The iterated local transitivity model for hypergraphs
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Complex networks are pervasive in the real world, capturing dyadic interactions between pairs of vertices, and a large corpus has emerged on their mining and modeling. However, many phenomena are comprised of polyadic interactions between more than two vertices. Such complex hypergraphs range from emails among groups of individuals, scholarly collaboration, or joint interactions of proteins in living cells. A key generative principle within social and other complex networks is transitivity, where friends of friends are more likely friends. The previously proposed Iterated Local Transitivity (ILT) model incorporated transitivity as an evolutionary mechanism. The ILT model provably satisfies many observed properties of social networks, such as densification, low average distances, and high clustering coefficients. We propose a new, generative model for complex hypergraphs based on transitivity, called the Iterated Local Transitivity Hypergraph (or ILTH) model. In ILTH, we iteratively apply the principle of transitivity to form new hypergraphs. The resulting model generates hypergraphs simulating properties observed in real-world complex hypergraphs, such as densification and low average distances. We consider properties unique to hypergraphs not captured by their 2-section. We show that certain motifs, which are specified subhypergraphs of small order, have faster growth rates in ILTH hypergraphs than in random hypergraphs with the same order and expected average degree. We show that the graphs admitting a homomorphism into the 2-section of the initial hypergraph appear as induced subgraphs in the 2-section of ILTH hypergraphs. We consider new and existing hypergraph clustering coefficients, and show that these coefficients have larger values in ILTH hypergraphs than in comparable random hypergraphs.
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