Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes

TL;DR

This paper introduces a framework for analyzing and constructing simplicial complexes, a generalized network structure, using configuration models and ensembles, with applications to understanding complex systems like brain and social networks.

Contribution

It develops the configuration model and canonical ensemble for simplicial complexes, including entropy evaluation and algorithms for their construction.

Findings

Derived the asymptotic expression for the number of simplicial complexes in the configuration model.

Provided algorithms for constructing simplicial complexes within these models.

Analyzed the natural correlations in simplicial complexes without structural cutoff.

Abstract

Simplicial complexes are generalized network structures able to encode interactions occurring between more than two nodes. Simplicial complexes describe a large variety of complex interacting systems ranging from brain networks, to social and collaboration networks. Here we characterize the structure of simplicial complexes using their generalized degrees that capture fundamental properties of one, two, three or more linked nodes. Moreover we introduce the configuration model and the canonical ensemble of simplicial complexes, enforcing respectively the sequence of generalized degrees of the nodes and the sequence of the expected generalized degrees of the nodes. We evaluate the entropy of these ensembles, finding the asymptotic expression for the number of simplicial complexes in the configuration model. We provide the algorithms for the construction of simplicial complexes belonging to the configuration model and the canonical ensemble of simplicial complexes. We give an expression for the structural cutoff of simplicial complexes that for simplicial complexes of dimension $d=1$ reduces to the structural cutoff of simple networks. Finally we provide a numerical analysis of the natural correlations emerging in the configuration model of simplicial complexes without structural cutoff.