从非回溯圈的拓扑观点看图的距离
Graph Distance from the Topological View of Non-backtracking Cycles
摘要：
无论是将网络相互比较还是与随机期望进行比较，测量相异性对于理解所研究的复杂现象至关重要。然而，确定网络之间的结构差异是一个定义不清的问题，因为没有规范的方法来比较两个网络。事实上，许多现有的网络比较方法在其原理、效率、可解释性和理论合理性方面存在差异。因此，距离的概念建立在理论上稳健的第一性原理上，并且可以解释复杂网络中无处不在的特征，这将允许在不同网络之间进行有意义的比较。在这里，我们介绍了一个理论上健全和有效的新措施的图形距离，基于“长度谱”功能从代数拓扑，它比较了两个无向，无权图的结构，考虑其非回溯循环。我们展示了这种距离如何与结构特征相关，例如通过所谓的非回溯矩阵的特征值的行为来表现枢纽和三角形的存在，并且我们展示了它在真实和合成数据集中区分网络的能力。通过对非回溯环的拓扑解释，本文提出了一种新的拓扑数据分析在复杂网络研究中的应用。
英文摘要：
Whether comparing networks to each other or to random expectation, measuring dissimilarity is essential to understanding the complex phenomena under study. However, determining the structural dissimilarity between networks is an ill-defined problem, as there is no canonical way to compare two networks. Indeed, many of the existing approaches for network comparison differ in their heuristics, efficiency, interpretability, and theoretical soundness. Thus, having a notion of distance that is built on theoretically robust first principles and that is interpretable with respect to features ubiquitous in complex networks would allow for a meaningful comparison between different networks. Here we introduce a theoretically sound and efficient new measure of graph distance, based on the "length spectrum" function from algebraic topology, which compares the structure of two undirected, unweighted graphs by considering their non-backtracking cycles. We show how this distance relates to structural features such as presence of hubs and triangles through the behavior of the eigenvalues of the so-called non-backtracking matrix, and we showcase its ability to discriminate between networks in both real and synthetic data sets. By taking a topological interpretation of non-backtracking cycles, this work presents a novel application of Topological Data Analysis to the study of complex networks.
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