On algebraic and combinatorial properties of weighted simplicial complexes

Abstract

Weighted simplicial complexes (WSCs) are powerful tools for describing weighted cloud data or networks with weighted nodes. In this paper, we propose a novel approach to study WSCs via the concept of polarization. Polarization of a WSC allows one to construct a new (unweighted) simplicial complex which coincides with an object called the mixed wreath product. This new construction preserves several properties and invariants of the underlying simplicial complex of a WSC. Our main focus is to analyze WSCs through their underlying simplicial complexes and mixed wreath products. Combinatorially, we investigate properties such as vertex-decomposability, shellability, constructibility; algebraically, we study Betti numbers, associated primes and primary decompositions of ideals associated to WSCs.