The impact of degree variability on connectivity properties of large networks

TL;DR

This paper investigates how increased degree variability influences the connectivity of large networks, revealing that higher variability can both enlarge or shrink the largest connected component depending on specific conditions.

Contribution

It provides a detailed analysis of the relationship between degree variability and network connectivity, including counterexamples and conditions affecting the size of the largest component.

Findings

Higher degree variability can lead to larger connected components in some cases.

Under certain conditions, increased variability decreases the size of the largest component.

Counterexamples challenge basic intuition about branching processes in network connectivity.

Abstract

The goal of is to study how increased variability in the degree distribution impacts the global connectivity properties of a large network. We approach this question by modeling the network as a uniform random graph with a given degree sequence. We analyze the effect of the degree variability on the approximate size of the largest connected component using stochastic ordering techniques. A counterexample shows that a higher degree variability may lead to a larger connected component, contrary to basic intuition about branching processes. When certain extremal cases are ruled out, the higher degree variability is shown to decrease the limiting approximate size of the largest connected component.