  One of the main characteristics of real-world networks is their large
clustering. Clustering is one aspect of a more general but much less studied
structural organization of networks, i.e. edge multiplicity, defined as the
number of triangles in which edges, rather than vertices, participate. Here we
show that the multiplicity distribution of real networks is in many cases
scale-free, and in general very broad. Thus, besides the fact that in real
networks the number of edges attached to vertices often has a scale-free
distribution, we find that the number of triangles attached to edges can have a
scale-free distribution as well. We show that current models, even when they
generate clustered networks, systematically fail to reproduce the observed
multiplicity distributions. We therefore propose a generalized model that can
reproduce networks with arbitrary distributions of vertex degrees and edge
multiplicities, and study many of its properties analytically.
