# Variational principle for scale-free network motifs

## TL;DR

This paper introduces an optimization-based method to identify dominant motifs in scale-free networks with degree distributions following a power law, providing explicit formulas for motif counts and fluctuations.

## Contribution

It presents a novel variational principle to determine the most probable motif structures in scale-free networks with degrees in (2,3), advancing understanding of their subgraph patterns.

## Key findings

- Derived explicit asymptotic formulas for motif counts.
- Classified motifs into categories with small and large fluctuations.
- Provided a method to identify dominant motif structures.

## Abstract

For scale-free networks with degrees following a power law with an exponent $\tau\in(2,3)$, the structures of motifs (small subgraphs) are not yet well understood. We introduce a method designed to identify the dominant structure of any given motif as the solution of an optimization problem. The unique optimizer describes the degrees of the vertices that together span the most likely motif, resulting in explicit asymptotic formulas for the motif count and its fluctuations. We then classify all motifs into two categories: motifs with small and large fluctuations.