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.

翻译：自利个体之间合作如何起源并持续存在，是社会科学与行为科学的核心问题。在Nowak与May（1992）提出的采用无条件模仿规则的经典二维空间囚徒困境中，基于Moore晶格的模拟显示合作水平在诱惑参数$T\approx5/3$附近急剧下降，但即便在此严苛条件下，合作结构仍可能涌现。然而，这些合作基元的成核速率及其在整个合作曲线中的贡献尚未被量化。本文通过大规模蒙特卡洛模拟结合自动聚类分类技术证明：在Moore晶格中，当$T\ge5/3$时，残余合作完全由$3\times3$（或更大）的矩形合作砖块维持；而在度数为8的随机正则图中，当$T\gtrsim1.5$时，合作主要由星型基元（1个中心节点+8个叶节点）主导。一旦动力学过程受限于成核机制，宏观合作水平便由少数具有异常韧性的形态统计特性所决定，而非多种不同的合作基元。进一步研究表明，晶格系统在$T=5/3$附近的合作崩塌属于动力学现象而非临界现象：合作减少并非源于矩形砖块生长能力的丧失，而是由于略低于此阈值时占主导的亚临界基元逐渐失稳。我们的结果表明，高诱惑参数下的残余合作是一种由少量拓扑陷阱主导的稀有事件成核现象，并凸显了基元层次分析对于解释及设计空间网络、社会网络与技术网络中合作机制的重要价值。