Optimal subgraph structures in scale-free configuration models

TL;DR

This paper analyzes the occurrence and structure of small subgraphs in scale-free networks with degree exponent between 2 and 3, revealing how subgraphs are formed among vertices with specific degree ranges.

Contribution

It introduces a novel optimization framework to determine the degrees of vertices forming subgraphs in scale-free configuration models, enabling precise asymptotic counts.

Findings

Subgraphs typically occur among vertices with specific degree ranges.

The paper provides asymptotic formulas for subgraph counts in scale-free networks.

It characterizes all subgraphs without double counting in the erased configuration model.

Abstract

Subgraphs reveal information about the geometry and functionalities of complex networks. For scale-free networks with unbounded degree fluctuations, we obtain the asymptotics of the number of times a small connected graph occurs as a subgraph or as an induced subgraph. We obtain these results by analyzing the configuration model with degree exponent $\tau\in(2,3)$ and introducing a novel class of optimization problems. For any given subgraph, the unique optimizer describes the degrees of the vertices that together span the subgraph. We find that subgraphs typically occur between vertices with specific degree ranges. In this way, we can count and characterize {\it all} subgraphs. We refrain from double counting in the case of multi-edges, essentially counting the subgraphs in the {\it erased} configuration model.