Logconcave Random Graphs

TL;DR

This paper introduces a flexible model of random graphs based on log-concave distributions, generalizing Erdős-Rényi graphs, and explores properties like connectivity thresholds and applications to combinatorial optimization problems.

Contribution

It proposes a new, general framework for random graphs using log-concave distributions, extending classical models and analyzing their fundamental properties.

Findings

Analyzed connectivity thresholds for general distributions.

Captured properties of triangle-free and weighted random graphs.

Provided insights into applications in combinatorial optimization.

Abstract

We propose the following model of a random graph on n vertices. Let F be a distribution in R_+^{n(n-1)/2} with a coordinate for every pair i$ with 1 \le i,j \le n. Then G_{F,p} is the distribution on graphs with n vertices obtained by picking a random point X from F and defining a graph on n vertices whose edges are pairs ij for which X_{ij} \le p. The standard Erd\H{o}s-R\'{e}nyi model is the special case when F is uniform on the 0-1 unit cube. We examine basic properties such as the connectivity threshold for quite general distributions. We also consider cases where the X_{ij} are the edge weights in some random instance of a combinatorial optimization problem. By choosing suitable distributions, we can capture random graphs with interesting properties such as triangle-free random graphs and weighted random graphs with bounded total weight.