Higher-order Connection Laplacians for Directed Simplicial Complexes

TL;DR

This paper introduces Higher-order Connection Laplacians for directed simplicial complexes, enabling the analysis of directional higher-order interactions in complex systems like brain networks and social systems.

Contribution

It formulates the first higher-order Connection Laplacians for directed simplicial complexes, extending the undirected framework to incorporate directionality and complex diffusion dynamics.

Findings

Defined Connection Laplacians for 2D directed simplicial complexes

Demonstrated higher-order diffusion dynamics with synthetic examples

Highlighted potential applications in real-world directional higher-order systems

Abstract

Higher-order networks encode the many-body interactions existing in complex systems, such as the brain, protein complexes, and social interactions. Simplicial complexes are higher-order networks that allow a comprehensive investigation of the interplay between topology and dynamics. However, simplicial complexes have the limitation that they only capture undirected higher-order interactions while in real-world scenarios, often there is a need to introduce the direction of simplices, extending the popular notion of direction of edges. On graphs and networks the Magnetic Laplacian, a special case of Connection Laplacian, is becoming a popular operator to treat edge directionality. Here we tackle the challenge of treating directional simplicial complexes by formulating Higher-order Connection Laplacians taking into account the configurations induced by the simplices' directions. Specifically, we define all the Connection Laplacians of directed simplicial complexes of dimension two and we discuss the induced higher-order diffusion dynamics by considering instructive synthetic examples of simplicial complexes. The proposed higher-order diffusion processes can be adopted in real scenarios when we want to consider higher-order diffusion displaying non-trivial frustration effects due to conflicting directionalities of the incident simplices.