Optimal subgraphs in geometric scale-free random graphs

TL;DR

This paper analyzes the occurrence and scaling of subgraphs in geometric scale-free random graphs, revealing how vertex degrees and distances influence subgraph patterns and providing precise asymptotics for specific subgraph types.

Contribution

It introduces a divide-and-conquer approach to count subgraphs, linking their scaling behavior to solutions of a mixed-integer linear program, and derives asymptotics for trees and Hamiltonian subgraphs.

Findings

Subgraph counts are dominated by vertices with specific degrees and distances.

Scaling behavior relates to solutions of a mixed-integer linear program.

Provides asymptotic formulas for trees and Hamiltonian subgraphs.

Abstract

Geometric scale-free random graphs are popular models for networks that exhibit as heavy-tailed degree distributions, small-worldness and high clustering. In these models, vertices have weights that cause the heavy-tailed degrees and are embedded in a metric space so that close-by groups of vertices tend to cluster. The interplay between the vertex weights and positions heavily affects the local structure of the random graph, in particular the occurrence of subgraph patterns, but the dependencies in these structures and weights make them difficult to analyze. In this paper we investigate subgraph counts using a \textit{divide et impera} strategy: first counting the number of subgraphs in specific classes of vertices; then computing which class yields maximum contribution. Interestingly, the scaling behavior of induced and general subgraphs in such geometric heavy-tailed random graphs is closely related to the solution of a mixed-integer linear program which also shows that subgraphs appear predominantly on vertices with some prescribed degrees and inter-distances. Finally, we derive precise asymptotics for trees and Hamiltonian subgraphs.