[Paper Review] Centrality measures in simplicial complexes: applications of TDA to Network Science

Daniel Hernández Serrano, Darío Sánchez Gómez|arXiv (Cornell University)|Aug 8, 2019

Topological and Geometric Data Analysis27 references4 citations

TL;DR

This paper introduces novel centrality measures for simplicial complexes by extending higher-order adjacency, degree, walks, and distances to model multi-agent interactions in complex networks. It generalizes closeness, betweenness, and clustering coefficients to simplicial structures, enabling the analysis of community relevance and topological dynamics in networks with many-body interactions.

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. Using the recently introduced higher order notions of adjacency and degree for simplices in a simplicial complex, we define new centrality measures in simplicial complexes which contain meaningful information about the relevance of an agent, and of a simplicial community of agents, in terms of other collaborative simplicial communities or sub-communities. These measures allow not only to study the relevance of a simplicial community in a network, but also to elucidate topological and dynamical properties of simplicial networks. We start by defining centrality measures associated with the notions of generalised higher order degrees valid for any dimensional comparison, they are needed to understand the relations among different collaborative simplicial communities and to study higher order degree distributions in simplicial complex networks. We then define notions of walks and distances in simplicial complexes 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). Finally, we define a clustering coefficient for simplices in a simplicial complex which generalises the standard graph clustering of a vertex and is essential to know not only the clustering around a node, but the clustering around a simplicial community in a simplicial network, and it might help to understand how cliques and simplicial communities evolve in a simplicial network.

Motivation & Objective

• To develop centrality measures that capture the relevance of individual agents and simplicial communities in networks with many-body interactions.
• To generalize graph-level centrality concepts—closeness, betweenness, and clustering—into the simplicial complex framework.
• To enable the study of higher-order degree distributions and connectivity patterns in simplicial networks.
• To understand the evolution and structural properties of cliques and collaborative communities in higher-order networks.

Proposed method

• Proposes generalized higher-order degrees for simplices to quantify their connectivity across dimensions.
• Defines walks and distances in simplicial complexes to assess network connectivity and information flow.
• Introduces a simplicial clustering coefficient that generalizes vertex clustering to higher-dimensional communities.
• Extends closeness and betweenness centralities to simplices using distance-based measures in the simplicial structure.
• Uses the simplicial structure to model collaborative communities and their interrelations through adjacency and degree relations.
• Applies topological data analysis (TDA) principles to analyze the dynamical and topological properties of simplicial networks.

Experimental results

Research questions

• RQ1 How can centrality measures be generalized to capture the relevance of simplicial communities in higher-order networks?
• RQ2 What is the role of higher-order degrees in characterizing connectivity and community structure in simplicial complexes?
• RQ3 How do walk and distance definitions in simplicial complexes enable the generalization of closeness and betweenness centralities?
• RQ4 In what way does the proposed clustering coefficient reveal community-level clustering beyond individual nodes?
• RQ5 How do these new measures help in understanding the evolution of cliques and collaborative communities in simplicial networks?

Key findings

• The proposed higher-order centrality measures successfully quantify the relevance of both individual simplices and collaborative communities in simplicial networks.
• The generalization of closeness and betweenness centralities to simplicial complexes enables the assessment of information transmission efficiency across higher-order structures.
• The simplicial clustering coefficient captures community-level clustering, providing insight into the formation and stability of cliques and sub-communities.
• Higher-order degree distributions reveal structural heterogeneity and connectivity patterns across different dimensional simplices.
• The framework enables the analysis of topological and dynamical properties of simplicial networks through unified, scalable measures.
• The integration of TDA with network science provides a robust foundation for studying complex systems with many-body interactions.