A note on loglog distances in a power law random intersection graph

TL;DR

This paper investigates the typical shortest path length in a power law random intersection graph with infinite second moment, providing an O(log log n) upper bound for the distance between vertices in the giant component.

Contribution

It establishes a probabilistic upper bound of O(log log n) for shortest path lengths in such graphs, advancing understanding of their connectivity properties.

Findings

Shortest path length between vertices is bounded by O(log log n) with high probability.

The result applies to graphs with infinite second moment degree distributions.

Provides insights into the structure of power law random intersection graphs.

Abstract

We consider the typical distance between vertices of the giant component of a random intersection graph having a power law (asymptotic) vertex degree distribution with infinite second moment. Given two vertices from the giant component we construct O(log log n) upper bound (in probability) for the length of the shortest path connecting them.