## Abstract

A family of models of growing hypergraphs with preferential rules of new linking is introduced and studied. The model hypergraphs evolve via the hyperedge-based growth as well as the node-based one, thus generalizing the preferential attachment models of scale-free networks. We obtain the degree distribution and hyperedge size distribution for various combinations of node- and hyperedge-based growth modes. We find that the introduction of hyperedge-based growth can give rise to power law degree distribution \(P(k)\sim k^{-\gamma }\) even without node-wise preferential attachments. The hyperedge size distribution *P*(*s*) can take diverse functional forms, ranging from exponential to power law to a nonstationary one, depending on the specific hyperedge-based growth rule. Numerical simulations support the mean-field theoretical analytical predictions.

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## Acknowledgements

We thank L. Hébert-Dufresne for insightful discussion. This work was supported in part by the National Research Foundation of Korea (NRF) grants funded by the Korea government (MSIT) (No. NRF-2020R1A2C2003669). D.R. is grateful for financial support from Hyundai Motor Chung Mong-Koo Foundation.

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## Appendix A: The master equation approach

### Appendix A: The master equation approach

The master equation analysis due to [15] has been utilized to obtain the exact solutions for power law distribution and the exponent. In this appendix, we briefly outline how the method is applied in our models.

Suppose at each time step a new node is introduced and makes *m* directed connections by preferential attachments and \(n_r\) directed connections by random attachments. After *t* steps, the system consists of *t* nodes and \((m+n_r)t\) directed connections. In the preferential attachment, a node *i* gains a new in-degree proportional to its *attractiveness*, \(A_i = d_i+A\) where \(d_i\) is the in-degree of node *i* and *A* is the *initial attractiveness*, a constant independent of *i*.

One can write the master equation for the in-degree distribution *P*(*d*, *t*) for this process, following [15], as

The left-hand side is the change in the number of nodes with in-degree *d* at time *t*. The first two terms on the right-hand side are contributions of random attachment, the next two terms are of preferential attachment and the last is a birth of a node. Using the Z-transform, \(\Phi (z) = \sum _{d=0}^{\infty } P(d) z^d\), and the expansion of hypergeometric function [15], the stationary solution for the distribution is obtained as

where \(a=(A+n_r)/m\) and \(\Gamma [\cdot ]\) is the gamma function. The asymptotic behavior of *P*(*d*) is of a power law form, \(P(d)\sim d^{-\gamma }\), with the power law exponent given by

Different model variations are characterized by the parameters *m*, \(n_r\), and *A*. First, throughout our models, \(A=2\); note that the node degree *k* relates to in-degree *d* as \(k=d+A=d+2\). Now for *M*-models, we have both for \(M_{PR}\)- and \(M_{PP}\)-models, \(m=1\) and \(n_r=0\), leading to Eq. (5); For \(C_{RR}\)-model, \(m=\langle s\rangle =3\) and \(n_r=1\), leading to Eq. (11); For \(C_{PR}\)-model, \(m=\langle s\rangle +1=4\) and \(n_r=0\), leading to Eq. (14); For \(C_{RP}\)-model, \(m=\langle s^2\rangle /\langle s\rangle\) diverges and \(n_r=1\). Therefore \(a\rightarrow 0\), leading to Eq. (21); For \(C_{PP}\)-model, \(m=\langle s^2\rangle /\langle s\rangle +1\) and \(n_r=0\), leading to Eq. (22). The analysis for the hyperedge size distribution *P*(*s*) can be applied in a similar way.

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Roh, D., Goh, K.I. Growing hypergraphs with preferential linking.
*J. Korean Phys. Soc.* **83**, 713–722 (2023). https://doi.org/10.1007/s40042-023-00909-4

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DOI: https://doi.org/10.1007/s40042-023-00909-4