We study stochastic pairwise interaction network systems whereby a finite population of agents, identified with the nodes of a graph, update their states in response to both individual mutations and pairwise interactions with their neighbors. The considered class of systems include the main epidemic models -such as the SIS, SIR, and SIRS models-, certain social dynamics models -such as the voter and anti-voter models-, as well as evolutionary dynamics on graphs. Since these stochastic systems fall into the class of finite-state Markov chains, they always admit stationary distributions. We analyze the asymptotic behavior of these stationary distributions in the limit as the population size grows large while the interaction network maintains certain mixing properties. Our approach relies on the use of Lyapunov-type functions to obtain concentration results on these stationary distributions. Notably, our results are not limited to fully mixed population models, as they do apply to a much broader spectrum of interaction network structures, including, e.g., Erd\"oos-R\'enyi random graphs.

翻译：我们研究随机成对交互网络系统，其中有限数量的智能体（与图的节点相对应）响应个体突变和与邻居的成对交互而更新其状态。所考虑的系统类别包括主要流行病模型（如SIS、SIR和SIRS模型）、某些社会动力学模型（如选民和反选民模型），以及图上的演化动力学。由于这些随机系统属于有限状态马尔可夫链类别，它们总是存在平稳分布。我们分析这些平稳分布在种群规模趋于无穷大而交互网络保持一定混合性质时的渐近行为。我们的方法依赖于使用李雅普诺夫型函数来获得这些平稳分布的集中性结果。值得注意的是，我们的结果不限于完全混合的种群模型，因为它们适用于更广泛的交互网络结构谱，例如Erdős-Rényi随机图。