Kannan Harish, Saucan Emil, Roy Indrava, Samal Areejit
The Institute of Mathematical Sciences (IMSc), Homi Bhabha National Institute (HBNI), Chennai, 600113, India.
Department of Applied Mathematics, ORT Braude College, Karmiel, 2161002, Israel.
Sci Rep. 2019 Sep 25;9(1):13817. doi: 10.1038/s41598-019-50202-3.
Topological data analysis can reveal higher-order structure beyond pairwise connections between vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and undirected networks using persistent homology. Leveraging on the features of discrete Morse theory, our method not only captures the topology of the clique complex of such graphs via the concept of critical simplices, but also achieves close to the theoretical minimum number of critical simplices in several analyzed model and real networks. This leads to a reduced filtration scheme based on the subsequence of the corresponding critical weights, thereby leading to a significant increase in computational efficiency. We have employed our filtration scheme to explore the persistent homology of several model and real-world networks. In particular, we show that our method can detect differences in the higher-order structure of networks, and the corresponding persistence diagrams can be used to distinguish between different model networks. In summary, our method based on discrete Morse theory further increases the applicability of persistent homology to investigate the global topology of complex networks.
拓扑数据分析可以揭示复杂网络中顶点之间成对连接之外的高阶结构。我们提出了一种基于离散莫尔斯理论的新方法，使用持久同调研究无加权无向网络的拓扑性质。利用离散莫尔斯理论的特性，我们的方法不仅通过临界单形的概念捕捉此类图的团复形的拓扑，而且在几个分析的模型网络和真实网络中实现了接近理论最小数量的临界单形。这导致基于相应临界权重子序列的简化过滤方案，从而显著提高计算效率。我们采用我们的过滤方案来探索几个模型网络和真实世界网络的持久同调。特别是，我们表明我们的方法可以检测网络高阶结构的差异，并且相应的持久图可用于区分不同的模型网络。总之，我们基于离散莫尔斯理论的方法进一步提高了持久同调在研究复杂网络全局拓扑方面的适用性。
