Generalized Topologies: Hypergraphs, Chemical Reactions, and Biological Evolution

doi:10.2174/9781681080529115020017

Cited by 4 publications

(1 citation statement)

References 57 publications

(81 reference statements)

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“…While graphs model pairwise interactions, hypergraphs generalize this concept by capturing higher-order interactions involving more than two elements. This extension provides a more expressive framework for modeling intricate dependencies and interactions in various fields, ranging from social network analysis (early acknowledged in Wolff, 1950) or co-authorship relations (Roy and Ravindran, 2015) to ecological systems (Muyinda et al, 2020), neurosciences (Chelaru et al, 2021) or even chemistry (Flamm et al, 2015). We refer to Battiston et al, 2020;Bick et al, 2023;Torres et al, 2021 for recent reviews on higher-order interactions.…”

Section: Introductionmentioning

confidence: 99%

Comparison of modularity-based approaches for nodes clustering in hypergraphs

Poda,

Matias

2024

Peer Community Journal

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Statistical analysis and node clustering in hypergraphs constitute an emerging topic suffering from a lack of standardization. In contrast to the case of graphs, the concept of nodes' community in hypergraphs is not unique and encompasses various distinct situations. In this work, we conducted a comparative analysis of the performance of modularity-based methods for clustering nodes in binary hypergraphs. To address this, we begin by presenting, within a unified framework, the various hypergraph modularity criteria proposed in the literature, emphasizing their differences and respective focuses. Subsequently, we provide an overview of the state-of-the-art codes available to maximize hypergraph modularities for detecting node communities in hypergraphs. Through exploration of various simulation settings with controlled ground truth clustering, we offer a comparison of these methods using different quality measures, including true clustering recovery, running time, (local) maximization of the objective, and the number of clusters detected. Our contribution marks the first attempt to clarify the advantages and drawbacks of these newly available methods. This effort lays the foundation for a better understanding of the primary objectives of modularity-based node clustering methods for binary hypergraphs.

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Section: Introductionmentioning

confidence: 99%

Comparison of modularity-based approaches for nodes clustering in hypergraphs

Poda,

Matias

2024

Peer Community Journal

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Thermodynamics of structure-forming systems

et al. 2021

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Structure-forming systems are ubiquitous in nature, ranging from atoms building molecules to self-assembly of colloidal amphibolic particles. The understanding of the underlying thermodynamics of such systems remains an important problem. Here, we derive the entropy for structure-forming systems that differs from Boltzmann-Gibbs entropy by a term that explicitly captures clustered states. For large systems and low concentrations the approach is equivalent to the grand-canonical ensemble; for small systems we find significant deviations. We derive the detailed fluctuation theorem and Crooks’ work fluctuation theorem for structure-forming systems. The connection to the theory of particle self-assembly is discussed. We apply the results to several physical systems. We present the phase diagram for patchy particles described by the Kern-Frenkel potential. We show that the Curie-Weiss model with molecule structures exhibits a first-order phase transition.

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Signal Processing on Simplicial Complexes With Vertex Signals

Kahn

Tay

2022

IEEE Access

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In classical graph signal processing (GSP), the underlying topological structures are restricted in terms of dimensionality. A graph or a 1-complex is a combinatorial object that models binary relations, which do not directly capture complex high arity relations. One possible high dimensional generalization of graphs is a simplicial complex. In this paper, we develop a signal processing framework on simplicial complexes with vertex signals, which recovers the traditional GSP theory. We introduce the concept of a generalized Laplacian, which allows us to embed a simplicial complex into a traditional graph and hence perform signal processing similar to traditional GSP. We show that the generalized Laplacian satisfies several desirable properties, same as the graph Laplacian. We propose a method to learn 2-complex structures and demonstrate how to perform signal processing by applying the generalized Laplacian on both synthetic and real data. We observe performance gains in our experiments when 2-complexes are used to model the data, compared to the traditional GSP approach of restricting to 1-complexes.

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Generalized Topologies: Hypergraphs, Chemical Reactions, and Biological Evolution

Cited by 4 publications

References 57 publications

Comparison of modularity-based approaches for nodes clustering in hypergraphs

Comparison of modularity-based approaches for nodes clustering in hypergraphs

Thermodynamics of structure-forming systems

Signal Processing on Simplicial Complexes With Vertex Signals

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