Higher-order homophily on simplicial complexes

Higher-order network models are becoming increasingly relevant for their ability to explicitly capture interactions between three or more entities in a complex system at once. In this paper, we study homophily, the tendency for alike individuals to form connections, as it pertains to higher-order interactions. We find that straightforward extensions of classical homophily measures to interactions of size 3 and larger are often inflated by homophily present in pairwise interactions. This inflation can even hide the presence of anti-homophily in higher-order interactions. Hence, we develop a structural measure of homophily, simplicial homophily, which decouples homophily in pairwise interactions from that of higher-order interactions. The definition applies when the network can be modeled as a simplicial complex, a mathematical abstraction which makes a closure assumption that for any higher-order relationship in the network, all corresponding subsets of that relationship occur in the data. Whereas previous work has used this closure assumption to develop a rich theory in algebraic topology, here we use the assumption to make empirical comparisons between interactions of different sizes. The simplicial homophily measure is validated theoretically using an extension of a stochastic block model for simplicial complexes and empirically in large-scale experiments across 16 datasets. We further find that simplicial homophily can be used to identify when node features are valuable for higher-order link prediction. Ultimately, this highlights a subtlety in studying node features in higher-order networks, as measures defined on groups of size k can inherit features described by interactions of size [Formula: see text].