# Variational principle for scale-free network motifs

Clara Stegehuis*, Remco van der Hofstad, Johan S. H. van Leeuwaarden

*Corresponding author for this work

### 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.
Original language English 6762 Scientific reports 9 https://doi.org/10.1038/s41598-019-43050-8 Published - 1 May 2019 Yes

### Keywords

• cs.SI
• physics.soc-ph