Self-Similarity in Fractal and Non-Fractal Networks

摘要

We study the origin of scale invariance(SI)of the degree distribution in scale-free(SF)networkswith a degree,exponent-y under coarse graining.A varying number of vertices belonging to acommunity or a box in a fractal analysis is grouped into a supernode,where the box mass Mfollows a power-law distribution,P_m(M)~ M~(-n).The renormalized degreek'of a supernode scaleswith its box mass M as k'~M~0.The two exponents nand 0 can be nontrivial as n≠y and 0<1.They act as relevant parameters in determining the self-similarity,i.e.,the SI of degreedistribution, as follows:The self-similarity appears either when y ≤ n or under the condition 0=(n-1)/(y1) when y>n,irrespective of whether the original SF network is fractal ornon-fractal.Thus, fractality and self-similarity are disparate notions in SF networks.