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

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10.1016/j.chaos.2020.109839
 
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Hernández Serrano, Daniel
Hernández Serrano, JuanMés informacióMés informacióMés informació
Sánchez Gómez, Darío
Document typeArticle
Defense date2020-08
Rights accessOpen Access
All rights reserved. This work is protected by the corresponding intellectual and industrial property rights. Without prejudice to any existing legal exemptions, reproduction, distribution, public communication or transformation of this work are prohibited without permission of the copyright holder
ProjectANALISIS FORENSE AVANZADO (MINECO-TEC2015-68734-R)
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.
CitationHernández, D.; Hernández-Serrano, J.; Sánchez, D. Simplicial degree in complex networks. Applications of topological data analysis to network science. "Chaos solitons and fractals", Agost 2020, vol. 137, p. 1-52. 
URIhttp://hdl.handle.net/2117/343541
DOI10.1016/j.chaos.2020.109839
ISSN0960-0779
Publisher versionhttps://www.sciencedirect.com/science/article/abs/pii/S0960077920302393
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  • Departament d'Enginyeria Telemàtica - Articles de revista [423]
  • ISG - Grup de Seguretat de la Informació - Articles de revista [50]
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