Abstract. 
In these talks I am going to report results on ergodic theoretical properties
of dynamical systems coupled on graphs. The local dynamics at each node
is uniformly expanding and coupled with other nodes according to the
edges of the graph. The attention is focused on the case of graphs with
heterogeneous degrees meaning that most of the nodes make a small
number of interactions, while a few hub nodes have very high degree.
For such high-dimensional systems there is a regime of the interaction
strength for which the coupling is small for poorly connected systems, and
large for the hub nodes. In particular, global hyperbolicity might be lost.
It is shown that, under certain hypotheses, the dynamics of the hub nodes
can be very well approximated by a low-dimensional system for exponentially
long time in the size of the network, and that the system exhibit hyperbolic
behaviour in this time window. Even if this describes only a long transient,
this is the behaviour that one expects to observe in experiments. Such a
description allows to establish the emergence of macroscopic behaviour
such as coherence of dynamics among hubs of the same connectivity layer
(i.e. with the same number of connections). In particular, the systems
studied provide a new paradigm to explain why and how the dynamics
of a network dynamical system can change across layers.