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Growth patterns and models of real-world hypergraphs

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Abstract

What kind of macroscopic structural and dynamical patterns can we observe in real-world hypergraphs? What can be underlying local dynamics on individuals, which ultimately lead to the observed patterns, beyond apparently random evolution? Graphs, which provide effective ways to represent pairwise interactions among entities, fail to represent group interactions (e.g., collaborations of three or more researchers, etc.). Regarded as a generalization of graphs, hypergraphs allowing for various sizes of edges prove fruitful in addressing this limitation. However, the increased complexity makes it challenging to understand hypergraphs as thoroughly as graphs. In this work, we closely examine seven structural and dynamical properties of real hypergraphs from six domains. To this end, we define new measures, extend notions of common graph properties to hypergraphs, and assess the significance of observed patterns by comparison with a null model and statistical tests. We also propose HyperFF, a stochastic model for generating realistic hypergraphs. Its merits are threefold: (a) Realistic: it successfully reproduces all seven patterns, in addition to five patterns established in previous studies, (b) Self-contained: unlike previously proposed models, it does not rely on oracles (i.e., unexplainable external information) at all, and it is parameterized by just two scalars, and (c) Emergent: it relies on simple and interpretable mechanisms on individual entities, which do not trivially enforce but surprisingly lead to macroscopic properties. While HyperFF is mathematically intractable, we provide theoretical justifications and mathematical analysis based on its simplified version.

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Notes

  1. This work is an extended version of [14], which was presented at the 20th IEEE International Conference on Data Mining. In this extended version, we introduce a simple version of HyperFF, and based on it, we theoretically justify densification and heavy-tailed degree distributions. For additional experiments, we test the stability of HyperFF (Fig. 9 in Sect. 6.2), examine the overlapping patterns of hyperedges (Fig. 11 in Sect. 6.2), and the occurrences of 26 hypergraph motifs in hypergraphs generated by HyperFF (Figs. 12 and 13 in Sect. 6.2). Additionally, we analyze the effect of the parameters p and q on the structures and dynamics of what HyperFF generates (Figs. 14 and 15 in Sect. 6.3). Lastly, we perform an ablation study, where we take six variants of HyperFF into consideration (Figs. 16 and 17 in Sect. 6.4).

  2. Linear interpolation is used to find such d.

  3. Note that, compared to that in static CGAH the tree distance between leaf nodes has doubled in dynamic CGAH. Thus, we divide the exponent by 2 when computing the probability.

  4. Formally, the \( n \)-level decomposed graph of a hypergraph \(G=(V, E)\) is defined as \(G_{(n)}=(V_{(n)}, E_{(n)})\) where

    $$\begin{aligned}&V_{(n)} := \{v_{(n)} \in 2^{V} : |v_{(n)}| = n \text { and } \exists e\in E \text { s.t. } v_{(n)} \subseteq e \}, \\&{E_{(n)} := \left\{ \{u_{(n)},v_{(n)}\} \in \left( {\begin{array}{c}V_{(n)}\\ 2\end{array}}\right) : \exists e\in E \text { s.t. } u_{(n)} \cup v_{(n)} \subseteq e \right\} .} \end{aligned}$$

  5. The egonet of a node v is defined as \(\{e \in E \mid v \in e\}\), the set of hyperedges containing v.

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Acknowledgements

This work was supported by Young Scientist Fellowship funded by the Institute for Basic Science (IBS-R029-Y2), National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2020R1C1C1008296), and Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No. 2019-0-00075, Artificial Intelligence Graduate School Program (KAIST)).

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  1. Jihoon Ko and Yunbum Kook have contributed equally to this study.

Authors and Affiliations

  1. Kim Jaechul Graduate School of AI, KAIST, Seoul, South Korea

    Jihoon Ko

  2. School of Computer Science, Georgia Institute of Technology, Atlanta, GA, USA

    Yunbum Kook

  3. Kim Jaechul Graduate School of AI and School of Electrical Engineering, KAIST, Seoul, South Korea

    Kijung Shin

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  1. Jihoon Ko
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Ko, J., Kook, Y. & Shin, K. Growth patterns and models of real-world hypergraphs. Knowl Inf Syst 64, 2883–2920 (2022). https://doi.org/10.1007/s10115-022-01739-9

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