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Complex networks have been mostly characterized from the point of view of thedegree distribution of their nodes and a few other motifs (or modules), with aspecial attention to triangles and cliques. The most exotic phenomena have beenobserved when the exponent $\gamma$ of the associated power lawdegree-distribution is sufficiently small. In particular, a zero percolationthreshold takes place for $\gamma<3$, and an anomalous critical behavior setsin for $\gamma<5$. In this Letter we prove that in sparse scale-free networkscharacterized by a cut-off scaling with the sistem size $N$, relativefluctuations are actually never negligible: given a motif $\Gamma$, we analyzethe 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 largestdegree of $\Gamma$, respectively. Remarkably, in $(\gamma_1,\gamma_2)$$R_{\Gamma}$ diverges, implying the instability of $\Gamma$ to smallperturbations.

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