基于调和函数的同调分布计算
Distributed computation of homology using harmonics
摘要：
提出了一种计算单纯复形第一同调的分布式算法。这类算法在传感器网络的拓扑分析中非常有用，如其覆盖特性。我们采用生成树计算代数1-圈的基础，然后使用谐波有效地确定可收缩和同源的循环。该算法的计算复杂性为$O(|P|^ω)$，其中$|P|$远小于边数，$ω$为矩阵矩阵的阶的复杂度阶数。对于几何图形，我们使用模拟表明，$|P|$是非常接近的第一个贝蒂数。
英文摘要：
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.
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