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Scaling laws for the spread of defection in evolutionary games on networks
Phys. Rev. E 114, 044301 – Published 1 October, 2026
DOI: https://doi.org/10.1103/25qf-sfsw
Abstract
We study the spread of defection in evolutionary games on networks using deterministic replicator dynamics posed on graphs. Each node hosts a mixed population of cooperators and defectors interacting through a Prisoner's Dilemma payoff matrix, with nearest-neighbor coupling determined by the network adjacency matrix. We develop two complementary analytical approximations that quantify how network architecture influences the timescale over which cooperation collapses. In a weak-coupling regime, internode interactions act as a small perturbation, yielding explicit approximate solutions on general graphs. For networks with strongly heterogeneous degree distributions, we derive a quasisteady approximation based on the natural separation of timescales between high- and low-degree nodes. Using these approximations, we introduce a defection spreading-time metric and show that the collapse of cooperation exhibits robust power-law scaling with system size across a broad range of network topologies. Analytical predictions are validated numerically on lattices, star graphs, incrementally connected graphs, small-world networks, and scale-free networks. The analysis shows that, to leading order, defection dynamics are controlled by local nodal degree, while network coupling and degree heterogeneity provide secondary corrections that influence the global collapse dynamics. We further show that increasing degree heterogeneity increases the average defection time, providing a complementary spectral interpretation through the principal eigenvalue of the adjacency matrix and related measures of graph irregularity. These results quantify how network architecture controls the transient dynamics of evolutionary collapse and establish new connections between evolutionary game dynamics and spreading processes on networks.
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