Topological traps in evolutionary games

TL;DR

This paper investigates how specific topological motifs sustain cooperation in spatial Prisoner's Dilemma games, revealing that rare nucleation events and resilient shapes govern high-temptation cooperation levels.

Contribution

It identifies key cooperator motifs responsible for residual cooperation and demonstrates that cooperation collapse is kinetically driven, not critical, emphasizing motif-level analysis.

Findings

Residual cooperation is maintained by specific motifs like 3x3 bricks and star-like structures.

Cooperation collapse near T=5/3 is due to motif destabilization, not loss of growth capacity.

High temptation cooperation is a rare-event nucleation phenomenon governed by topological traps.

Abstract

How cooperation originates and persists among self-interested individuals is a central question in the social and behavioural sciences. In the canonical two-dimensional spatial Prisoner's Dilemma with unconditional imitation introduced by Nowak and May (1992), simulations on a Moore lattice show an abrupt drop in cooperation near the temptation $T\approx5/3$, yet even under these harsh conditions cooperative structures can still arise. However, the nucleation rates of these motifs, and their contribution along the full cooperation curve had not been quantified. Here we show, using large-scale Monte Carlo simulations combined with automatic cluster classification, that on the Moore lattice for $T\ge5/3$ residual cooperation is sustained exclusively by $3\times3$ (or larger) rectangular cooperator bricks, whereas on degree-8 random-regular graphs for $T\gtrsim1.5$ it is dominated by star-like motifs (1 hub + 8 leaves). Once the dynamics becomes nucleation limited, the macroscopic cooperation level is therefore governed by the statistics of a few exceptionally resilient shapes, rather than by many different cooperator motifs. Furthermore, we show that the lattice cooperation collapse near $T=5/3$ is kinetic rather than critical: the reduction in cooperation is not due to a loss of growth capacity of rectangular bricks, but to the progressive destabilisation of the subcritical motifs that dominate just below this threshold. Our results show that residual cooperation at high temptation is a rare-event nucleation phenomenon governed by a small set of topological traps, and highlight the value of motif-level analysis for explaining and engineering cooperation in spatial, social, and technological networks.