Model-based clustering for random hypergraphs

A probabilistic model for random hypergraphs is introduced to represent unary, binary and higher order interactions among objects in real-world problems. This model is an extension of the latent class analysis model that introduces two clustering structures for hyperedges and captures variation in the size of hyperedges. An expectation maximization algorithm with minorization maximization steps is developed to perform parameter estimation. Model selection using Bayesian Information Criterion is proposed. The model is applied to simulated data and two real-world data sets where interesting results are obtained.

Panchromatic colorings of random hypergraphs

We study the threshold probability for the existence of a panchromatic coloring with r colors for a random k-homogeneous hypergraph in the binomial model H(n, k, p), that is, a coloring such that each edge of the hypergraph contains the vertices of all r colors. It is shown that this threshold probability for fixed r < k and increasing n corresponds to the sparse case, i. e. the case p = c n / ( n k ) $p = cn/{n \choose k}$ for positive fixed c. Estimates of the form c 1(r, k) < c < c 2(r, k) for the parameter c are found such that the difference c 2(r, k) − c 1(r, k) converges exponentially fast to zero if r is fixed and k tends to infinity.