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Distributed computation of homology using harmonics

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 $O(|P|^ω)$, where $|P|$ is much smaller than the number of edges, and $ω$ is the complexity order of matrix multiplication. For geometric graphs, we show using simulations that $|P|$ is very close to the first Betti number.