Statistical Physics and Laplacian Renormalization Group for Complex Networks

A team of researchers at CREF has developed an innovative method based on renormalization to study complex networks with heterogeneous nodes. The approach iteratively simplifies the network at different scales, identifying groups of strongly connected nodes and analyzing their dynamic properties. Published in Nature Physics (May 2023), the work offers a new perspective on mesoscopic analysis, overcoming the limitations of traditional null models. The Likelihood Renormalization Group (LRG) reveals hidden structures and modularity, with applications in biology, sociology, and neuroscience. By collaborating with IMT Lucca and Ca’ Foscari, the researchers have optimized critical processes like the spread of epidemics or energy management. In parallel, with the University of Rome “Tor Vergata,” they’ve created stochastic models inspired by constrained maximum entropy, which are useful for building null models and reconstructing real networks from partial data.

These advancements merge statistical physics and network theory, opening up new avenues for targeted interventions in complex systems.

Complex Systems Approach for Biological, Ecological, and Climate Systems

A collaboration between our team, IMT Lucca, and the University of Venice has explored how the spatial arrangement of plants influences the stability of ecosystems, using cutting-edge tools from statistical physics and complex network theory. Meanwhile, in the field of neuroscience, we’ve developed innovative methodologies with IMT Lucca and Bocconi to interpret brain signals from functional magnetic resonance imaging. With the Italian National Institute of Health and the BioBizkaia center in Bilbao, we are refining brain data analysis through machine learning techniques.

The pandemic prompted a fruitful collaboration with the Croatian Ruđer Bošković Institute to create more precise epidemiological models that account for the impact of diagnostic testing. This work has evolved to include the study of complex social interactions in viral spread, together with IFISC in Mallorca. We recently extended our research into the climate field, collaborating with the Copernicus Climate Change Service to apply our analytical methods to interpreting environmental data. These projects demonstrate how integrating different scientific disciplines can provide powerful tools to address crucial challenges, from ecosystem conservation and public health to understanding climate change.

The project has two main goals:

• Development of General Methods: To formulate and study models and methods based on statistical physics—both in and out of equilibrium—to provide innovative theoretical and numerical tools. These will be applicable in various areas of the natural sciences, with a particular focus on the biomedical field, as well as in the quantitative social sciences for analyzing complex networks and extracting signals from correlated time series.

• High-Impact Applications: To develop practical applications of these methods in the biomedical, social, ecological, and environmental sectors. There is also the potential to extend these tools to other research areas within the institution, such as new artificial intelligence systems based on Hopfield networks.

A special focus will be placed on the further development of the LRG (Likelihood Renormalization Group) technique, advancing the theoretical analysis of complex networks. This will enable the study of the organization and intrinsic correlation properties of natural systems, such as biological and ecological networks, and characterize their dynamic evolution. In addition, its applications will be extended to different fields, such as social and economic networks, and to new artificial intelligence systems.

Developments and Applications of the Laplacian Renormalization Group for Complex Networks

The LRG theory, by drawing a parallel with the Renormalization Group in statistical physics, has made it possible to study the multi-scale properties of a generic complex network. In short, the LRG is a fundamental tool for understanding and optimizing the function of complex systems.

The future development of this approach will focus on three main directions:

• Extension of the Statistical-Physical Theory: The plan is to complete the statistical-physical understanding of the method, with a particular focus on the local formulation of the theory, by extending the Fluctuation-Dissipation relations to the LRG. This will make it possible to study time-dependent diffusion and to locally detect correlation scales and the microscopic structure of the network. Additionally, the theory will be extended to the case of signed networks, a crucial problem for understanding spin glasses.

• Reformulation of the Clustering Problem: A new approach for detecting the optimal partition of the network will be developed, optimizing current methodologies to manage large networks, up to millions of nodes. This approach will allow for the real-time analysis of data and time series, leveraging diffusion time and statistical dynamic models, such as epidemic models and the Kuramoto model for synchronization. The goal is to study the behavior of extensive systems under well-defined scale transformations within the statistical-physical framework of the LRG.

• Characterization of Scale-Invariant Disordered Systems: After identifying the non-trivial fixed points of the LRG, the next step will be to analyze their stability and dynamic implications. This will enable the study of regular fractal networks and multi-scale systems, with the goal of exploring different local and global dimensionalities.

Andrea Gabrielli (Professor of Physics at University Roma 3, Head of Research Line)

Pablo Villegas