Various coarse-grained models have been proposed to study the spreading dynamics in the network. A microscopic theory is needed to connect the spreading dynamics with the individual behaviors. In this letter, we unify the description of different spreading dynamics on complex networks by decomposing the microscopic dynamics into two basic processes, the aging process and the contact process. A microscopic dynamical equation is derived to describe the dynamics of individual nodes on the network. The hierarchy of a duration coarse-grained (DCG) approach is obtained to study duration-dependent processes, where the transition rates depend on the duration of an individual node on a state. Applied to the epidemic spreading, such formalism is feasible to reproduce different epidemic models, e.g., the susceptible-infected-recovered and the susceptible-infected-susceptible models, and to associate with the corresponding macroscopic spreading parameters with the microscopic transition rate. The DCG approach enables us to obtain the steady state of the general SIS model with arbitrary duration-dependent recovery and infection rates. The current hierarchical formalism can also be used to describe the spreading of information and public opinions, or to model a reliability theory in networks.

中文翻译：

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复杂网络中依赖于持续时间的传播动力学的分层粗粒度方法

已经提出了各种粗粒度模型来研究网络中的传播动态。需要微观理论将传播动态与个体行为联系起来。在这封信中，我们通过将微观动力学分解为老化过程和接触过程这两个基本过程，统一了对复杂网络上不同传播动力学的描述。推导出微观动力学方程来描述网络上单个节点的动力学。获得持续时间粗粒度 (DCG) 方法的层次结构以研究持续时间相关过程，其中转换率取决于状态上单个节点的持续时间。应用于流行病传播，这种形式主义可以复制不同的流行病模型，例如，易感-感染-恢复和易感-感染-易感模型，并将相应的宏观传播参数与微观转变速率相关联。DCG 方法使我们能够获得具有任意持续时间相关恢复和感染率的一般 SIS 模型的稳态。当前的等级形式主义也可用于描述信息和舆论的传播，或为网络中的可靠性理论建模。