[Paper Review] Higher order degree in simplicial complexes, multi combinatorial Laplacian and applications of TDA to complex networks

Daniel Hernández Serrano, Darío Sánchez Gómez|arXiv (Cornell University)|Aug 2, 2019

Topological and Geometric Data Analysis22 references4 citations

TL;DR

This paper introduces higher-order adjacency degrees—lower, upper, and generalized—for simplices in simplicial complexes, enabling cross-dimensional comparison of agents and communities. It proposes multi-parameter boundary and coboundary operators and a novel multi-combinatorial Laplacian, which generalize classical operators and effectively compute higher-order degrees, offering a computationally feasible framework for analyzing multi-agent collaboration in complex networks via topological data analysis.

ABSTRACT

Many real networks in social, biological or computer sciences have an inherent structure of a simplicial complex, which reflects the multi interactions among agents (and groups of agents) and constitutes the basics of Topological Data Analysis. Normally, the relevance of an agent in a network of graphs is given in terms of the number of edges incident to it, 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 lower, upper and generalised adjacency degrees for simplices in a simplicial complex, allowing any dimensional comparison among them and their faces. New multi parameter boundary and coboundary operators in an oriented simplicial complex are also given and a novel multi combinatorial Laplacian is defined. These operators generalise the known ones and are proved to be an effective tool for calculating the higher order degrees here presented. Thus, this mathematical framework allows us to elucidate the relevance not only of an agent, but of a bunch of them as a simplicial community, and also to study the degree of collaboration between different communities in a simplicial complex. In addition, they are effective and programmable computational techniques. Some potential applications to simplicial Network Science are also proposed.

Motivation & Objective

• To address the lack of cross-dimensional comparison methods for simplex relevance in simplicial complexes.
• To define new notions of adjacency and degree that apply across different simplex dimensions.
• To develop multi-parameter boundary and coboundary operators for oriented simplicial complexes.
• To construct a novel multi-combinatorial Laplacian that generalizes classical Laplacians.
• To enable computational analysis of community relevance and inter-community collaboration in complex networks.

Proposed method

• Proposes higher-order lower, upper, and generalized adjacency degrees for simplices, allowing comparison across dimensions.
• Introduces multi-parameter boundary and coboundary operators in oriented simplicial complexes.
• Defines a multi-combinatorial Laplacian that generalizes the standard combinatorial Laplacian.
• Uses the new Laplacian to compute higher-order degrees efficiently.
• Applies the framework to analyze relevance of individual agents and simplicial communities.
• Ensures the framework is computationally effective and programmable for real-world network applications.

Experimental results

Research questions

• RQ1 How can simplex relevance be meaningfully compared across different dimensions in a simplicial complex?
• RQ2 What is the role of multi-parameter boundary and coboundary operators in higher-order network analysis?
• RQ3 How does the proposed multi-combinatorial Laplacian improve upon classical Laplacians in capturing higher-order interactions?
• RQ4 In what ways can the framework quantify collaboration between simplicial communities?
• RQ5 What are the computational and practical implications of applying this framework to real complex networks?

Key findings

• The proposed higher-order degrees enable direct comparison of simplices regardless of their dimension, overcoming limitations of traditional degree notions.
• The multi-parameter boundary and coboundary operators generalize classical operators and are compatible with the new Laplacian structure.
• The multi-combinatorial Laplacian effectively computes higher-order degrees, validating its utility in topological network analysis.
• The framework supports the identification of relevant agents and simplicial communities, including collaborative structures across groups.
• The method is computationally effective and programmable, enabling practical applications in network science.
• The approach provides a robust mathematical foundation for studying multi-agent interactions in complex networks through topological data analysis.