Distinguishing power-law uniform random graphs from inhomogeneous random graphs through small subgraphs

TL;DR

This paper develops an algorithm to distinguish power-law uniform random graphs from inhomogeneous random graphs by analyzing the frequency of specific small induced subgraphs, revealing structural differences related to degree distributions.

Contribution

It introduces a linear-time randomized algorithm that leverages subgraph frequency differences to differentiate between uniform and inhomogeneous random graph models.

Findings

Induced subgraphs appear on vertices with specific degrees.

The algorithm distinguishes graph types based on subgraph frequency.

Power-law uniform random graphs have distinct subgraph patterns.

Abstract

We investigate the asymptotic number of induced subgraphs in power-law uniform random graphs. We show that these induced subgraphs appear typically on vertices with specific degrees, which are found by solving an optimization problem. Furthermore, we show that this optimization problem allows to design a linear-time, randomized algorithm that distinguishes between uniform random graphs and random graph models that create graphs with approximately a desired degree sequence: the power-law rank-1 inhomogeneous random graph. This algorithm uses the fact that some specific induced subgraphs appear significantly more often in uniform random graphs than in rank-1 inhomogeneous random graphs.