  We study expanding circle maps interacting in a heterogeneous random network.
Heterogeneity means that some nodes in the network are massively connected,
while the remaining nodes are only poorly connected. We provide a probabilistic
approach which enables us to describe the effective dynamics of the massively
connected nodes when taking a weak interaction limit. More precisely, we show
that for almost every random network and almost all initial conditions the high
dimensional network governing the dynamics of the massively connected nodes can
be reduced to a few macroscopic equations. Such reduction is intimately related
to the ergodic properties of the expanding maps. This reduction allows one to
explore the coherent properties of the network.
