# Quantifying Dismantlement in Disconnected Networks

## TL;DR

This paper introduces a new measure for network dismantlement that accounts for the number and size variance of disconnected components, improving upon traditional measures focused solely on the largest component.

## Contribution

It adapts the uniformity measure from Szemeredi's Regularity Lemma to better quantify network fragmentation, considering component count and size variance.

## Key findings

- The new measure increases with more disconnected components.
- It decreases as the variance in component sizes increases.
- It offers a refined way to evaluate network dismantlement.

## Abstract

We propose a novel measure to quantify dismantlement of a fragmented network. The existing measure of dismantlement used to study problems like optimal percolation is usually the size of the largest component of the network. We modify the measure of uniformity used to prove the Szemeredi's Regularity Lemma to obtain the proposed measure. The proposed measure incorporates the notion that the measure of dismantlement increases as the number of disconnected components increase and decreases as the variance of sizes of these components increases.