On the complexity of color-avoiding site and bond percolation

TL;DR

This paper investigates the computational complexity of color-avoiding percolation in networks, where shared vulnerabilities among vertices or edges affect robustness, providing insights into different modeling approaches.

Contribution

It analyzes various models of shared vulnerabilities and determines their computational complexity in finding robustly connected components.

Findings

Different modeling approaches vary significantly in complexity.

Some approaches are computationally hard, others more tractable.

The study advances understanding of network robustness under shared vulnerabilities.

Abstract

The mathematical analysis of robustness and error-tolerance of complex networks has been in the center of research interest. On the other hand, little work has been done when the attack-tolerance of the vertices or edges are not independent but certain classes of vertices or edges share a mutual vulnerability. In this study, we consider a graph and we assign colors to the vertices or edges, where the color-classes correspond to the shared vulnerabilities. An important problem is to find robustly connected vertex sets: nodes that remain connected to each other by paths providing any type of error (i.e. erasing any vertices or edges of the given color). This is also known as color-avoiding percolation. In this paper, we study various possible modeling approaches of shared vulnerabilities, we analyze the computational complexity of finding the robustly (color-avoiding) connected components. We find that the presented approaches differ significantly regarding their complexity.