Large fluctuations in anti-coordination games on scale-free graphs

Abstract

We study the influence of the complex topology of scale-free graphs on the dynamics of anti-coordination games (e.g. snowdrift games). These reference models are characterized by the coexistence (evolutionary stable mixed strategy) of two competing species, say "cooperators" and "defectors", and, in finite systems, by metastability and large-fluctuation-driven fixation. In this work, we use extensive computer simulations and an effective diffusion approximation (in the weak selection limit) to determine under which circumstances, depending on the individual-based update rules, the topology drastically affects the long-time behavior of anti-coordination games. In particular, we compute the variance of the number of cooperators in the metastable state and the mean fixation time when the dynamics is implemented according to the voter model (death-first/birth-second process) and the link dynamics (birth/death or death/birth at random). For the voter update rule, we show that the scale-free topology effectively renormalizes the population size and as a result the statistics of observables depend on the network's degree distribution. In contrast, such a renormalization does not occur with the link dynamics update rule and we recover the same behavior as on complete graphs.

WKB theory of large deviations in stochastic populations

“…Refs. [140][141][142][143] studied evolutionary models where the network nodes represent either a wild-type individual A, or a mutant B, and their interactions such as A + B → A + A or A + B → B + B determine the evolutionary dynamics. The main questions of interest here are in calculating the fixation probability of each species and the mean fixation time.…”

How does homophily shape the topology of a dynamic network?

“…An important line of research has focused on dynamical processes on networks, particularly on opinion dynamics [2] and evolutionary processes [9]. In this context, the dynamics of paradigmatic statistical physics models have been studied on complex networks whose structure is random but static, see, e.g., [10][11][12][13][14][15][16][17]. In other models, collective phenomena emerge from the interactions between agents whose links evolve while the state of the nodes (e.g., representing an agent's "opinion") remain static.…”

Effects of homophily and heterophily on preferred-degree networks: mean-field analysis and overwhelming transition

“…Here, we focus on an evolving network model in which the opinions of the nodes remain fixed while the links, hence the entries of A, evolve stochastically and their number fluctuates endlessly. This may be viewed as the complement of most voter models where only the nodes are dynamic, see, e.g., [18][19][20][21][22][23][24][25]. In opinion dynamics, our fixed-opinion agents are often referred to as "zealots" [42,[66][67][68][69], see also Refs.…”

“…As a result, there is an intense field of research dedicated to the study of social networks by means of models and tools borrowed from statistical physics [15][16][17]. In particular, various dynamical processes have been studied on complex networks whose structure is random but static, see, e.g., [18][19][20][21][22][23][24][25], while in other models agents and links co-evolve [26][27][28][29][30][31][32]. Naturally, collective phenomena such as phase transitions and polarization that emerge from the agent interactions have received great attention.…”