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Centrality measures in simplicial complexes: Applications of topological data analysis to network science

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  • Hernández Serrano, Daniel
  • Sánchez Gómez, Darío

Abstract

Many real networks in social sciences, biological and biomedical sciences or computer science have an inherent structure of simplicial complexes reflecting many-body interactions. Therefore, to analyse topological and dynamical properties of simplicial complex networks centrality measures for simplices need to be proposed. Many of the classical complex networks centralities are based on the degree of a node, so in order to define degree centrality measures for simplices (which would characterise the relevance of a simplicial community in a simplicial network), a different definition of adjacency between simplices is required, since, contrarily to what happens in the vertex case (where there is only upper adjacency), simplices might also have other types of adjacency. The aim of these notes is threefold: first we will use the recently introduced notions of higher order simplicial degrees to propose new degree based centrality measures in simplicial complexes. These theoretical centrality measures, such as the simplicial degree centrality or the eigenvector centrality would allow not only to study the relevance of a simplicial community and the quality of its higher-order connections in a simplicial network, but also they might help to elucidate topological and dynamical properties of simplicial networks; sencond, we define notions of walks and distances in simplicial complexes in order to study connectivity of simplicial networks and to generalise, to the simplicial case, the well known closeness and betweenness centralities (needed for instance to study the relevance of a simplicial community in terms of its ability of transmitting information); third, we propose a new clustering coefficient for simplices in a simplicial network, different from the one knows so far and which generalises the standard graph clustering of a vertex. This measure should be essential to know the density of a simplicial network in terms of its simplicial communities.

Suggested Citation

  • Hernández Serrano, Daniel & Sánchez Gómez, Darío, 2020. "Centrality measures in simplicial complexes: Applications of topological data analysis to network science," Applied Mathematics and Computation, Elsevier, vol. 382(C).
    • Handle: RePEc:eee:apmaco:v:382:y:2020:i:c:s0096300320302976
    DOI: 10.1016/j.amc.2020.125331
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    References listed on IDEAS

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    1. Gert Sabidussi, 1966. "The centrality index of a graph," Psychometrika, Springer;The Psychometric Society, vol. 31(4), pages 581-603, December.
    2. Wang, Juan & Li, Chao & Xia, Chengyi, 2018. "Improved centrality indicators to characterize the nodal spreading capability in complex networks," Applied Mathematics and Computation, Elsevier, vol. 334(C), pages 388-400.
    3. Maletić, Slobodan & Rajković, Milan, 2014. "Consensus formation on a simplicial complex of opinions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 397(C), pages 111-120.
    4. Iacopo Iacopini & Giovanni Petri & Alain Barrat & Vito Latora, 2019. "Simplicial models of social contagion," Nature Communications, Nature, vol. 10(1), pages 1-9, December.
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    1. Kovalenko, K. & Romance, M. & Vasilyeva, E. & Aleja, D. & Criado, R. & Musatov, D. & Raigorodskii, A.M. & Flores, J. & Samoylenko, I. & Alfaro-Bittner, K. & Perc, M. & Boccaletti, S., 2022. "Vector centrality in hypergraphs," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    2. Li, Zhaohui & Li, Xinyu & Li, Mindi & Zhang, Kexin & Zhang, Xi & Zhou, Xiaoxia, 2024. "Evaluation of human epileptic brain networks by constructing simplicial complexes," Chaos, Solitons & Fractals, Elsevier, vol. 189(P1).
    3. Václav Snášel & Pavla Dráždilová & Jan Platoš, 2021. "Cliques Are Bricks for k-CT Graphs," Mathematics, MDPI, vol. 9(11), pages 1-9, May.
    4. Shang, Yilun, 2022. "Sombor index and degree-related properties of simplicial networks," Applied Mathematics and Computation, Elsevier, vol. 419(C).
    5. Serrano, Daniel Hernández & Villarroel, Javier & Hernández-Serrano, Juan & Tocino, Ángel, 2023. "Stochastic simplicial contagion model," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    6. Hernández Serrano, Daniel & Hernández-Serrano, Juan & Sánchez Gómez, Darío, 2020. "Simplicial degree in complex networks. Applications of topological data analysis to network science," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).

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