Construction of simplicial complexes with prescribed degree-size sequences

TL;DR

This paper investigates the conditions under which simplicial complexes with specific degree and facet size sequences can be constructed, providing polynomial algorithms for most cases, and explores how degree influences the complex's topology.

Contribution

It introduces a recursive polynomial-time algorithm for constructing simplicial complexes with prescribed sequences and analyzes the impact of node degrees on complex topology.

Findings

Most degree-size sequences are realizable in polynomial time.

Increasing node degrees reduces the number of loops in complexes.

Provides an efficient sampler for simplicial ensembles.

Abstract

We study the realizability of simplicial complexes with a given pair of integer sequences, representing the node degree distribution and the facet size distribution, respectively. While the $s$-uniform variant of the problem is $\mathsf{NP}$-complete when $s \geq 3$, we identify two populations of input sequences, most of which can be solved in polynomial time using a recursive algorithm that we contribute. Combining with a sampler for the simplicial configuration model [J.-G. Young $\textit{et al.}$, Phys. Rev. E $\textbf{96}$, 032312 (2017)], we facilitate the efficient sampling of simplicial ensembles from arbitrary degree and size distributions. We find that, contrary to expectations based on dyadic networks, increasing the nodes' degrees reduces the number of loops in simplicial complexes. Our work unveils a fundamental constraint on the degree-size sequences and sheds light on further analysis of higher-order phenomena based on local structures.