Percolation on dense random graphs with given degrees

TL;DR

This paper investigates the size of the largest connected component in random graphs with fixed degree sequences under bond percolation, revealing new threshold phenomena and complex behaviors such as multiple phase transitions.

Contribution

It introduces new threshold phenomena for the largest component in degree-constrained random graphs under percolation, including examples with multiple phase transitions.

Findings

Identification of new threshold phenomena

Existence of degree sequences with multiple jumps in component size

Behavior of the largest component can be highly complex

Abstract

In this paper, we study the order of the largest connected component of a random graph having two sources of randomness: first, the graph is chosen randomly from all graphs with a given degree sequence, and then bond percolation is applied. Far from being able to classify all such degree sequences, we exhibit several new threshold phenomena for the order of the largest component in terms of both sources of randomness. We also provide an example of a degree sequence for which the order of the largest component undergoes an unbounded number of jumps in terms of the percolation parameter, giving rise to a behavior that cannot be observed without percolation.