Hypernetwork Science: From Multidimensional Networks to Computational Topology

As data structures and mathematical objects used for complex systems modeling, hypergraphs sit nicely poised between on the one hand the world of network models, and on the other that of higher-order mathematical abstractions from algebra, lattice theory, and topology. They are able to represent complex systems interactions more faithfully than graphs and networks, while also being some of the simplest classes of systems representing topological structures as collections of multidimensional objects connected in a particular pattern. In this paper we discuss the role of (undirected) hypergraphs in the science of complex networks, and provide a mathematical overview of the core concepts needed for hypernetwork modeling, including duality and the relationship to bicolored graphs, quantitative adjacency and incidence, the nature of walks in hypergraphs, and available topological relationships and properties. We close with a brief discussion of two example applications: biomedical databases for disease analysis, and domain-name system (DNS) analysis of cyber data.

[1]  Herbert Edelsbrunner,et al.  Computational Topology - an Introduction , 2009 .

[2]  Daniel B. Larremore,et al.  Efficiently inferring community structure in bipartite networks , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Alice Patania,et al.  The shape of collaborations , 2017, EPJ Data Science.

[4]  M. Barber.  Modularity and community detection in bipartite networks. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Guillermo Restrepo,et al.  Formal structure of periodic system of elements , 2019, Proceedings of the Royal Society A.

[6]  Garry Robins,et al.  Small Worlds Among Interlocking Directors: Network Structure and Distance in Bipartite Graphs , 2004, Comput. Math. Organ. Theory.

[7]  Martin Schmidt,et al.  Functorial Approach to Graph and Hypergraph Theory , 2019 .

[8]  Rob H. Bisseling,et al.  Parallel hypergraph partitioning for scientific computing , 2006, Proceedings 20th IEEE International Parallel & Distributed Processing Symposium.

[9]  F. Chung.  The Laplacian of a Hypergraph. , 1992 .

[10]  Giorgio Gallo,et al.  Directed Hypergraphs and Applications , 1993, Discret. Appl. Math..

[11]  Dustin Arendt,et al.  Hypergraph Analytics of Domain Name System Relationships , 2020, WAW.

[12]  Vipin,et al.  Multilevel kway Hypergraph Partitioning * , 1999 .

[13]  Mark Minas.  Hypergraphs as a Uniform Diagram Representation Model , 1998, TAGT.

[14]  Cliff Joslyn,et al.  Hypernetwork science via high-order hypergraph walks , 2019, EPJ Data Science.

[15]  Steffen Klamt,et al.  Hypergraphs and Cellular Networks , 2009, PLoS Comput. Biol..

[16]  David I. Spivak,et al.  Hypergraph Categories , 2018, Journal of Pure and Applied Algebra.

[17]  Matthieu Latapy,et al.  Basic notions for the analysis of large two-mode networks , 2008, Soc. Networks.

[18]  Benny Sudakov,et al.  Approximate coloring of uniform hypergraphs , 1998, J. Algorithms.

[19]  J. A. Rodríguez-Velázquez,et al.  Complex Networks as Hypergraphs , 2005, physics/0505137.

[20]  A. Bonato,et al.  Graphs and Hypergraphs , 2021, Clustering.

[21]  D. A. Waller,et al.  A category-theoretical approach to hypergraphs , 1980 .

[22]  Steve Kirkland,et al.  Two-mode networks exhibiting data loss , 2018, J. Complex Networks.

[23]  Jeffrey Johnson,et al.  Hypernetworks in the Science of Complex Systems , 2013, Series on Complexity Science.

[24]  Xavier Pérez-Giménez,et al.  Subhypergraphs in non-uniform random hypergraphs , 2017, Internet Math..

[25]  Vojtech Rödl,et al.  Regularity Lemma for k‐uniform hypergraphs , 2004, Random Struct. Algorithms.

[26]  Daniele Frigioni,et al.  Directed Hypergraphs: Problems, Algorithmic Results, and a Novel Decremental Approach , 2001, ICTCS.

[27]  Noga Alon,et al.  Transversal numbers of uniform hypergraphs , 1990, Graphs Comb..

[28]  Vito Latora,et al.  Simplicial models of social contagion , 2018, Nature Communications.

[29]  Joshua N. Cooper,et al.  Spectra of Uniform Hypergraphs , 2011, 1106.4856.

[30]  Linyuan Lu,et al.  On a hypergraph probabilistic graphical model , 2018, Annals of Mathematics and Artificial Intelligence.

[31]  Vipin Kumar,et al.  Multilevel k-way Hypergraph Partitioning , 2000, VLSI Design.

[32]  J. A. Rodríguez-Velázquez,et al.  Subgraph centrality and clustering in complex hyper-networks , 2006 .