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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 $γ$ of the associated power law degree-distribution is sufficiently small. In particular, a zero percolation threshold takes place for $γ<3$, and an anomalous critical behavior sets in for $γ<5$. 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 $Γ$, we analyze the relative fluctuations $R_Γ$ of the associated density of $Γ$, and we show that there exists an interval in $γ$, $[γ_1,γ_2]$, where $R_Γ$ does not go to zero in the thermodynamic limit, where $γ_1\approx k_{\mathrm{min}}$ and $γ_2\approx 2 k_{\mathrm{max}}$, $k_{\mathrm{min}}$ and $k_{\mathrm{max}}$ being the smallest and the largest degree of $Γ$, respectively. Remarkably, in $(γ_1,γ_2)$ $R_Γ$ diverges, implying the instability of $Γ$ to small perturbations.