  Multiplex networks describe a large number of complex social, biological and
transportation networks where a set of nodes is connected by links of different
nature and connotation. Here we uncover the rich community structure of
multiplex networks by associating a community to each multilink where the
multilinks characterize the connections existing between any two nodes of the
multiplex network. Our community detection method reveals the rich interplay
between the mesoscale structure of the multiplex networks and their
multiplexity. For instance some nodes can belong to many layers and few
communities while others can belong to few layers but many communities.
Moreover the multilink communities can be formed by a different number of
relevant layers. These results point out that mesoscopically there can be large
differences in the compressibility of multiplex networks.
