  Complex networks play an important role in human society and in nature.
Stochastic multistate processes provide a powerful framework to model a variety
of emerging phenomena such as the dynamics of an epidemic or the spreading of
information on complex networks. In recent years, mean-field type
approximations gained widespread attention as a tool to analyze and understand
complex network dynamics. They reduce the model's complexity by assuming that
all nodes with a similar local structure behave identically. Among these
methods the approximate master equation (AME) provides the most accurate
description of complex networks' dynamics by considering the whole neighborhood
of a node. The size of a typical network though renders the numerical solution
of multistate AME infeasible. Here, we propose an efficient approach for the
numerical solution of the AME that exploits similarities between the
differential equations of structurally similar groups of nodes. We cluster a
large number of similar equations together and solve only a single lumped
equation per cluster. Our method allows the application of the AME to
real-world networks, while preserving its accuracy in computing estimates of
global network properties, such as the fraction of nodes in a state at a given
time.
