Simplicial degree in complex networks. Applications of topological data analysis to network science

Daniel Hernández Serrano, Juan Hernández-Serrano and Darío Sánchez Gómez

Chaos, Solitons & Fractals, 2020, vol. 137, issue C

Abstract: Network Science provides a universal formalism for modelling and studying complex systems based on pairwise interactions between agents. However, many real networks in the social, biological or computer sciences involve interactions among more than two agents, having thus an inherent structure of a simplicial complex. The relevance of an agent in a graph network is given in terms of its degree, and in a simplicial network there are already notions of adjacency and degree for simplices that, as far as we know, are not valid for comparing simplices in different dimensions. We propose new notions of higher-order degrees of adjacency for simplices in a simplicial complex, allowing any dimensional comparison among them and their faces. We introduce multi-parameter boundary and coboundary operators in an oriented simplicial complex and also a novel multi-combinatorial Laplacian is defined. As for the graph or combinatorial Laplacian, the multi-combinatorial Laplacian is shown to be an effective tool for calculating the higher-order degrees presented here. To illustrate the potential applications of these theoretical results, we perform a structural analysis of higher-order connectivity in simplicial-complex networks by studying the associated distributions with these simplicial degrees in 17 real-world datasets coming from different domains such as coauthor networks, cosponsoring Congress bills, contacts in schools, drug abuse warning networks, e-mail networks or publications and users in online forums. We find rich and diverse higher-order connectivity structures and observe that datasets of the same type reflect similar higher-order collaboration patterns. Furthermore, we show that if we use what we have called the maximal simplicial degree (which counts the distinct maximal communities in which our simplex and all its strict sub-communities are contained), then its degree distribution is, in general, surprisingly different from the classical node degree distribution.

Keywords: Complex networks; Simplicial complexes; Combinatorial laplacian; Topological data analysis; Network science; Statistical mechanics (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:137:y:2020:i:c:s0960077920302393

DOI: 10.1016/j.chaos.2020.109839

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