Perform inference on systems involving interactions between groups of nodes of any arbitrary size.
In recent years, real-world data from diverse domains, including social and biological systems, have revealed interactions that go beyond pairwise connections, involving groups of nodes of various sizes. Hypergraphs provide a versatile and comprehensive framework for characterizing systems where such higher-order interactions are relevant. To analyze such data, models must extend beyond conventional dyadic interactions (represented as \(A_{ij}\)) and instead focus on hyperedges, referred to as \(d\)-dimensional interactions \(A_e\), where \(d = \|e\|\).
We developed two distinct probabilistic models designed to perform inference on hypergraphs while capturing their hidden structural organization. These models, named Hypergraph-MT(Contisciani et al., 2022) and Hy-MMSBM(Ruggeri et al., 2023), posit the existence of a mixed-membership community structure as the main generative process, and utilize Poisson distributions to model the hyperedges. The main difference between the two approaches lies in how they integrate latent variables into the model, leading to two different assumptions about data generation.
Specifically, Hypergraph-MT describes a hyperedge through the product of the memberships of all nodes involved, assuming exclusively assortative community structures to make this computation feasible. On the other hand, Hy-MMSBM relaxes the assortativity constraint and flexibly captures various community structures, such as disassortative and core-periphery. It achieves this by employing a bilinear form to link hyperedge probabilities and node community memberships.
These models, along with a broad range of other tools and algorithms for handling data with higher-order interactions, are available in the Python library hypergraphx(Lotito et al., 2023).
Main takeaways
Our models reliably predict the existence of higher-order interactions of arbitrary size.
Hypergraph-MT detects communities that reliable depict the information carried by hyperedges and exhibit robustness against the addition of noisy interactions.
Hy-MMSBM accurately retrieves the planted communities in scenarios with varying hyperedges sizes, effectively captures assortative and disassortative community structures, and correctly represents core-periphery configurations.
References
[6]
Inference of hyperedges and overlapping communities in hypergraphs
Martina Contisciani, Federico Battiston, and Caterina De Bacco
Hypergraphs, encoding structured interactions among any number of system units, have recently proven a successful tool to describe many real-world biological and social networks. Here we propose a framework based on statistical inference to characterize the structural organization of hypergraphs. The method allows to infer missing hyperedges of any size in a principled way, and to jointly detect overlapping communities in presence of higher-order interactions. Furthermore, our model has an efficient numerical implementation, and it runs faster than dyadic algorithms on pairwise records projected from higher-order data. We apply our method to a variety of real-world systems, showing strong performance in hyperedge prediction tasks, detecting communities well aligned with the information carried by interactions, and robustness against addition of noisy hyperedges. Our approach illustrates the fundamental advantages of a hypergraph probabilistic model when modeling relational systems with higher-order interactions.
@article{contisciani2022inference,title={Inference of hyperedges and overlapping communities in hypergraphs},author={Contisciani, Martina and Battiston, Federico and De Bacco, Caterina},journal={Nature Communications},volume={13},number={1},pages={7229},year={2022},month=nov,publisher={Nature Publishing Group UK London},}
[9]
Community detection in large hypergraphs
Nicolò Ruggeri, Martina Contisciani, Federico Battiston, and Caterina De Bacco
Hypergraphs, describing networks where interactions take place among any number of units, are a natural tool to model many real-world social and biological systems. Here, we propose a principled framework to model the organization of higher-order data. Our approach recovers community structure with accuracy exceeding that of currently available state-of-the-art algorithms, as tested in synthetic benchmarks with both hard and overlapping ground-truth partitions. Our model is flexible and allows capturing both assortative and disassortative community structures. Moreover, our method scales orders of magnitude faster than competing algorithms, making it suitable for the analysis of very large hypergraphs, containing millions of nodes and interactions among thousands of nodes. Our work constitutes a practical and general tool for hypergraph analysis, broadening our understanding of the organization of real-world higher-order systems. A principled and fast model accurately detects mixed-membership communities in hypergraphs.
@article{ruggeri2023community,title={Community detection in large hypergraphs},author={Ruggeri, Nicol{\`o} and Contisciani, Martina and Battiston, Federico and De Bacco, Caterina},journal={Science Advances},volume={9},number={28},pages={eadg9159},year={2023},month=jul,publisher={American Association for the Advancement of Science},}
[8]
Hypergraphx: a library for higher-order network analysis
Quintino Francesco Lotito, Martina Contisciani, Caterina De Bacco, Leonardo Di Gaetano, Luca Gallo, Alberto Montresor, Federico Musciotto, Nicolò Ruggeri, and Federico Battiston
From social to biological systems, many real-world systems are characterized by higher-order, non-dyadic interactions. Such systems are conveniently described by hypergraphs, where hyperedges encode interactions among an arbitrary number of units. Here, we present an open-source python library, hypergraphx (HGX), providing a comprehensive collection of algorithms and functions for the analysis of higher-order networks. These include different ways to convert data across distinct higher-order representations, a large variety of measures of higher-order organization at the local and the mesoscale, statistical filters to sparsify higher-order data, a wide array of static and dynamic generative models, and an implementation of different dynamical processes with higher-order interactions. Our computational framework is general, and allows to analyse hypergraphs with weighted, directed, signed, temporal and multiplex group interactions. We provide visual insights on higher-order data through a variety of different visualization tools. We accompany our code with an extended higher-order data repository and demonstrate the ability of HGX to analyse real-world systems through a systematic analysis of a social network with higher-order interactions. The library is conceived as an evolving, community-based effort, which will further extend its functionalities over the years. Our software is available at https://github.com/HGX-Team/hypergraphx.
@article{lotito2023hypergraphx,title={Hypergraphx: a library for higher-order network analysis},author={Lotito, Quintino Francesco and Contisciani, Martina and De Bacco, Caterina and Di Gaetano, Leonardo and Gallo, Luca and Montresor, Alberto and Musciotto, Federico and Ruggeri, Nicol{\`o} and Battiston, Federico},journal={Journal of Complex Networks},volume={11},number={3},pages={cnad019},year={2023},month=may,publisher={Oxford University Press},}