  We consider the problem of predicting the time evolution of influence, the
expected number of activated nodes, given a set of initially active nodes on a
propagation network. To address the significant computational challenges of
this problem on large-scale heterogeneous networks, we establish a system of
differential equations governing the dynamics of probability mass functions on
the state graph where the nodes each lumps a number of activation states of the
network, which can be considered as an analogue to the Fokker-Planck equation
in continuous space. We provides several methods to estimate the system
parameters which depend on the identities of the initially active nodes,
network topology, and activation rates etc. The influence is then estimated by
the solution of such a system of differential equations. This approach gives
rise to a class of novel and scalable algorithms that work effectively for
large-scale and dense networks. Numerical results are provided to show the very
promising performance in terms of prediction accuracy and computational
efficiency of this approach.
