Distributed computation of homology using harmonics

TL;DR

This paper introduces a distributed algorithm leveraging spanning trees and harmonic analysis to efficiently compute the first homology of simplicial complexes, aiding topological analysis in sensor networks.

Contribution

It proposes a novel distributed method using harmonics for homology computation, improving efficiency and applicability in sensor network analysis.

Findings

Algorithm has complexity $O(|P|^ ext{)}$, with $|P|$ much smaller than total edges.

Simulations show $|P|$ closely approximates the first Betti number in geometric graphs.

Method effectively identifies contractible and homologous cycles.

Abstract

We present a distributed algorithm to compute the first homology of a simplicial complex. Such algorithms are very useful in topological analysis of sensor networks, such as its coverage properties. We employ spanning trees to compute a basis for algebraic 1-cycles, and then use harmonics to efficiently identify the contractible and homologous cycles. The computational complexity of the algorithm is $O(|P|^\omega)$, where $|P|$ is much smaller than the number of edges, and $\omega$ is the complexity order of matrix multiplication. For geometric graphs, we show using simulations that $|P|$ is very close to the first Betti number.