  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.
