Networks with arbitrary edge multiplicities

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. Introduction. – Real networks, where nodes (or vertices) are intricately connected by links (or edges), are characterized by complex topological properties such as a scale-free distribution of the degree (number of edges reaching a vertex), degree-degree correlations, and nonvanishing degree-dependent clustering (density of triangles reaching a vertex) [1]. Understanding the structural and dynamical properties of complex networks strongly relies on the possibility to investigate theoretical models which are both realistic and analytically solvable. Several analytically solvable models reproducing the most important local property of real networks, i.e. the degree distribution, have been proposed [1]. However, models reproducing higher-order properties including clustering (also called transitivity) are only a few and are either entirely computational [2, 3] (i.e. not analytically solvable) or solvable only for particular cases, e.g. when triangles are non-overlapping [4–7] or when the network is made by cliques [8] or other subgraphs [9] embedded in a treelike skeleton. Unfortunately, real networks generally violate the above particular conditions, as empirical analyses have revealed and as we will further show in what follows. Moreover, it has been shown that clustering is only one aspect of a more general topological organization which is best captured by edge multiplicity [4, 5, 10], i.e. the number of triangles in which edges, rather than vertices, participate. Besides being more informative than vertexbased clustering, edge multiplicity strongly determines the percolation properties of networks [11] and their community structure [12]. A model with arbitrary edge multiplicities. – In order to overcome these limitations, here we propose an analytically solvable model of networks with no restriction on their clustering properties, and able to generate edges of any multiplicity. Let us denote by m(i, j) the multiplicity of the edge (i, j), i.e. m(i, j) = ∑ k 6=i,j aikakj where aij = 1 if a link between vertices i and j is there, and aij = 0 otherwise. In our model we allow each vertex i to have k (0) i edges of zero multiplicity, k (1) i edges of multiplicity 1 and so on, up to k (M) i edges of multiplicity M , where M = N−2 is the maximum possible multiplicity in a network with N vertices. Thus each vertex i is assigned a (N − 1)-dimensional vector ki ≡ (k (0) i , ..., k (M) i ), that