Fluctuations of motifs and non–self-averaging in complex networks: A self- vs. non–self-averaging phase transition scenario
Cooperative Association for Internet Data Analysis, San Diego Supercomputer Center, UCSD San Diego, CA, USA
Received: 14 September 2013
Accepted: 17 January 2014
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 , and an anomalous critical behavior sets in for . 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 of the associated density of Γ, and we show that there exists an interval in γ, , where does not go to zero in the thermodynamic limit, where and , and being the smallest and the largest degree of Γ, respectively. Remarkably, in diverges, implying the instability of Γ to small perturbations.
PACS: 89.75.Fb – Structures and organization in complex systems / 89.75.Hc – Networks and genealogical trees / 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion
© EPLA, 2014