Complex networks of interactions permeate reality. Examples are all around us—the Internet, international trade, ecosystems, online and offline social networks—and even within us—biochemical interactions in our cells, the brain connectome. Surprisingly, complex networks speak a common language: they exhibit universal features regardless of their origin. In particular, real networks are small worlds, where any pair of nodes is separated by only a few intermediate links. This property suggests an apparent lack of metric structure, making such networks difficult to map in geometries consistent with connectivity. Yet, many networks are sustained by a hidden geometry that allows them to be embedded in a low-dimensional hyperbolic space. The discovery and analysis of such hidden geometry has become a cornerstone of modern network science, giving rise to the field of network geometry.

One of the key breakthroughs in network geometry is the identification of multiscale self-similarity as a common symmetry in real networks, revealed through geometric renormalization transformations. The renormalization group, a powerful theoretical framework in physics, enables the systematic study of systems with many degrees of freedom across different scales. It describes how configurations and associated model parameters evolve as resolution is changed and helps identify critical points and system behavior near phase transitions. However, applying renormalization techniques to complex networks is challenging due to their small-world nature, which induces non-locality, and their heterogeneous and hierarchical organization. Geometric renormalization (GR) overcomes these challenges. This method builds progressively coarse-grained maps of real networks in hyperbolic space with rescaled interactions, revealing an inherent self-similarity in the multiscale unfolding of real networks. These findings also extend to multiscale anatomical reconstructions of the human brain connectome and to the growth patterns of other networked systems, whose evolution can be effectively modeled through a reverse renormalization process. Practical applications of GR include the production of scaled up and scaled down network replicas, useful for a wide range of downstream tasks such as studying dynamical processes where network size is relevant.

This master class will explore the key ideas driving these developments, including the principles of network geometry, challenges of hyperbolic embedding, and the fundamentals and applications of GR theory to complex networks. We will examine the multiscale architecture and self-similarity of real-world networks, with a focus on the brain connectome as a case study. Participants will engage with both theoretical foundations and hands-on computational implementations, gaining tools to analyze, visualize, and manipulate networks across multiple scales.

• Network geometry
  M Boguna, I Bonamassa, M De Domenico, S Havlin, D Krioukov, MÁ Serrano
  Nature Reviews Physics 3 (2), 114-135, 2021
• Multiscale unfolding of real networks by geometric renormalization
  G García-Pérez, M Boguñá, MÁ Serrano
  Nature Physics 14 (6), 583-589, 2018
• Scaling up real networks by geometric branching growth
  M Zheng, G García-Pérez, M Boguñá, MÁ Serrano
  Proceedings of the National Academy of Sciences 118 (21), e2018994118, 2021
• Geometric renormalization of weighted networks
  M Zheng, G García-Pérez, M Boguñá, MÁ Serrano
  Communications Physics 7 (1), 97, 2024
• The D-Mercator method for the multidimensional hyperbolic embedding of real networks
  R Jankowski, A Allard, M Boguñá, MÁ Serrano
  Nature Communications 14 (1), 7585, 2023
• Mercator: uncovering faithful hyperbolic embeddings of complex networks
  G García-Pérez, A Allard, MÁ Serrano, M Boguñá
  New Journal of Physics 21 (12), 123033, 2019
• Geometric renormalization unravels self-similarity of the multiscale human connectome
  M Zheng, A Allard, P Hagmann, Y Alemán-Gómez, MÁ Serrano
  Proceedings of the National Academy of Sciences 117 (33), 20244-20253, 2020
• The multiscale self-similarity of the weighted human brain connectome
  L Barjuan, M Zheng, M Serrano
  PLOS Computational Biology 2025
• Network renormalization
  A Gabrielli, D Garlaschelli, SP Patil, MÁ Serrano
  Nature Reviews Physics, 1-17, 2025