Hierarchical Coarse-grained Approach to the Duration-dependent Spreading Dynamics in Complex Networks

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.