Simplicial complexes: higher-order spectral dimension and dynamics

Joaquín J Torres,Ginestra Bianconi

doi:10.1088/2632-072x/ab82f5

Abstract

Simplicial complexes constitute the underlying topology of interacting complex systems including among the others brain and social interaction networks. They are generalized network structures that allow to go beyond the framework of pairwise interactions and to capture the many-body interactions between two or more nodes strongly affecting dynamical processes. In fact, the simplicial complexes topology allows to assign a dynamical variable not only to the nodes of the interacting complex systems but also to links, triangles, and so on. Here we show evidence that the dynamics defined on simplices of different dimensions can be significantly different even if we compare dynamics of simplices belonging to the same simplicial complex. By investigating the spectral properties of the simplicial complex model called ‘network geometry with flavor’ (NGF) we provide evidence that the up and down higher-order Laplacians can have a finite spectral dimension whose value depends on the order of the Laplacian. Finally we discuss the implications of this result for higher-order diffusion defined on simplicial complexes showing that the n-order diffusion dynamics have a return type distribution that can depends on n as it is observed in NGFs.

Highlights

Simplicial complexes are generalized network structures that allow to capture the many body interactions existing between the constituents of complex systems [1,2,3]
The investigation of the spectral properties of the higher-order Laplacian is rather crucial to reveal the properties of higher-order dynamical processes on simplicial complexes
In this work we reveal that the higher-order up and down-Laplacian can display a finite spectral dimension by providing a concrete example where this phenomenon is displayed, the simplicial complex model called “Network Geometry with Flavor”

Summary

Introduction

Simplicial complexes are generalized network structures that allow to capture the many body interactions existing between the constituents of complex systems [1,2,3]. Being built by geometrical building blocks, simplicial complexes represent an ideal setting to investigate the properties of emergent network geometry and topology in complex systems [1, 15,16,17]. They reveal the rich interplay between network geometry and dynamics [18,19,20, 26,27,28,29]. The presence of a finite spactral dimension is not an exclusive property of NGFs but it has been observed in a recent modelling framework that uses hyperbolic and small-world simplicial complex to model nano-networks [32]

Methods

Results

Conclusion