# Hyper-Null Models and Their Applications

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## Abstract

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## 1. Introduction

- We propose a new method to construct null models for hypergraphs and summarize the relationship between our proposed hypergraph models.
- We introduce the concept of entropy to quantify the randomness of networks, retaining different statistical properties, and explore the relationship between randomness and the network structure.
- We utilize topological statistics analysis, network dismantling, and epidemic contagion to showcase the universality of the framework employed in the original network and its hyper-null models.

## 2. Methods

#### 2.1. Hypergraphs

#### 2.2. Hyperedges and Hypertriangles

#### 2.3. Statistics of Hypergraphs

**Hyperdegree**: The hyperdegree refers to the node degree in hypergraphs. It represents the number of hyperedges that a node is located in.

**Hyperedge degree**: The hyperedge degree denotes the number of nodes that are contained in a hyperedge.

**Co-average hyperdegree**: The co-average hyperdegree of node i represents the average hyperdegree of node i’s neighbors.

**Hyperdegree distribution**: The hyperdegree distribution denotes the probability that a randomly chosen node across the entire hypergraph will have a hyperdegree of k.

**Joint hyperdegree distribution**: The joint hyperdegree distribution denotes the distribution of hyperdegrees among nodes within each hyperedge.

**Hypergraph clustering coefficient**: The clustering coefficient of a node represents the ratio of the number of existing hyperedges between its neighbors to the number of all possible hyperedges. The clustering coefficient of the hypergraph is the average value of the clustering coefficients of all nodes in the hypergraph.

**Average neighbor degree**: The average neighbor degree of a node denotes the average of the neighbors’ degree of it.

**Strength**: The strength of node i represents the total number of hyperedges that node i shares with any other node in the hypergraph.

**Assortativity**: Based on the assortativity r proposed by Newman [31], we calculate the assortativity for hypergraphs as

#### 2.4. Matrices of Hypergraphs

#### 2.4.1. Hyperdegree Matrices

#### 2.4.2. Hyperedge degree Matrices

#### 2.4.3. Incidence Matrices

#### 2.4.4. Adjacency Matrices

#### 2.5. Randomness

#### 2.6. Hyper-Null Models Based on Hyperedge Swapping

#### 2.6.1. Hyper-0k Null Model

#### 2.6.2. Hyper-1k Null Model with Hyperdegree Constant

#### 2.6.3. Hyper-1k Null Model with Hyperedge Degree Constant

#### 2.6.4. Hyperdegree–Hyperedge Degree Null Model

#### 2.6.5. Hyper-2k Null Model

#### 2.6.6. Hyper-2.25k Null Model

## 3. Results

#### 3.1. Data Description

**Algebra**: A question–answer network, which is collected from MathOverflow.net, where the nodes denote users, and the users who answered the same question are enclosed in a hyperedge.**Bars-Rev**: A review hypergraph collected from Yelp.com, where a hyperedge consists of the users who reviewed the same bars.**iAF1260b and iJO1366**: The metabolic hypergraph where nodes denote metabolites and the hyperedges represent the reaction that is involved in the same metabolic.

#### 3.2. Statistical Analysis Based on Hyper-Null Models

#### 3.3. Hypergraph Dismantling

#### 3.3.1. Hypergraph Dismantling by Removing Nodes

#### 3.3.2. Hypergraph Dismantling by Removing Hyperedges

#### 3.4. Hypergraph Epidemic Contagion

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Barabási, A.L. Network Science; Cambridge University Press: Cambridge, UK, 2016. [Google Scholar]
- Newman, M. Networks; Oxford University Press: Oxford, UK, 2018. [Google Scholar]
- Majhi, S.; Perc, M.; Ghosh, D. Dynamics on higher-order networks: A review. J. R. Soc. Interface
**2022**, 19, 20220043. [Google Scholar] [CrossRef] [PubMed] - Battiston, F.; Cencetti, G.; Iacopini, I.; Latora, V.; Lucas, M.; Patania, A.; Young, J.G.; Petri, G. Networks beyond pairwise interactions: Structure and dynamics. Phys. Rep.
**2020**, 874, 1–92. [Google Scholar] - Battiston, F.; Amico, E.; Barrat, A.; Bianconi, G.; Ferraz de Arruda, G.; Franceschiello, B.; Iacopini, I.; Kéfi, S.; Latora, V.; Moreno, Y. The physics of higher-order interactions in complex systems. Nat. Phys.
**2021**, 17, 1093–1098. [Google Scholar] [CrossRef] - Zeng, Y.; Huang, Y.; Ren, X.L.; Lü, L. Identifying vital nodes through augmented random walks on higher-order networks. arXiv
**2023**, arXiv:2305.06898. [Google Scholar] - Bianconi, G. Higher-Order Networks; Cambridge University Press: Cambridge, UK, 2021. [Google Scholar]
- Berge, C. Hypergraphs: Combinatorics of Finite Sets; Elsevier: Amsterdam, The Netherlands, 1984; Volume 45. [Google Scholar]
- LaRock, T.; Lambiotte, R. Encapsulation Structure and Dynamics in Hypergraphs. arXiv
**2023**, arXiv:2307.04613. [Google Scholar] - Feng, Y.; You, H.; Zhang, Z.; Ji, R.; Gao, Y. Hypergraph neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, Hilton, HI, USA, 27 January–1 February 2019; Volume 33, pp. 3558–3565. [Google Scholar]
- Liao, X.; Xu, Y.; Ling, H. Hypergraph neural networks for hypergraph matching. In Proceedings of the IEEE/CVF International Conference on Computer Vision, Montreal, BC, Canada, 11–17 October 2021; pp. 1266–1275. [Google Scholar]
- Milo, R.; Shen-Orr, S.; Itzkovitz, S.; Kashtan, N.; Chklovskii, D.; Alon, U. Network motifs: Simple building blocks of complex networks. Science
**2002**, 298, 824–827. [Google Scholar] [CrossRef] - Milo, R.; Itzkovitz, S.; Kashtan, N.; Levitt, R.; Shen-Orr, S.; Ayzenshtat, I.; Sheffer, M.; Alon, U. Superfamilies of evolved and designed networks. Science
**2004**, 303, 1538–1542. [Google Scholar] [CrossRef] [PubMed] - Liu, B.; Xu, S.; Li, T.; Xiao, J.; Xu, X.K. Quantifying the effects of topology and weight for link prediction in weighted complex networks. Entropy
**2018**, 20, 363. [Google Scholar] [CrossRef] - Orsini, C.; Dankulov, M.M.; Colomer-de Simón, P.; Jamakovic, A.; Mahadevan, P.; Vahdat, A.; Bassler, K.E.; Toroczkai, Z.; Boguná, M.; Caldarelli, G.; et al. Quantifying randomness in real networks. Nat. Commun.
**2015**, 6, 8627. [Google Scholar] [CrossRef] - Gjoka, M.; Kurant, M.; Markopoulou, A. 2.5K-graphs: From sampling to generation. In Proceedings of the 2013 IEEE INFOCOM, Turin, Italy, 14–19 April 2013; pp. 1968–1976. [Google Scholar]
- Mahadevan, P.; Hubble, C.; Krioukov, D.; Huffaker, B.; Vahdat, A. Orbis: Rescaling degree correlations to generate annotated internet topologies. In Proceedings of the 2007 Conference on Applications, Technologies, Architectures, and Protocols for Computer Communications, Kyoto, Japan, 27–31 August 2007; pp. 325–336. [Google Scholar]
- Zeng, Y.; Huang, Y.; Wu, Q.; Lü, L. Influential Simplices Mining via Simplicial Convolutional Network. arXiv
**2023**, arXiv:2307.05841. [Google Scholar] - Mahadevan, P.; Krioukov, D.; Fall, K.; Vahdat, A. Systematic topology analysis and generation using degree correlations. ACM Sigcomm Comput. Commun. Rev.
**2006**, 36, 135–146. [Google Scholar] [CrossRef] - Klimm, F.; Deane, C.M.; Reinert, G. Hypergraphs for predicting essential genes using multiprotein complex data. J. Complex Netw.
**2021**, 9, cnaa028. [Google Scholar] [CrossRef] - Chodrow, P.S. Configuration models of random hypergraphs. J. Complex Netw.
**2020**, 8, cnaa018. [Google Scholar] [CrossRef] - Miyashita, R.; Nakajima, K.; Fukuda, M.; Shudo, K. Randomizing Hypergraphs Preserving Two-mode Clustering Coefficient. In Proceedings of the 2023 IEEE International Conference on Big Data and Smart Computing (BigComp), Jeju, Republic of Korea, 13–16 February 2023; pp. 316–317. [Google Scholar]
- Nakajima, K.; Shudo, K.; Masuda, N. Randomizing hypergraphs preserving degree correlation and local clustering. IEEE Trans. Netw. Sci. Eng.
**2021**, 9, 1139–1153. [Google Scholar] [CrossRef] - Omar, Y.M.; Plapper, P. A survey of information entropy metrics for complex networks. Entropy
**2020**, 22, 1417. [Google Scholar] [CrossRef] - Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Fortunato, S.; Barthelemy, M. Resolution limit in community detection. Proc. Natl. Acad. Sci. USA
**2007**, 104, 36–41. [Google Scholar] [CrossRef] [PubMed] - Newman, M.E.; Girvan, M. Finding and evaluating community structure in networks. Phys. Rev. E
**2004**, 69, 026113. [Google Scholar] [CrossRef] [PubMed] - Hu, D.; Li, X.L.; Liu, X.G.; Zhang, S.G. Extremality of graph entropy based on degrees of uniform hypergraphs with few edges. Acta Math. Sin. Engl. Ser.
**2019**, 35, 1238–1250. [Google Scholar] [CrossRef] - Berge, C. Graphs and Hypergraphs; North-Holland Publishing Company: Amsterdam, The Netherlands, 1973. [Google Scholar]
- Estrada, E.; Rodríguez-Velázquez, J.A. Subgraph centrality and clustering in complex hyper-networks. Phys. A Stat. Mech. Its Appl.
**2006**, 364, 581–594. [Google Scholar] [CrossRef] - Newman, M.E. Assortative Mixing in Networks. Phys. Rev. Lett.
**2002**, 89, 208701. [Google Scholar] [CrossRef] [PubMed] - Amburg, I.; Veldt, N.; Benson, A. Clustering in graphs and hypergraphs with categorical edge labels. In Proceedings of the Web Conference 2020, Taipei, Taiwan, 20–24 April 2020; pp. 706–717. [Google Scholar]
- King, Z.A.; Lu, J.; Dräger, A.; Miller, P.; Federowicz, S.; Lerman, J.A.; Ebrahim, A.; Palsson, B.O.; Lewis, N.E. BiGG Models: A platform for integrating, standardizing and sharing genome-scale models. Nucleic Acids Res.
**2016**, 44, D515–D522. [Google Scholar] [CrossRef] [PubMed] - Peng, H.; Qian, C.; Zhao, D.; Zhong, M.; Ling, X.; Wang, W. Disintegrate hypergraph networks by attacking hyperedge. J. King Saud-Univ.-Comput. Inf. Sci.
**2022**, 34, 4679–4685. [Google Scholar] [CrossRef] - Bodó, Á.; Katona, G.Y.; Simon, P.L. SIS epidemic propagation on hypergraphs. Bull. Math. Biol.
**2016**, 78, 713–735. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**A hypergraph and its hyper-null models. (

**a**,

**d**) denote the original hypergraph and its incidence matrix, (

**b**,

**e**) represent the H0k null model and its incidence matrix, (

**c**,

**f**) denote the H1k-HD null model and its incidence matrix, (

**g**,

**j**) denote the H1k-HED null model and its incidence matrix, (

**h**,

**k**) denote the HD-HED null model and its incidence matrix, (

**i**,

**l**) denote the H2k null model and its incidence matrix. The green columns represent the randomly chosen hyperedges, while the red boxes denote the nodes that undergo a single swap during the transformation process.

**Figure 2.**The relationship between hyper-null models. From the inside to the outside are the null models of different orders. $HDk=\mathcal{H}$ denotes the original network.

**Figure 3.**The trend of randomness on four datasets. (

**a**–

**d**) demonstrate how the randomness of different null models changes as the times of swapped hyperedges increase in datasets Algebra, Bar-Rev, iAF1260b and iJO13660. The x-axis denotes the times of hyperedge swapping, while the y-axis represents the randomness (degree distribution entropy) of the network. The subfigures in the top right of a, b, c, d are the enlarged drawing of HD-HED, H2k and H2.25k. The red dotted line denotes actual swapped times. Here, we choose 10 times the number of hyperedges.

**Figure 4.**Statistical indices of network iAF1260b and its hull models. The x-axis in four rows from top to bottom represents the distribution of hyperdegree (HD), hyperedge degree (HED), clustering coefficient (CC) and co-average hyperdegree (CHD). The y-axis of each subgraph represents the proportion p of all nodes that are under the current value.

**Figure 5.**Dismantling networks by removing nodes (

**a**–

**d**) and hyperedges (

**e**–

**h**). The x-axis denotes the number of removed nodes (

**a**–

**d**) or hyperedges (

**e**–

**h**). N denotes the number of removed nodes while ${N}_{H}$ represents the number of removed hyperedges. The y-axis represents the GCC size of the network. The small panels inside (

**a**–

**h**) denote the GCC-AUC of each null model in four datasets. The GCC-AUC of each null model is shown at the right end of the column.

**Figure 6.**SIR epidemic contagion across four datasets with different infection rates. The first (

**a**–

**d**), second (

**e**–

**h**), and third (

**i**–

**l**) rows denote the epidemic results for 1% initial infected nodes selected randomly, based on degree, and using hyperdegree, respectively. The x-axis represents the different infection rates $\beta $ and the y-axis denotes the recovery number ratio at the steady state.

Networks | n | e | $\u2329\mathit{k}\u232a$ | $\u2329{\mathit{k}}_{\mathit{H}}\u232a$ | C | Node | Hyperedge |
---|---|---|---|---|---|---|---|

Algebra | 423 | 1268 | 78.90 | 19.53 | 0.79 | User | Question |

Bars-Rev | 1234 | 1194 | 174.30 | 9.62 | 0.58 | User | Bar review |

iAF1260b | 1668 | 2351 | 13.26 | 5.46 | 0.55 | Metabolite | Metabolic interaction |

iJO1366 | 1805 | 2546 | 16.92 | 5.55 | 0.58 | Metabolite | Metabolic interaction |

**Table 2.**The assortativity, clustering coefficient, average neighbor degree and their $\mu $ of each network.

Statistics | Network | Original | H0k | H1k-HED | H1k-HD | HD-HED | H2k | H2.5k |
---|---|---|---|---|---|---|---|---|

Assortativity ($\mu $) | Algebra | −0.10 | −0.02 (0.20) | −0.03 (0.30) | −0.05 (0.50) | −0.08 (0.80) | −0.09(0.90) | −0.09(0.90) |

Bars-Rev | 0.30 | −0.002 (−0.01) | 0.08 (0.27) | 0.26 (0.87) | 0.27 (0.90) | 0.29 (0.97) | 0.29 (0.97) | |

iAF1260b | −0.30 | −0.03 (0.10) | −0.16 (0.53) | −0.22 (0.73) | −0.26 (0.87) | −0.28 (0.93) | −0.28 (0.93) | |

iJO1366 | −0.29 | −0.03 (0.10) | −0.14 (0.48) | −0.23 (0.79) | −0.25 (0.86) | −0.27 (0.93) | −0.28 (0.97) | |

Clustering coefficient ($\mu $) | Algebra | 0.80 | 0.66 (0.83) | 0.75 (0.93) | 0.76 (0.95) | 0.79 (0.99) | 0.80 (1.00) | 0.80 (1.00) |

Bars-Rev | 0.58 | 0.28 (0.48) | 0.44 (0.76) | 0.49 (0.84) | 0.57 (0.98) | 0.58 (1.00) | 0.58 (1.00) | |

iAF1260b | 0.55 | 0.46 (0.84) | 0.54 (0.98) | 0.55 (1.00) | 0.55 (1.00) | 0.55 (1.00) | 0.55 (1.00) | |

iJO1366 | 0.58 | 0.45 (0.77) | 0.55 (0.95) | 0.57 (0.98) | 0.57 (0.98) | 0.57 (0.98) | 0.58 (0.98) | |

Average Neighbor Degree ($\mu $) | Algebra | 44.35 | 22.88 (0.52) | 38.15 (0.86) | 42.12 (0.95) | 44.13 (0.99) | 44.25 (0.99) | 44.35 (1.00) |

Bars-Rev | 13.28 | 8.94 (0.67) | 12.32 (0.93) | 11.95 (0.90) | 12.29 (0.93) | 13.28 (1.00) | 13.28 (1.00) | |

iAF1260b | 141.425 | 25.66 (0.18) | 37.21 (0.26) | 135.27 (0.96) | 139.49 (0.99) | 141.395 (1.00) | 141.425 (1.00) | |

iJO1366 | 159.46 | 30.85 (0.19) | 150.43 (0.94) | 42.79 (0.27) | 157.45 (0.99) | 157.79 (0.99) | 159.46 (1.00) |

Origin | H0k | H1k-HED | H1k-HD | HD-HED | H2K | H2.25k | |
---|---|---|---|---|---|---|---|

Algebra | 0.15 | 0.81 | 0.42 | 0.45 | 0.41 | 0.36 | 0.39 |

Bars-Rev | 0.43 | 0.83 | 0.55 | 0.54 | 0.49 | 0.49 | 0.51 |

iAF1260b | 0.04 | 0.20 | 0.04 | 0.04 | 0.05 | 0.04 | 0.05 |

iJO1366 | 0.29 | 0.65 | 0.31 | 0.28 | 0.27 | 0.29 | 0.30 |

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**MDPI and ACS Style**

Zeng, Y.; Liu, B.; Zhou, F.; Lü, L.
Hyper-Null Models and Their Applications. *Entropy* **2023**, *25*, 1390.
https://doi.org/10.3390/e25101390

**AMA Style**

Zeng Y, Liu B, Zhou F, Lü L.
Hyper-Null Models and Their Applications. *Entropy*. 2023; 25(10):1390.
https://doi.org/10.3390/e25101390

**Chicago/Turabian Style**

Zeng, Yujie, Bo Liu, Fang Zhou, and Linyuan Lü.
2023. "Hyper-Null Models and Their Applications" *Entropy* 25, no. 10: 1390.
https://doi.org/10.3390/e25101390