Long lived topological features are distinguished from short lived ones
(considered as topological noise) in simplicial complexes constructed from
complex networks. A new topological invariant, persistent homology, is
determined and presented as a parametrized version of a Betti number. Complex
networks with distinct degree distributions exhibit distinct persistent
topological features. Persistent toplogical attributes, shown to be related to
robust quality of networks, also reflect defficiency in certain connectivity
properites of networks. Random networks, networks with exponential conectivity
distribution and scale-free networks were considered for homological
persistency analysis.

该研究介绍了一个新的拓扑不变量 —— 持续同调，它代表了一个 Betti 数的参数化版本，可以用于区分长期和短期的拓扑特征，并且可以展示出不同网络拓扑特征之间的差异，从而反映出网络的强韧性与连通性等特性。

复杂网络的持久同调

Persistent Homology of Complex Networks

Each complex network (or class of networks) presents specific topological
features which characterize its connectivity and highly influence the dynamics
of processes executed on the network. The analysis, discrimination, and
synthesis of complex networks therefore rely on the use of measurements capable
of expressing the most relevant topological features. This article presents a
survey of such measurements. It includes general considerations about complex
network characterization, a brief review of the principal models, and the
presentation of the main existing measurements. Important related issues
covered in this work comprise the representation of the evolution of complex
networks in terms of trajectories in several measurement spaces, the analysis
of the correlations between some of the most traditional measurements,
perturbation analysis, as well as the use of multivariate statistics for
feature selection and network classification. Depending on the network and the
analysis task one has in mind, a specific set of features may be chosen. It is
hoped that the present survey will help the proper application and
interpretation of measurements.

本文对复杂网络的测量进行了综述，包括对其特征的一般考虑、主要模型的简要回顾和主要现有的测量的介绍。同时，本文讨论了复杂网络在多种测量空间下的轨迹表示、传统测量之间的相关性分析、紊动分析以及多元统计用于特征选择和网络分类等相关问题。根据具体的网络和分析任务，可以选择特定的特征集合。希望本文可以帮助正确应用和解释测量结果。