Logconcave Random Graphs
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 i,j n. Then GF,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 Xij p. The standard Erdos-R\'enyi 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 Xij 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.