Hypergraph categories

Fong, Brendan; Spivak, David I.

doi:10.1016/j.jpaa.2019.02.014

Cited by 26 publications

(20 citation statements)

References 11 publications

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“…[WP13], which abandons the input/output distinction: our span formalization of machines is already symmetric in that sense. This is also related to machines giving an immediate instance of a hypergraph category, recently shown to be equivalent to algebras for an operad of cospans [FS18]. On the other hand, total machines in our context seem to be strongly associated to the concept of open maps in the theory of bisimulation, see e.g.…”

mentioning

confidence: 65%

Dynamical Systems and Sheaves

Schultz

Spivak

Vasilakopoulou

2019

Appl Categor Struct

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A. A categorical framework for modeling and analyzing systems in a broad sense is proposed. These systems should be thought of as 'machines' with inputs and outputs, carrying some sort of signal that occurs through some notion of time. Special cases include continuous and discrete dynamical systems (e.g. Moore machines). Additionally, morphisms between the different types of systems allow their translation in a common framework. A central goal is to understand the systems that result from arbitrary interconnection of component subsystems, possibly of different types, as well as establish conditions that ensure totality and determinism compositionally. The fundamental categorical tools used here include lax monoidal functors, which provide a language of compositionality, as well as sheaf theory, which flexibly captures the crucial notion of time.

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mentioning

confidence: 65%

Dynamical Systems and Sheaves

Schultz

Spivak

Vasilakopoulou

2019

Appl Categor Struct

Self Cite

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“…For the purposes of this work, gearing our exposition towards hypergraphs rather than bicolored graphs is more natural because our approach is set-theoretic. d Beyond these bijective correspondences, mathematical research on hypergraph categories and their isomorphisms requires careful consideration [13][14][15].…”

Section: Definition 2 the Incidence Matrix S Of A Hypergraphmentioning

confidence: 99%

Hypernetwork science via high-order hypergraph walks

et al. 2020

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We propose high-order hypergraph walks as a framework to generalize graph-based network science techniques to hypergraphs. Edge incidence in hypergraphs is quantitative, yielding hypergraph walks with both length and width. Graph methods which then generalize to hypergraphs include connected component analyses, graph distance-based metrics such as closeness centrality, and motif-based measures such as clustering coefficients. We apply high-order analogs of these methods to real world hypernetworks, and show they reveal nuanced and interpretable structure that cannot be detected by graph-based methods. Lastly, we apply three generative models to the data and find that basic hypergraph properties, such as density and degree distributions, do not necessarily control these new structural measurements. Our work demonstrates how analyses of hypergraph-structured data are richer when utilizing tools tailored to capture hypergraph-native phenomena, and suggests one possible avenue towards that end.

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“…Therefore, Euler ℎ : ⇒ D is a natural transformation. This natural transformation defines a functor between the hypergraph categories defined by and D (see [5]) which in turn induces the transformation Euler ℎ (see [6]).…”

Section: Julia Implementationmentioning

confidence: 99%

“…The algebra Dynam D is induced by the 1-equivalence between hypergraph categories and Cospan(FinSet) algebras defined in [6]. Proof.…”

Section: A Proofsmentioning

confidence: 99%

Operadic Modeling of Dynamical Systems: Mathematics and Computation

Libkind¹,

Baas²,

Patterson³

et al. 2021

Preprint

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Dynamical systems are ubiquitous in science and engineering as models of phenomena that evolve over time. Although complex dynamical systems tend to have important modular structure, conventional modeling approaches suppress this structure. Building on recent work in applied category theory, we show how deterministic dynamical systems, discrete and continuous, can be composed in a hierarchical style. In mathematical terms, we reformulate some existing operads of wiring diagrams and introduce new ones, using the general formalism of -sets (copresheaves). We then establish dynamical systems as algebras of these operads. In a computational vein, we show that Euler's method is functorial for undirected systems, extending a previous result for directed systems. All the ideas in this paper are implemented as practical software using Catlab and the AlgebraicJulia ecosystem, written in the Julia programming language for scientific computing.

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Hypergraph categories

Cited by 26 publications

References 11 publications

Dynamical Systems and Sheaves

Dynamical Systems and Sheaves

Hypernetwork science via high-order hypergraph walks

Operadic Modeling of Dynamical Systems: Mathematics and Computation

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