An Invitation to Network Renormalization
The renormalization group (RG) method, rooted in the statistical physics of the 1960s and 70s, has unraveled the scaling properties of critical systems that span a range of length scales. As complex systems are inherently multiscale, networked, and exhibit emergent properties such as (approximate) scale-freeness and self-similarity across scales, recent years have seen a surge of activity in developing robust frameworks for renormalization techniques applied to complex networks.
A diversity of network renormalization techniques and applications have been proposed over the last few years, including principled approaches, such as geometric renormalization, Laplacian renormalization, and multiscale renormalization, as well as phenomenological approaches applied to time-series, complex networks and simplicial complexes. These methods have been instrumental in uncovering multiscale structures, scaling laws, and hierarchical organization in various complex systems, e.g., from brain networks, financial networks, and machine learning.
In this satellite meeting, we aim to bring together scientists working on network renormalization from various angles to reconcile different techniques and discuss applications and generalizations of network renormalization methods. This gathering will strengthen the community, clarify the connections between different renormalization schemes, and help build a vision for the future of research on networks spanning multiple scales.
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Topics of Interest:
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Geometric renormalization
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Laplacian renormalization
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Multiscale renormalization
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Phenomenological renormalization group approaches in time series analysis
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Scaling relations and universality classes in complex networks
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Coarse-graining methods and multiscale analysis
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Applications of network renormalization to neuroscience and neural networks
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Renormalization in social, financial, and biological networks
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Higher-order Renormalization of simplicial complexes
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Computational tools and algorithms for network renormalization
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Empirical studies demonstrating scaling behaviors in real-world networks
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Challenges and limitations of network renormalization techniques
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Future directions and open questions in network renormalization​
Confirmed invited speakers
Giovanni Petri
(Northeastern University London)
Ginestra Bianconi
(Queen Mary University of London)
Diego Garlaschelli
(IMT Advanced School Lucca & Leiden University)
Fernando N. Santos
(University of Amsterdam)
Muhua Zheng
(Jiangsu University)
Format and Registration
Half-day event with invited contributions and a panel discussion. All satellite participants and presenters should be registered, either for the satellites only or for the full conference. Please register through the conference website.
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Preliminary program:
Session 1 (8:30 – 10:30)
8:30 - 9:00 Walk-in and introduction by organizers
9:00 – 9:30 Diego Garlaschelli
9:30 – 10:00 Muhua Zheng
10:00 – 10:30 Giovanni Petri
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Coffee break 10:30 – 11:00
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Session 2 (11:00 – 12:30)
11:00 – 11:30 Fernando N. Santos
11:30 - 12:00 Ginestra Bianconi
12:00 – 12:30 Discussion session
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Talk abstracts:
Diego Garlaschelli - Multiscale network renormalization
The internal architecture of all physical systems depends on the resolution scale at which they are represented, yet their observable macroscopic behaviour is independent of that scale. In spatially embedded systems, geometric coordinates indicate a natural way of consistently coarse-graining both the real system and any model of the latter, a property that underlies 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 recently designed to model the same network consistently under arbitrary aggregations of nodes. It is based on a probability distribution over graphs that is invariant under node aggregation, thereby representing a fixed point of a certain RG flow. The model successfully replicates, at multiple hierarchical levels, the properties of several real-world networks [2,3,4,5]. 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]. In particular, 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. 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. Finally, a purely abstract, annealed version of the model successfully replicates several real-world network properties without any fitting parameters [2,6].
[1] Nature Reviews Physics 7, 203-219 (2025).
[2] Physical Review Research 5, 4, 043101 (2023).
[3] arXiv:2403.00235 (2024).
[4] arXiv:2412.16122 (2024).
[5] arXiv:2412.04354 (2024).
[6] arXiv:2212.08462 (2022).
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Muhua Zheng - Geometric renormalization of complex networks
Understanding large networked complex systems requires reducing their complexity. The renormalization group offers a powerful framework for this purpose. In this talk, we provide a framework for investigating complex networks at different resolutions. Firstly, we introduce a geometric renormalization (GR) protocol by decreasing the resolution in a complex network. Then, we show that the GR model produces a multiscale unfolding of the network in scaled-down replicas, and predict the multiscale self-similar properties of human connectomes. Finally, we extend the geometric renormalization framework to weighted networks, where the intensities of the interactions play a crucial role in their structural organization and function. Our approach allows straightforward derivation of scaled-down weighted networks, facilitating the investigation of various size-dependent phenomena in downstream applications.
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Giovanni Petri - Renormalization and Higher-Order Interactions: Bridging Structure and Dynamics in Complex Systems
I will explore the interplay between renormalization techniques and higher-order correlations in both structural and dynamical models. By examining how renormalization can be extended to capture higher-order interactions in complex systems, I aim to highlight the crucial role these correlations play in shaping emergent phenomena. Through recent advancements in higher-order network theory and models of dynamical systems, I will demonstrate how incorporating higher-order structural correlations refines our understanding of collective dynamics. This synthesis provides a pathway to unify topological insights and renormalization group methods, offering a deeper framework for analyzing systems that exhibit both strong interdependence and hierarchical organization. The implications for both theoretical approaches and empirical models will be discussed, with an emphasis on future research directions.
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Fernando N. Santos - Interdependent scaling exponents in the human brain
We apply the phenomenological renormalization group to resting-state fMRI time series of brain activity in a large population. By recursively coarse-graining the data, we compute scaling exponents for the series variance, log probability of silence, and largest covariance eigenvalue. The exponents clearly exhibit linear interdependencies, which we derive analytically in a mean-field approach. We find a significant correlation of exponent values with the gray matter volume and cognitive performance. Akin to scaling relations near critical points in thermodynamics, our findings suggest scaling interdependencies are intrinsic to brain organization and may also exist in other complex systems.
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Ginestra Bianconi - What is universal about the spectral dimension of simple and higher-order networks?
In network theory examples of scaling behaviour that do not correspond to strictly speaking RG universality classes are ubiquitous. For instance power-law exponents of the scale-free distributions can vary widely, and they do not have specific values. Here we provide evidence that also the spectral dimension of simple and higher-order networks does not take specific values but can vary widely.
By considering examples drawn from the very comprehensive non-equilibrium model for simple and higher-order network called Network Geometry with Flavor (NGF), we show that the spectral dimension can be tuned by considering:
(i) different level of stochasticity in the growth of simple and higher-order networks;
(ii) different nature of their building blocks, including different topological dimension (triangles, versus tetrahedra,etc.) and their geometry (tetrahedra versus hypercubes ect.);
(iii) different order of the considered Laplacian.
When dynamical process are defined on top of simple and higher-order networks, however, specific range of values of the spectral dimension can give rise to different universality classes of the dynamic behaviour as it occurs for the Ising and the Kuramoto model.
This highlights the fact that possibly universality classes and behaviors need to be studied and addressed at the dynamical level rather than at the structural level while at the structural level the most important question is what are the mechanisms that affect the values of the spectral dimension.​​
Organisers
Fernando N. Santos
(University of Amsterdam)
Wout Merbis
(University of Amsterdam, Statistics Netherlands)
M. Ángeles Serrano
(Universitat de Barcelona)
