurcations and Limit Cycles in Evolutionary Network Dynamics

“…Figure 3.5 shows phase portraits of the dynamics for various choices of µ, illustrating the Hopf bifurcation. In [22] we proved that stable limit cycles of the dynamics exist in a wide region of parameter space for circulant payoff matrices B, for which the directed cycle topology in Figure 3.3(e) is a special case. Here we state two of the main results from [22] that provide necessary and sufficient conditions for the existence of stable limit cycles for (2.2) with circulant payoff matrix given by…”

“…There are 73 corresponding non-isomorphic graph topologies in the set [22]. Figure 6.1 shows stable limit cycles for four topologies in this set corresponding to noncirculant payoff matrices.…”

“…Analysis for N = 3 Strategies. To build intuition for our general results in §4, we summarize here the main results from our recent paper [22], which focuses on the bifurcations of the dynamics (2.2) as a function of the bifurcation parameter µ for N = 3 strategies. Because the simplex is two-dimensional for N = 3 (2.3), it is easier than in higher dimensions to prove necessary and sufficient conditions for limit cycles and to visualize codimension-one bifurcations.…”

“…In §3 we show motivating simulations and illustrate bifurcations of the dynamics for N = 3 strategies, summarizing our previous work in [22]. In §4 we present our main result proving Hopf bifurcations for N ≥ 3 strategies and a two-parameter family of circulant fitness matrices.…”

Introduction to Complex Networks: Structure and Dynamics

“…We turn here our attention to the so-called replicator-mutator model, which describes the dynamics of complex adaptive systems, such as in population genetics, autocatalytic reaction networks and the evolution of languages [27,38,41,42]. Let us consider a series V D fv 1 ; v 2 ; : : : ; v n g of n agents such that each agent plays one of the n behaviours or strategies available at b D fb 1 ; b 2 ; : : : ; b n g. Let x D fx 1 ; x 2 ; : : : ; x n g be a vector such that 0 Ä x i Ä 1 is the fraction of individuals using the i th behaviour.…”