Relationships between node degrees and hyperedge sizes in empirical hypergraphs

We investigate networks represented as hypergraphs and propose a a novel measure that captures the relationship between their node degrees and hyperedge sizes. We test the presence of such an association in 36 empirical hypergraphs from diverse domains, with a focus on social networks. Using nested model comparisons, we classify each such relationship as linear, monotonic, non-monotonic, or absent. Results reveal that true absence of this relationship is rare, while nearly half exhibit non-monotonic patterns. We evaluate three correlation measures of this association and find that Pearson correlation best aligns with relationship direction. We also consider three ways to capture this relationship (called: bipartite, node-centric or edge-centric) and show that the bipartite one yields most consistent results. We discuss the implications of existence of relationship between node degrees and hyperedge sizes for dynamic processes on social systems.