  We analyze a class of stochastic processes, referred to as asynchronous and
semi-anonymous dynamics (ASD), over directed labeled random networks. These
processes are a natural tool to describe general best-response and noisy
best-response dynamics in network games where each agent, at random times
governed by independent Poisson clocks, can choose among a finite set of
actions. The payoff is determined by the relative popularity of different
actions among neighbors, while being independent of the specific identities of
neighbors.
  Using a mean-field approach, we prove that, under certain conditions on the
network and initial node configuration, the evolution of ASD can be
approximated, in the limit of large network sizes, by the solution of a system
of non-linear ordinary differential equations. Our framework is very general
and applies to a large class of graph ensembles for which the typical random
graph locally behaves like a tree. In particular, we will focus on labeled
configuration-model random graphs, a generalization of the traditional
configuration model which allows different classes of nodes to be mixed
together in the network, permitting us, for example, to incorporate a community
structure in the system. Our analysis also applies to configuration-model
graphs having a power-law degree distribution, an essential feature of many
real systems. To demonstrate the power and flexibility of our framework, we
consider several examples of dynamics belonging to our class of stochastic
processes. Moreover, we illustrate by simulation the applicability of our
analysis to realistic scenarios by running our example dynamics over a real
social network graph.
