Partite, 3-uniform hypergraphs are 3-uniform hypergraphs in which each hyperedge contains exactly one point from each of the 3 disjoint vertex classes. We consider the degree sequence problem of partite, 3-uniform hypergraphs, that is, to decide if such a hypergraph with prescribed degree sequences exists. We prove that this decision problem is NP-complete in general, and give a polynomial running time algorithm for third almost-regular degree sequences, that is, when each degree in one of the vertex classes is k or k − 1 for some fixed k, and there is no restriction for the other two vertex classes. We also consider the sampling problem, that is, to uniformly sample partite, 3-uniform hypergraphs with prescribed degree sequences. We propose a Parallel Tempering method, where the hypothetical energy of the hypergraphs measures the deviation from the prescribed degree sequence. The method has been implemented and tested on synthetic and real data. It can also be applied for χ² testing of contingency tables. We have shown that this hypergraph-based χ² test is more sensitive than the standard χ² test. The extra sensitivity is especially advantageous on small data sets, where the proposed Parallel Tempering method shows promising performance.

In this paper, we present algorithms that learn and update temporal node embeddings on the fly for tracking and measuring node similarity over time in graph streams. Recently, several representation learning methods have been proposed that are capable of embedding nodes in a vector space in a way that captures the network structure. Most of the known techniques extract embeddings from static graph snapshots. By contrast, modeling the dynamics of the nodes in temporal networks requires evolving node representations. In order to update node representations that reflect the temporal changes in the local graph structure, we rely on ideas for data stream algorithms. For example, we assess neighborhood overlap by a MinHash fingerprint-based algorithm. To evaluate our methods, in addition to the standard link prediction task, we provide dynamic ground truth data for the quantitative evaluation of similarity search by using online updated node embeddings. In our experiments, we constructed tennis tournament Twitter mention graphs as edge streams and compiled dynamic ground truth by using tournament schedule as external source. Our new algorithms outperformed snapshot-based batch methods for both link prediction and similarity search.