Evolution of cooperation on graphs with degree-dependent inertia.

Cooperation is fundamental to biological and social systems, yet its evolutionary success depends critically on population structure. In reality, individuals often exhibit inertia, retaining current behaviors even when imitation is beneficial. The synergistic effects between heterogeneous inertia and network structure have been explored primarily through numerical simulations. In this work, we develop a mathematical framework incorporating degree-dependent inertia, wherein the tendency to maintain one's strategy is determined by node degree, and self-loops are permitted, and self-loops are permitted. Using a coalescent-theoretic approach, we derive the threshold benefit-to-cost ratio favoring cooperation in the donation game on arbitrary graphs in the weak-selection limit. Compared with the no-inertia, uniform-inertia, and inverse degree-dependent inertia cases, positive degree-dependent inertia markedly reduces this critical threshold in disassortative networks. An analytical expression for multi-star networks further reveals that, architectures with fewer hubs and more leaves promote cooperation most effectively. While prior studies emphasize that slower updating by high-degree nodes and faster updating by low-degree nodes can foster cooperation, we propose a refined perspective: cooperation is especially favored when high-degree nodes update slowly and their neighbors are predominantly low-degree individuals. This alignment of inertia and neighborhood composition provides a mechanistic explanation for the emergence of cooperation in structured populations.