Fundamental Science for Society FS2
Network Geometry. Theory and Applications
By M. Ángeles Serrano and Marián Boguñá
Boston, MA, United States, June 1
Room TBA
About the Workshop
This workshop aims to bring together researchers who are working on and/or interested in various aspects of Network Geometry. Broadly defined, the field of Network Geometry encompasses any studies in Network Science that either explicitly use geometry or have geometric interpretations. The scope of this workshop extends well beyond geometric network models and the problems of geometric network representations (aka network embeddings) and includes network topology of higher-order interactions, network geometry in machine learning, the emergence of network geometry from network dynamics, and geometric interpretations of network inference methods. Of particular interest are the applications of Network Geometry. Examples of disciplines that benefit from these results are Neuroscience, Biomedicine, Social Sciences, and Telecommunications.
Invited Speakers
Katy Börner
Indiana University
Marta Gonzalez
University of California
Andrea Gabrielli
University "Roma Tre" of Rome
Diego Garlaschelli
IMT Lucca, Leiden University
Filippo Radicchi
Indiana University
Melanie Weber
Harvard University
Pim van der Hoorn
Eindhoven University of Technology
Jasper van der Kolk
Central European University
Schedule
9:00 - 9:45
Katy Börner
The HuBMAP Human Reference Atlas (HRA) effort aims to develop a common coordinate framework (CCF) for the healthy human body, see HRA Portal at https://humanatlas.io. An international team of organ experts across 25+ consortia are authoring so called ASCT+B tables that capture the partonomy and typology information for anatomical structures, cell types, and biomarkers used to identify cell types. The ASCT+B tables are used to revise and extend existing CCF-relevant ontologies. In close collaboration with NIAID at NIH, a 3D Reference Object Library was compiled (https://3d.nih.gov/collections/hra) that provides semantically annotated 3D representations of major anatomical structures captured in the ASCT+B tables. The HRA can be extended and explored using several interactive user interfaces: The Registration User Interface (RUI, https://apps.humanatlas.io/rui) supports tissue data registration and annotation across 50+ 3D reference organs. The Exploration User Interface (EUI, https://apps.humanatlas.io/eui) supports exploration of semantically and spatially explicit data—from the whole body to the single cell level. The Cell Distance Explorer (CDE, https://apps.humanatlas.io/cde) computes and visualizes distance distributions between different cells, cell types, and anatomical structures and cell types and morphological features. For an introduction to HuBMAP goals, data, and code visit the Visible Human MOOC (VHMOOC, https://expand.iu.edu/browse/sice/cns/courses/hubmap-visible-human-mooc). This talk presents first multiscale, 3D visualizations of the HRA together with network geometry challenges and opportunities.
9:45 - 10:30
Filippo Radicchi
Cities around the world face the same fundamental challenge: allocating limited transport resources to serve highly heterogeneous mobility demands. In this talk, I present an empirical characterization of the properties of various types of real-world urban transit networks, then I show that the observed patterns can be coherently interpreted as the outcomes of relatively simple optimization principles. More specifically, in the first part of the talk, I consider a multiplex network model with a fast layer embedded in a slow one. The model is intended to describe the simultaneous presence and interaction of two distinct modes of transportation, for example cars and subways. To move between any pair of nodes, one can then use either the fast or slow layer, or both, with a switching cost when going from one layer to the other. I take advantage of analytical and numerical arguments to show that the optimal structure minimizing the transit time in the network is characterized by symmetry breaking, indicating that it is sometimes better to avoid serving a whole area in order to save on switching costs, at the expense of using more the slow layer. I then discuss the relevance of the model to estimate the efficiency of the subway systems in the cities of Atlanta, Boston, and Toronto. In the second part of the presentation, I analyze thousands of bus routes across twenty major cities worldwide and uncover a universal scaling relation linking service frequency to demand and route length. I then propose a model where such a scaling emerges from the minimization of passengers’ waiting time under a fixed operational budget. I conclude showing how quantitative predictions of the model could offer practical insights for improving the efficiency of resource-constrained public transport systems.
10:30 - 11:00
Coffee break
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11:00 - 11:45
Diego Garlaschelli
The representation of the internal architecture of all physical systems depends on the resolution at which they are observed. In spatially embedded systems, geometric coordinates provide a natural way to change the resolution level, allowing consistent mappings across scales that lie at the foundation of the renormalization group (RG). By contrast, for complex networks with no explicit spatial embedding, multiple renormalization schemes exist [1], resulting in non-unique representations of the same system across different scales. The Multiscale Network Renormalization approach [2,3] has been introduced as a way to design random graph models that can represent the same network consistently under arbitrary aggregations of nodes. It seeks a probability distribution over graphs that is invariant under node aggregation (see Fig. 1), thereby representing a nontrivial fixed point of some RG flow. The model comes with node variables that are additive upon node aggregation. It successfully replicates, at multiple hierarchical levels, the properties of several real-world networks [2,4,5] and lends itself naturally to the renormalization of directed graphs [3], which is otherwise problematic. It also allows one to infer the structural properties of a network at a hierarchical level that is different from the one at which empirical observations are available, opening new avenues for cross-scale network reconstruction [4]. Moreover, the approach can be applied to Machine Learning algorithms that take a graph as input and encode its structure onto output vectors that represent nodes in an abstract space [5,6]. Under arbitrary coarse-grainings of the input graph, the multiscale method ensures statistical consistency of the embedding vector of a block-node with the sum of the embedding vectors of its constituent nodes. This guarantee enables the interpretable application of the basic properties of vector spaces (i.e. sum of vectors and multiplication of a vector by a scalar) to the latent space where node embeddings are identified. It turns out that several key network properties, including a large number of triangles, are successfully replicated already from embeddings of very low dimensionality, allowing for the generation of faithful replicas of the original networks at arbitrary resolution levels [5,6]. Finally, an annealed version of the model leads to infinite-mean node variables and spontaneously replicates several real-world network properties, such as power-law degree distributions [2,7], disassortativity profiles [2], and the surprising coexistence of sparsity and clustering, without geometry or edge dependencies [8]. Moreover, since node aggregation invariance is a form of discrete scale invariance, several unique properties emerge in the spectrum of the adjancency matrix, such as log-periodicity and complex scaling exponents [9].
[1] A. Gabrielli, D. Garlaschelli, S. P. Patil, M. A. Serrano, Nature Reviews Physics 7, 203-219 (2025). ́
[2] E. Garuccio, M. Lalli, D. Garlaschelli, Physical Review Research 5(4), 043101 (2023).
[3] M. Lalli and D. Garlaschelli, https://arxiv.org/abs/2403.00235 (2024).
[4] L. N. Ialongo, S. Bangma, F. Jansen, D. Garlaschelli, https://arxiv.org/abs/2412.16122 (2024).
[5] R. Milocco, F. Jansen, D. Garlaschelli, https://arxiv.org/abs/2412.04354 (2024).
[6] R. Milocco, F. Jansen, D. Garlaschelli, https://arxiv.org/abs/2508.20706 (2025).
[7] L. Avena, D. Garlaschelli., R. S. Hazra, M. Lalli, Journal of Applied Probability, 1-26 (2025).
[8] A. Catanzaro, R. van der Hofstad, D. Garlaschelli, https://arxiv.org/abs/2603.13159 (2026).
[9] A. Catanzaro, R. S. Hazra, D. Garlaschelli, https://arxiv.org/abs/2509.12407 (2025).
11:45 - 12:30
Andrea Gabrielli
Heterogeneous and complex networks represent the intertwined interactions between real-world elements or agents. Determining the multi-scale mesoscopic organization of clusters and intertwined structures is still a fundamental and open problem of complex network theory. By taking advantage of the recent Laplacian Renormalization Group [1-5] approach , we scrutinize information diffusion pathways throughout networks to shed further light on this issue. Based on internode communicability, our definition provides a clear-cut framework for resolving the multi- scale mesh of structures in complex networks, disentangling their intrinsic arboreal architecture. Then we move to adapt the LRG framework to signed networks up to show its usefulness to tackle the issue connected with balancing, frustration and spin glass transition [6].
[1] P. Villegas, T. Gili, G. Caldarelli, A. Gabrielli, Laplacian renormalization group for heterogeneous networks, Nature Physics 19 (3), 445-450 (2023)
[2] A. Gabrielli, D. Garlaschelli, S.P. Patil & M.Á. Serrano, Network renormalization, Nature Reviews Physics 7, 203–219 (2025)
[3] A. Poggialini, P. Villegas, M.A. Munoz, A. Gabrielli, Networks with Many Structural Scales: A Renormalization Group Perspective, Phys. Rev. Lett. 134, 057401 (2025)
[4] P. Villegas, A. Gabrielli, A. Poggialini, T. Gili, Multi-scale Laplacian community detection in heterogeneous networks, Phys. Rev. Res. 7, 013065 (2025)
[5] G. Iannelli, P. Villegas, T. Gili, A. Gabrielli, Topological Symmetry Breaking in Antagonistic Dynamics, https://arxiv.org/abs/2009.11024
12:30 - 14:30
Lunch
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14:30 - 15:15
Pim van der Hoorn
The theory of graph limits has been one of the major developments in the last two decades. It started with dense graph limits, described as symmetric functions on the unit square (Graphons). Later, sparse graph limits were characterized as functions on the positive quadrant of the Euclidean plane that were invariant under measure preserving transformations (Graphexes). However, for a long time no proper limit notion was available for ultra-sparse graphs, where nodes and edges are of the same order. The introduction of local convergence seemed to address this and is by now an established method to study limit properties of ultra- sparse graphs. Unfortunately, local convergence fails to provide a statistical model from which finite graphs can be sampled. Something that both Graphons and Graphexes do have, and which makes them so powerful. Geometry is now presenting itself as a possible solution to this important gap.
In this talk, I will present a general framework, based on the concept of projective limits, that describes both Graphons and Graphexes and show how the inclusion of geometry to this framework opens the door to new powerful notions of graph limits for ultra-sparse graphs. Example that are covered include classical Random Geometric Graphs, but also more complex models such as Geometric Inhomogeneous Random Graphs and many more models with a geometric component.
This is based on joint work with: Remco van der Hofstad, Dmitri Krioukov, Neeladri Maitra and Huck Stepanyants.
15:15 - 16:00
Jasper van der Kolk
Geometric network models successfully reproduce many observed network properties such as small-worldness, high clustering and degree heterogeneity. In these models, nodes lie in an underlying metric space where the distances between them determine their connection probability. One key feature is that the coupling between the network’s underlying geometry and the final topology can be tuned. If the coupling is strong, only nodes that lie close together can be connected. If the coupling is weak, further away nodes also have a chance to connect.
In this talk, I will focus on this latter regime. Several important network quantities like clustering vanish here in the thermodynamic limit. However, the decay is very slow, making this region relevant for finite real networks. I will ask how a weak geometric coupling affects network structure, embedding and renormalization, and explore the effects on dynamics running atop the network. Additionally, I will investigate how these observations extend to other structures, like multiplexes and networks with higher order interactions.
These results clarify the role of geometry in shaping complex systems and highlight a regime that remains both theoretically subtle and empirically relevant.
16:00 - 16:30
Coffee break
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16:30 - 17:15
Melanie Weber
Community detection is a fundamental task in unsupervised graph learning and network analysis. In this talk we take a geometric perspective on community detection in complex networks based on notions of discrete Ricci curvature and associated geometric flows. Such methods rely on the key observation that edges connecting different communities tend to have low curvature, while edges within the same community exhibit higher curvature. This yields a principled approach to analyzing community structure through curvature-based thresholding, reweighting, and flow-based algorithms. We will discuss recent curvature-based algorithms and their theoretical underpinnings. Beyond the classical single-membership setting, we will also discuss the recovery of mixed-membership structure in overlapping communities, as well as community detection in complex networks with side information.
17:15 - 18:00
Marta Gonzalez
Organizers
Antoine Allard
Université Laval
Contact Us
Dima Krioukov, dima@northeastern.edu
M. Ángeles Serrano, marian.serrano@ub.edu
