  The aim of this paper is to rigorously study dynamics of Heterogeneously
Coupled Maps (HCM). Such systems are determined by a network with heterogeneous
degrees. Some nodes, called hubs, are very well connected while most nodes
interact with few others. The local dynamics on each node is chaotic, coupled
with other nodes according to the network structure. Such high-dimensional
systems are hard to understand in full, nevertheless we are able to describe
the system over exponentially large time scales. In particular, we show that
the dynamics of hub nodes can be very well approximated by a low-dimensional
system. This allows us 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), and chaotic behaviour of the poorly connected
nodes. The HCM we study provide a paradigm to explain why and how the dynamics
of the network can change across layers.
