On Random Graph Properties

TL;DR

This paper investigates various properties of labeled random graphs, analyzes their relationships, and demonstrates how certain properties can be predicted from others, enabling efficient estimation of complex graph metrics.

Contribution

It provides an analysis of 15 graph properties, establishes the effectiveness of Erdős–Rényi models for larger graphs, and develops predictive models for graph property estimation.

Findings

Erdős–Rényi graphs model well the space of all labeled graphs with fixed vertices.

Pairs and triples of properties can predict each other with high accuracy.

Predictive models enable estimation of complex properties from simpler ones.

Abstract

We consider 15 properties of labeled random graphs that are of interest in the graph-theoretical and the graph mining literature, such as clustering coefficients, centrality measures, spectral radius, degree assortativity, treedepth, treewidth, etc. We analyze relationships and correlations between these properties. Whereas for graphs on a small number of vertices we can exactly compute the average values and range for each property of interest, this becomes infeasible for larger graphs. We show that graphs generated by the \ErdosRenyi graph generator with $p = 1/2$ model well the underlying space of all labeled graphs with a fixed number of vertices. The later observation allows us to analyze properties and correlations between these properties for larger graphs. We then use linear and non-linear models to predict a given property based on the others and for each property, we find the most predictive subset. We experimentally show that pairs and triples of properties have high predictive power, making it possible to estimate computationally expensive to compute properties with ones for which there are efficient algorithms.