Finite-size scaling of geometric renormalization flows in complex networks

Recently, the concept of geometric renormalization group provides a good approach for studying the structural symmetry and functional invariance of complex networks. Along this line, we systematically investigate the finite-size scaling of structural and dynamical observables in geometric renormalization flows of synthetic and real evolutionary networks. Our results show that these observables can be well characterized by a certain scaling function. Specifically, we show that the critical exponent implied by the scaling function is independent of these observables but only depends on the small-world properties of the network, namely, all networks located in the small-world phase have a uniform scaling exponent, while those located in the non-small-world phase and in their critical regions have another uniform scaling. More importantly, we perform extensive experiments on real evolutionary networks with small-world characteristics, and our results show that these observables also have uniform scaling in their geometric renormalization flows. Therefore, in a sense this exponent can be used as an effective measure for classifying universal small-world and non-small-world network models.