We analyze the late-time relaxation dynamics for a general contagion model. In this model, nodes are either active or failed. Active nodes can fail either “spontaneously” at any time or “externally” if their neighborhoods are sufficiently damaged. Failed nodes may always recover spontaneously. At late times, the breaking of time-translation invariance is a necessary condition for physical aging. We observe that time-translational invariance is lost for initial conditions that lie between the basins of attraction of the model's two stable stationary states. Based on corresponding mean-field predictions, we characterize the observed model behavior in terms of a phase diagram spanned by the fractions of spontaneously and externally failed nodes. For the square lattice, the phases in which the dynamics approaches one of the two stable stationary states are not linearly separable due to spatial correlation effects. Our results provide insights into aging and relaxation phenomena that are observable in a model of social contagion processes.

中文翻译：

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双稳态传染动力学的衰老和平衡

我们分析了一般传染模型的后期松弛动力学。在此模型中，节点处于活动状态或故障状态。如果活动节点的邻域受到充分破坏，则它们可以在任何时候“自发”或“外部”失效。发生故障的节点可能始终会自发恢复。在较晚的时间，打破时移不变性是物理老化的必要条件。我们观察到，初始条件位于模型的两个稳定稳态的吸引盆之间时，时间平移不变性会丢失。基于相应的平均场预测，我们通过相图来描述观察到的模型行为，相图由自发和外部故障节点的分数所覆盖。对于方格，由于空间相关效应，动力学接近两个稳定稳态之一的阶段无法线性分离。我们的结果提供了对社会传播过程模型中可观察到的衰老和松弛现象的见解。