1306.5565 (Massimo Ostilli)

Fluctuations of motifs and non self-averaging in complex networks    [PDF]

Massimo Ostilli

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 seen to occur 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 large cut-off, 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 [\gamma_1,\gamma_2] where R_{\Gamma} does not go to zero in the thermodynamic limit, where \gamma_1 ~ k_min and \gamma_2 ~ 2 k_max, k_min and k_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.

View original: http://arxiv.org/abs/1306.5565