Complex networks often exhibit a rich architecture organized over multiple intertwined scales. Information pathways are expected to pervade these scales, reflecting structural insights that are not manifest from analyses of the network topology. Moreover, small-world effects correlate the different network hierarchies making the identification of coexisting mesoscopic structures and functional cores a difficult task. We first present a thermodynamic interpretation of effective information pathways throughout complex networks based on information diffusion and statistical mechanics to illuminate these issues [1]. This leads us to a formulation of a new and general Renormalization Group scheme for heterogeneous networks that permits to change resolution scale in a physically motivated way. The Renormalization Group (RG) is the cornerstone of the modern theory of scale transformation, universality, and phase transitions, a powerful tool to scrutinize symmetries and organizational scales in dynamical systems. However, its network counterpart is particularly challenging due to correlations and small-world coupling between intertwined scales. Here, we propose a Laplacian Renormalization Group (LRG) diffusion-based approach to complex networks, defining the coarse-grained supernodes and superedges concept à la Kadanoff, the equivalent of the momentum space RG procedure à la Wilson for graphs and applying this RG scheme to real networks in a natural and parsimonious way to define proper scale transformation at arbitrarily resolution scale, study the topological organisation of the network [2] and detect characteristic structures [3].

[1] P. Villegas, A. Gabrielli, G. Caldarelli, T. Gili, Laplacian paths in complex networks: Information core emerges from entropic transitions, Physical Review Research 4, 033196 (2022).

[2] P. Villegas, T. Gili, G. Caldarelli, A. Gabrielli, Laplacian Renormalization Group for heterogeneous networks, Nature Physics 19, 445–450 (2023).

[3] P. Villegas, A. Gabrielli, A. Poggialini, T. Gili, Multi-scale Laplacian community detection in heterogeneous networks, https://arxiv.org/abs/2301.04514