Fragmentation properties of two-dimensional Proximity Graphs considering random failures and targeted attacks

TL;DR

This paper investigates how different two-dimensional proximity graphs fragment under random failures and targeted attacks, analyzing the critical thresholds and structural changes during the process.

Contribution

It provides a detailed analysis of the fragmentation process of proximity graphs under various node removal strategies, including phase transition characterization.

Findings

Fragmentation follows a second order phase transition.

Thresholds depend on graph type and removal strategy.

Finite-size scaling estimates critical removal fractions.

Abstract

The pivotal quality of proximity graphs is connectivity, i.e. all nodes in the graph are connected to one another either directly or via intermediate nodes. These types of graphs are robust, i.e., they are able to function well even if they are subject to limited removal of elementary building blocks, as it may occur for random failures or targeted attacks. Here, we study how the structure of these graphs is affected when nodes get removed successively until an extensive fraction is removed such that the graphs fragment. We study different types of proximity graphs for various node removal strategies. We use different types of observables to monitor the fragmentation process, simple ones like number and sizes of connected components, and more complex ones like the hop diameter and the backup capacity, which is needed to make a network N-1 resilient. The actual fragmentation turns out to be described by a second order phase transition. Using finite-size scaling analyses we numerically assess the threshold fraction of removed nodes, which is characteristic for the particular graph type and node deletion scheme, that suffices to decompose the underlying graphs.