The problem of multiscale descriptions of complex networks [1] presents strict analogies with studying critical phenomena and scale invariance in statistical physics [2]. In this context, the problem has been solved by the introduction of the Renormalization Group (RG) both in real space à la Kadanoff [3] and in the wave mode space à la Wilson [4]. The essential idea behind RG is to study under scale transformations the changes in the physical behavior of the system about criticality.  In such a procedure, elementary degrees of freedom are progressively integrated out to define mesoscale interacting physical variables at larger and larger scales. This permits the detection of the presence of characteristic scales and to study how these diverge when the critical point is approached.

The main obstacle to extending such an approach to general networks is given their complex geometrical and topological structure. Indeed, in real systems, very often, if not always, such interactions are characterized by the presence of hidden multiple scales and related substructures whose identification is as fundamental as difficult by using ordinary statistical network tools.

Directly inspired by the statistical physics ideas, we have introduced a new scale-transformation procedure for networks based on the Laplacian diffusion dynamics as a detector of the inner scales of the complex interacting system in the paper:

 Laplacian Renormalization group for heterogeneous networks 

on Nature Physics. 

This novel method proved essential to efficiently and elegantly uncover complex networks' multiple-scale organization and detect scale-invariant features when present. Moreover, it defines a universal procedure of rescaling networks which is, on one side, beneficial for data analysis of large natural networked systems and can be seen as the direct generalization to complex network geometry of the original Kadanoff’s and Wilson’s ideas.

[1] Barabási, A-L, Network Science CUP (2016); Caldarelli, G, Scale-Free Networks OUP (2007).

[2] Mandelbrot, B, The Fractal Geometry of Nature (1982).

[3] Kadanoff, LP, "Scaling laws for Ising models near Tc". Physics Physique Fizika. 2, 263 (1966).

[4] Wilson, KG and Kogut, J, "The renormalization group and the e-expansion". Physics Reports 12(2), 75 (1975).