Seminar by Guillermo García-Perez

Multiscale unfolding of networks by geometric renormalization

 Renormalization has proven to be a very powerful tool for a systematic investigation of systems--from condensed matter to quantum field theory--as viewed at different scales. In the field of complex networks, some topological renormalization schemes based on shortest paths between nodes have been proposed in the past. However, the collection of shortest-path lengths, albeit a well-defined metric space, is a poor source of length-based scaling factors in networks due to their homogeneity and the small-world property.

We present empirical evidence that real-world complex networks are invariant under geometric length scale transformations when embedded in an underlying hidden metric space. We prove analytically that the embedding model is also renormalizable in the same framework and take it as new evidence supporting our conjecture that hidden metric spaces underlie real networks. The congruency between real networks and the geometric model enables to define a multiscale unfolding of complex networks that allows the study of their self-similarity properties.

As applications, this geometric renormalization scheme yields a natural way of building smaller-scale replicas of real networks while simultaneously preserving their statistical properties, which can be extremely useful in the study of dynamical processes on large networks. The hidden metric space renormalization group can also be applied to design a new greedy routing protocol in hyperbolic space which exploits the multiscale unfolding of complex networks increasing the success rate of single-scale versions.