The authors are hosting an AMS sponsored Mathematics Research Community (MRC) focusing on two themes that have garnered intense attention in network models of com- plex relational data: (1) how to faithfully model multi-way relations in hypergraphs, rather than only pairwise inter- actions in graphs; and (2) challenges posed by modeling networks with extreme sparsity. Here we introduce and explore these two themes and their challenges. We hope to generate interest from researchers in pure and applied mathematics and computer science.