Counting simplicial pairs in hypergraphs
<|reference_start|>Counting simplicial pairs in hypergraphs: We present two ways to measure the simplicial nature of a hypergraph: the simplicial ratio and the simplicial matrix. We show that the simplicial ratio captures the frequency, as well as the rarity, of simplicial interactions in a hypergraph while the simplicial matrix provides more fine-grained details. We then compute the simplicial ratio, as well as the simplicial matrix, for 10 real-world hypergraphs and, from the data collected, hypothesize that simplicial interactions are more and more deliberate as edge size increases. We then present a new Chung-Lu model that includes a parameter controlling (in expectation) the frequency of simplicial interactions. We use this new model, as well as the real-world hypergraphs, to show that multiple stochastic processes exhibit different behaviour when performed on simplicial hypergraphs vs. non-simplicial hypergraphs.<|reference_end|>
@article{barrett2024counting,
    title={Counting simplicial pairs in hypergraphs},
    author={Jordan Barrett, Pawe{l} Pra{l}at, Aaron Smith and Franc{c}ois
  Th'eberge},
    journal={arXiv preprint arXiv:2408.11806},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.11806},
    primaryClass={cs.SI cs.DM math.CO}
}