# Simplicial degree in complex networks. Applications of Topological Data Analysis to Network Science

## TL;DR

This paper introduces new higher-order adjacency notions and Laplacian operators for simplicial complexes, enabling advanced structural analysis of complex networks with multi-agent interactions across various real-world datasets.

## Contribution

It proposes novel multi-parameter boundary, coboundary, and Laplacian operators for simplicial complexes, extending network analysis beyond pairwise interactions.

## Key findings

- Higher-order connectivity structures are rich and diverse across datasets.
- Datasets of similar types show comparable higher-order collaboration patterns.
- Maximal simplicial degree distributions differ significantly from classical node degree distributions.

## 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. 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, which as far as we know were lacked in the literature. We introduce multi-parameter boundary and coboundary operators in an oriented simplicial complex and also a novel multi-combinatorial Laplacian is defined, which generalises the graph and combinatorial Laplacian. 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.