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We show that renormalization group (RG) theory applied to complex networksare useful to classify network topologies into universality classes in thespace of configurations. The RG flow readily identifies a small-world/fractaltransition by finding (i) a trivial stable fixed point of a complete graph,(ii) a non-trivial point of a pure fractal topology that is stable or unstableaccording to the amount of long-range links in the network, and (iii) anotherstable point of a fractal with short-cuts that exists exactly at thesmall-world/fractal transition. As a collateral, the RG technique explains thecoexistence of the seemingly contradicting fractal and small-world phases andallows to extract information on the distribution of short-cuts in real-worldnetworks, a problem of importance for information flow in the system.

Small world-Fractal Transition in Complex Networks: Renormalization Group Approach