## Abstract

In the future, analysis of social networks will conceivably move from graphs to hypergraphs. However, theory has not yet caught up with this type of data organizational structure. By introducing and analyzing a general model of preferential attachment hypergraphs, this paper makes a step towards narrowing this gap. We consider a random preferential attachment model H(p,Y) for network evolution that allows arrivals of both nodes and hyperedges of random size. At each time step t, two possible events may occur: (1) [vertex arrival event:] with probability p > 0 a new vertex arrives and a new hyperedge of size Yt, containing the new vertex and Yt − 1 existing vertices, is added to the hypergraph; or (2) [hyperedge arrival event:] with probability 1−p, a new hyperedge of size Yt, containing Yt existing vertices, is added to the hypergraph. In both cases, the involved existing vertices are chosen independently at random according to the preferential attachment rule, i.e., with probability proportional to their degree, where the degree of a vertex is the number of edges containing it. Assuming general restrictions on the distribution of Yt, we prove that the H(p,Y) model generates power law networks, i.e., the expected fraction of nodes with degree k is proportional to k^{−}1−^{Γ}, where Γ = limt→∞ ^{P}_{t}(_{E}^{t i=0 −}_{[Y}^{1}_{t}^{E}_{]−}^{[Y}_{p}^{i})] ∈ (0,∞). This extends the special case of preferential attachment graphs, where Yt = 2 for every t, yielding Γ = 2/(2 − p). Therefore, our results show that the exponent of the degree distribution is sensitive to whether one considers the structure of a social network to be a hypergraph or a graph. We discuss, and provide examples for, the implications of these considerations.

Original language | English |
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Title of host publication | Proceedings of the 2019 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, ASONAM 2019 |

Editors | Francesca Spezzano, Wei Chen, Xiaokui Xiao |

Publisher | Association for Computing Machinery, Inc |

Pages | 398-405 |

Number of pages | 8 |

ISBN (Electronic) | 9781450368681 |

DOIs | |

State | Published - 27 Aug 2019 |

Externally published | Yes |

Event | 11th IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, ASONAM 2019 - Vancouver, Canada Duration: 27 Aug 2019 → 30 Aug 2019 |

### Publication series

Name | Proceedings of the 2019 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, ASONAM 2019 |
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### Conference

Conference | 11th IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, ASONAM 2019 |
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Country/Territory | Canada |

City | Vancouver |

Period | 27/08/19 → 30/08/19 |

### Bibliographical note

Publisher Copyright:© 2019 Association for Computing Machinery.

### Funding

This research was supported in part by the Israel Science Foundation (grant 1549/13). David Peleg was supported in part by I-CORE program of the Israel PBC and ISF (grant 4/11). Part of this work was done while Chen Avin was a long term visitor at ICERM, Brown University. 2014. It may be interesting to study next a setting where p is not constant but rather a sequence, pt, which depends on the time t and tends to 0, i.e., limt→∞pt = 0. We find this to be an exciting new research direction, which can provide explanations to a variety of phenomena that are not captured in the current model. Acknowlegments: This research was supported in part by the Israel Science Foundation (grant 1549/13). David Peleg was supported in part by I-CORE program of the Israel PBC and ISF (grant 4/11). Part of this work was done while Chen Avin was a long term visitor at ICERM, Brown University. 2014.

Funders | Funder number |
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ICERM | |

Israel PBC | |

Brown University | |

Israel Science Foundation | 1549/13, 4/11 |

## Keywords

- Degree distribution
- Preferential attachment
- Random hypergraphs
- Social networks