网络演化博弈中的广义Hamilton动力学与混沌
Generalized Hamiltonian Dynamics and Chaos in Evolutionary Games on Networks
摘要：
研究了$2 \times 2$对称对策的网络复制子方程，刻画了它在任意图结构上的不动点。我们展示了网络复制子的渐近行为和图中独立顶点集的存在之间的关系，并表明在$2 \times 2$游戏中不会出现复杂的行为。这将动力系统的性质与组合图形性质联系起来。我们对比这表明，普通的石头剪刀布（RPS）表现出混沌的3圈和一般的图与$\geq 3$顶点的网络复制与RPS是一个广义的哈密顿系统。这与RPS在标准复制子动力学或双矩阵复制子动力学中不表现出混沌的既定事实形成鲜明对比，双矩阵复制子动力学等价于具有一条边和两个顶点的图（$K_2$）上的网络复制子。
英文摘要：
We study the network replicator equation and characterize its fixed points on arbitrary graph structures for $2 \times 2$ symmetric games. We show a relationship between the asymptotic behavior of the network replicator and the existence of an independent vertex set in the graph and also show that complex behavior cannot emerge in $2 \times 2$ games. This links a property of the dynamical system with a combinatorial graph property. We contrast this by showing that ordinary rock-paper-scissors (RPS) exhibits chaos on the 3-cycle and that on general graphs with $\geq 3$ vertices the network replicator with RPS is a generalized Hamiltonian system. This stands in stark contrast to the established fact that RPS does not exhibit chaos in the standard replicator dynamics or the bimatrix replicator dynamics, which is equivalent to the network replicator on a graph with one edge and two vertices ($K_2$).
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