Expansion in Supercritical Random Subgraphs of Expanders and its Consequences

TL;DR

This paper investigates the expansion properties of the giant component in supercritical random subgraphs of expanders, revealing its diameter, mixing time, and subgraph expansion characteristics, with implications for understanding random graph behavior.

Contribution

It establishes new asymptotic expansion properties of the giant component in supercritical random subgraphs of expanders, including diameter and mixing time bounds.

Findings

Giant component has vertex expansion factor of O(psilon^2)

Diameter of the giant is typically O(psilon  log n)

Lazy random walk mixing time is O(psilon  log^2 n)

Abstract

In 2004, Frieze, Krivelevich and Martin [17] established the emergence of a giant component in random subgraphs of pseudo-random graphs. We study several typical properties of the giant component, most notably its expansion characteristics. We establish an asymptotic vertex expansion of connected sets in the giant by a factor of $\tilde{O}\left(\epsilon^2\right)$. From these expansion properties, we derive that the diameter of the giant is typically $O_{\epsilon}\left(\log n\right)$, and that the mixing time of a lazy random walk on the giant is asymptotically $O_{\epsilon}\left(\log^2 n\right)$. We also show similar asymptotic expansion properties of (not necessarily connected) linear sized subsets in the giant, and the typical existence of a large expander as a subgraph.