The distribution of the number of node neighbors in random hypergraphs

Hypergraphs, graph generalizations where edges are conglomerates of $r$ nodes called hyperedges of rank $r\geq 2$, are excellent models to study systems with interactions that are beyond the pairwise level. For hypergraphs, the node degree $\ell$ (number of hyperedges connected to a node) and the number of neighbors $k$ of a node differ from each other in contrast to the case of graphs. Here, I calculate the distribution of the number of node neighbors in random hypergraphs in which hyperedges of uniform rank $r$ have a homogeneous probability $p$ to appear. This distribution is equivalent to the degree distribution of ensembles of projected graphs from hypergraph or bipartite network ensembles, where the projection connects any two nodes in the projected graph when they are also connected in the hypergraph or bipartite network. The calculation is non-trivial due to the possibility that neighbor nodes belong simultaneously to multiple hyperedges (node overlaps). From the exact results, the traditional sparse (small $p$) asymptotic approximation to the distribution is rederived and improved; the approximation exhibits Poisson-like behavior accompanied by strong fluctuations modulated by power-law decays in the system size $N$ with decay exponents equal to the minimum number of overlapping nodes possible for a given number of neighbors. It is shown that the dense limit cannot be explained if overlaps are ignored, and the correct asymptotic distribution is provided. The neighbor distribution requires the calculation of a new combinatorial coefficient $Q_{r-1}(k,\ell)$, counting the number of distinct labelled hypergraphs of $k$ nodes, $\ell$ hyperedges of rank $r-1$, and where every node is connected to at least one hyperedge. Some identities of $Q_{r-1}(k,\ell)$ are derived and applied to the verification of normalization and the calculation of moments of the neighbor distribution.