Distributed computation of homology using harmonics

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 , where is much smaller than the number of edges, and is the complexity order of matrix multiplication. For geometric graphs, we show using simulations that is very close to the first Betti number.
View on arXivComments on this paper