Networks with arbitrary edge multiplicities

Abstract

-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][5][6][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.

The Kuramoto model in complex networks

“…There are, however, other network models [99,109,111,[117][118][119][120] that go beyond the low-clustering regime. For instance, an interesting generalization of the clustered random network [109,110] was introduced in [119], where networks can be constructed not only by single-edges and triangles, but also with arbitrary distributions of different kinds of subgraphs.…”

“…However, the implementation and analytical tractability of the model greatly increase as the connectivity pattern of the subgraphs becomes more complex. Another interesting model that can be suitably used to evaluate the dynamics of networks with similar topology as real structures was proposed in [120]. Instead of focusing on how many triangles are attached to a given node, the model in [99,111,120] is based on the concept of edge multiplicity, which is the number of triangles that a given edge participates.…”

“…The great advantage of the model based on edge multiplicities over other random models is that, while the evaluation of subgraph distributions in real networks is a potentially expensive task depending on the number of nodes, the distribution of edge multiplicities is easily calculated from real data, which can then be used as an input in the random model [120]. The analysis of the Kuramoto model and the development of MFAs in these and other random network models [121][122][123][124][125][126][127][128][129][130] for clustered networks are promising directions for future research.…”

“…Another interesting related work considers the influence of feedback. With an additional feedback, the system has a stronger symmetry than the one without feedback (120). The corresponding dynamics with feedback is governed by [187]θ…”

Latent geometry of bipartite networks

“…We also note that similar clustering-related heterogeneity has also been observed in unipartite networks. Many real unipartite networks have been shown to exhibit power-law distributions of edge multiplicity, defined as the number of triangles shared by edges [13]. Here we show that the observed common properties of real bipartite networks can be explained by the existence of latent geometric spaces underlying these networks.…”