Fluctuations of motifs and non self-averaging in complex networks. A self- vs non-self-averaging phase transition scenario

TL;DR

This paper investigates fluctuations of motifs in scale-free networks, revealing a phase transition where certain motifs become unstable due to non-vanishing fluctuations in the thermodynamic limit.

Contribution

It demonstrates the existence of a non-self-averaging phase in scale-free networks, identifying the specific gamma interval where motif fluctuations do not diminish as network size grows.

Findings

Relative fluctuations of motifs do not vanish in certain gamma ranges.

Motif instability occurs due to diverging fluctuations within specific gamma intervals.

Identifies critical gamma values related to motif degrees where fluctuations diverge.

Abstract

Complex networks have been mostly characterized from the point of view of the degree distribution of their nodes and a few other motifs (or modules), with a special attention to triangles and cliques. The most exotic phenomena have been observed when the exponent $\gamma$ of the associated power law degree-distribution is sufficiently small. In particular, a zero percolation threshold takes place for $\gamma<3$, and an anomalous critical behavior sets in for $\gamma<5$. In this Letter we prove that in sparse scale-free networks characterized by a cut-off scaling with the sistem size $N$, relative fluctuations are actually never negligible: given a motif $\Gamma$, we analyze the relative fluctuations $R_{\Gamma}$ of the associated density of $\Gamma$, and we show that there exists an interval in $\gamma$, $[\gamma_1,\gamma_2]$, where $R_{\Gamma}$ does not go to zero in the thermodynamic limit, where $\gamma_1\approx k_{\mathrm{min}}$ and $\gamma_2\approx 2 k_{\mathrm{max}}$, $k_{\mathrm{min}}$ and $k_{\mathrm{max}}$ being the smallest and the largest degree of $\Gamma$, respectively. Remarkably, in $(\gamma_1,\gamma_2)$ $R_{\Gamma}$ diverges, implying the instability of $\Gamma$ to small perturbations.