Renormalization flows in complex networks

Filippo Radicchi1, José J. Ramasco1, Alain Barrat2,3 and Santo Fortunato1
1Complex Systems Lagrange Laboratory, Complex Networks (CNLL), ISI Foundation, Turin I-10133, Italy.
2Laboratoire de Physique Theórique (CNRS UMR 8627), Universit&ecaute; de Paris-Sud, France
3CPT (CNRS UMR 6207), Luminy Case 907, F-13288 Marseille Cedex 9, France

(May 2008)

Recently, it has been claimed that some complex networks are self-similar under a convenient renormalization procedure. We present a general method to study renormalization flows in graphs. We find that the behavior of some variables under renormalization, such as the maximum number of connections of a node, obeys simple scaling laws, characterized by critical exponents. This is true for any class of graphs, from random to scale-free networks, from lattices to hierarchical graphs. Therefore, renormalization flows for graphs are similar as in the renormalization of spin systems. An analysis of classic renormalization for percolation and the Ising model on the lattice confirms this analogy. Critical exponents and scaling functions can be used to classify graphs in universality classes, and to uncover similarities between graphs that are inaccessible to a standard analysis.