# Large Fluctuations in Anti-Coordination Games on Scale-Free Graphs

## TL;DR

This paper investigates how the complex topology of scale-free networks influences the dynamics of anti-coordination games, revealing that different update rules lead to distinct effects on metastability and fixation times.

## Contribution

It demonstrates that the topology significantly affects game dynamics under voter updates but not under link dynamics, highlighting the role of network structure and update rules.

## Key findings

- Voter update rule causes topology-dependent behavior.
- Link dynamics results in behavior similar to complete graphs.
- Scale-free topology renormalizes population size under voter updates.

## 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.