Motif enrichment as a driver of scale-free behavior in rewired random regular graphs

We study the statistics of rewired random regular graphs (RRGs) in a mixed ensemble, where the average number of triangles is controlled by the fugacity $\lambda$, while the number of vertices and the vertex degree are fixed. This model exhibits a phase transition at critical fugacity $\lambda_{cr}$ from a triangle-poor phase (TPP), in which the number of triangles is independent of the system size, to a triangle-rich phase (TRP), in which the number of triangles scales linearly with the system size. We estimate $\lambda_{cr}$ by comparing the entropy of TPP with the energy of TRP. Above $\lambda_{cr}$, the RRG becomes a two-phase system in which dense clusters are connected by a sparse scale-free sub-network characterized by a degree distribution, $P(d) \sim d^{-\gamma}$, with $\gamma \approx 2$, independent of the size of the whole graph and its degree. We attribute this behavior to an "emergent preferential attachment" induced by triangle motifs, describe the mechanism underlying its formation, and derive the exponent $\gamma$ within a mean-field approach. We show that most inter-cluster triangles are isosceles, with the base lying inside one cluster and the apex belonging to the inter-cluster network. Finally, we speculate on a possible connection between these triangles and Efimov states in a conformally invariant potential.