Stochastic Homology. Reduction Formulas for Computing Stochastic Betti Numbers of Maximal Random Complexes with Discrete Probabilities. Computation and Applications

TL;DR

This paper introduces methods to compute stochastic Betti numbers for random complexes with cells appearing independently based on discrete probability distributions, advancing the understanding of topological properties in probabilistic settings.

Contribution

It develops reduction formulas for calculating stochastic homology in maximal random complexes with discrete probabilities, providing new tools for probabilistic topological analysis.

Findings

Derived explicit formulas for stochastic Betti numbers

Demonstrated applications in computational topology

Enhanced understanding of topological invariants in random complexes

Abstract

Given a chain complex with the only modification that each cell of the complex has a probability distribution assigned. We will call this complex - a random complex and what should be understood in practice, is that we have a classical chain complex whose cells appear and disappear according to some probability distributions. In this paper, we will try to find the stochastic homology of random complex, whose simplices have independent discrete distributions.