Effective Resistance in Simplicial Complexes: Generalizations and Properties

The concept of effective resistance, originally developed in electrical network theory, has become a powerful tool for studying the structure of graphs. It captures both direct and indirect connections between vertices, relates to random walks and spanning trees, and underlies applications ranging from graph sparsification to community detection.

In recent years, several matrix expressions have been introduced aiming at extending the notion of effective resistance from graphs to simplicial complexes. In this talk, I will present a basis-free definition of effective resistance, rooted in the original definition motivated by physics, which unifies existing approaches, allows the extension of graph theoretic results, and reveals new structural insights.

This is joint work with Claudia Landi, Sarah Percival, Anda Skeja, Bei Wang and Ling Zhou.