Abstract

Measuring robustness is a fundamental task for analysing the structure of complex networks. Indeed, several approaches to capture the robustness properties of a network have been proposed. In this paper we focus on spectral graph theory where robustness is measured by means of a graph invariant called Kirchhoff index, expressed in terms of eigenvalues of the Laplacian matrix associated to a graph. This graph metric is highly informative as a robustness indicator for several real-world networks that can be modeled as graphs. We discuss a methodology aimed at obtaining some new and tighter bounds of this graph invariant when links are added or removed. We take advantage of real analysis techniques, based on majorization theory and optimization of functions which preserve the majorization order. Applications to simulated graphs and to empirical networks generated by collecting assets of the S&P 100 show the effectiveness of our bounds, also in providing meaningful insights with respect to the results obtained in the literature.

Graphical abstract

中文翻译：

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通过主化技术在拓扑变化下的复杂网络具有强大的鲁棒性

摘要

测量鲁棒性是分析复杂网络结构的基本任务。实际上，已经提出了几种捕获网络的鲁棒性的方法。在本文中，我们专注于频谱图理论，其中，通过称为Kirchhoff指数的图不变性来衡量鲁棒性，该不变性用与图相关的拉普拉斯矩阵的特征值表示。该图形指标非常有用，可以作为几个可以建模为图形的现实网络的鲁棒性指标。我们讨论一种旨在在添加或删除链接时获得该图不变性的新的和更严格的边界的方法。我们基于专业化理论和功能优化（保留专业化顺序），利用了实际的分析技术。标准普尔100指数显示了我们的优势，对于就文献中获得的结果提供了有意义的见解。

图形概要