Topological traps in evolutionary games

TL;DR

This study investigates how cooperation persists in spatial Prisoner’s Dilemma dynamics under unconditional imitation on two topologies: a Moore lattice and a degree-8 random-regular graph, using large-scale Monte Carlo simulations and automatic motif classification. It shows that residual cooperation at high temptation $T$ is sustained by a small set of motifs: $3\times3$ bricks on the lattice for $T\ge 5/3$, and completed star motifs on RR graphs for $T\gtrsim 1.5$, with dynamics that are nucleation-limited and kinetically controlled rather than governed by a critical phase transition. The authors introduce a detailed motif-analysis framework (shape, size, stability) and demonstrate that the macroscopic cooperation curve is shaped by the availability and growth of a few topological traps, not by a diversity of cooperator motifs. These findings highlight the role of topology and update rules in engineering cooperation and provide a path toward motif-aware design of cooperative networks. The work bridges spatial game theory and network science, offering actionable insights for managing cooperation in social, technological, and biological networks where local patterns govern global outcomes.

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.

Figures (10)

• Figure 1: Final-step ($t = 99$) network-level metrics as a function of temptation $T$ for 2D lattices and random-regular graphs. Lines show means across repetitions; shaded regions indicate $\pm 1$ standard deviation.
• Figure 2: Mean number of stable ($N_s$) and unstable ($N_u$) cooperator clusters at $t = 99$ as a function of temptation $T$, for 2D lattices and random-regular graphs. Shaded regions indicate $\pm 1$ standard deviation.
• Figure 3: Share of stable and unstable clusters by motif type and temptation at $t = 99$ for 2D lattices and Random-Regular graphs.
• Figure 7: Time evolution of cooperator clusters on a $100\times100$ lattice for $T=1.7$. Snapshots at time steps $t=0,1,2,3,4,99$ show an abrupt halt to all growth. Only 3x3 (or larger bricks) are stable.
• Figure 8: Stable cooperator cluster motif composition at $t = 99$ for 2D lattices. Each cell shows the share of clusters of a given motif at temptation $T$; light grey cells correspond to zero frequency.