Abstract

The second smallest eigenvalue of the Laplacian is known as the algebraic connectivity of a graph. It shows degree of graph connectivity. However, this metric does not take into account possible changes in the graph. The removal of even one node or edge can make it disconnected. This work is devoted to the development of a metric that should describe robustness of a graph to such changes. All proposed metrics are based on the algebraic connectivity. In addition, we generalize some well-known optimization methods for our robust modifications of the algebraic connectivity. The paper also reports results of some numerical experiments demonstrating the efficiency of the proposed approaches.

中文翻译：

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鲁棒的代数连通性

摘要

拉普拉斯算子的第二小特征值称为图的代数连通性。它显示了图的连通程度。但是，该指标并未考虑图表中可能发生的变化。即使删除一个节点或边缘也可能使其断开连接。这项工作致力于开发一种指标来描述图表对此类变化的鲁棒性。所有提出的指标都基于代数连通性。此外，我们还推广了一些众所周知的优化方法来对代数连通性进行稳健的修改。该论文还报告了一些数值实验的结果，证明了所提出方法的效率。