Analyzing Network Robustness via Residual Closeness

TL;DR

This paper studies the robustness of networks by analyzing closeness and residual closeness in middle graph representations, deriving exact formulas, bounds, and an algorithm for computation.

Contribution

It introduces new formulas and bounds for residual closeness in middle graphs and proposes an algorithm for efficient closeness computation.

Findings

Derived exact closeness expressions for middle graphs.

Established bounds for residual closeness under vertex failures.

Proposed an algorithm with performance analysis for computing closeness.

Abstract

Networks are inherently vulnerable to vertex failures, making the analysis of their structural robustness a fundamental problem in graph theory. In this study, we investigate the closeness and vertex residual closeness of graphs, with a particular focus on the middle graph representations of certain special graph classes, which provide a richer structural framework for analysis. We derive exact expressions for the closeness values of these middle graphs and determine their residual closeness under vertex failures. By utilizing results obtained from specific graph families, we establish several general bounds for broader graph classes. Furthermore, by exploiting the relationship between the closeness of a graph, its line graphs, and middle graphs, we obtain new results that relate these three structures. In addition, we propose an algorithm for computing closeness in middle graphs and provide a detailed analysis of its performance.