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

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.