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].

Implicit feedback policies for COVID-19: why "zero-COVID" policies remain elusive.

Successful epidemic modeling requires understanding the implicit feedback control strategies used by populations to modulate the spread of contagion. While such strategies can be replicated with intricate modeling assumptions, here we propose a simple model where infection dynamics are described by a three parameter feedback policy. Rather than model individuals as directly controlling the contact rate which governs the spread of disease, we model them as controlling the contact rate's derivative, resulting in a dynamic rather than kinematic model. The feedback policy used by populations across the United States which best fits observations is proportional-derivative control, where learned parameters strongly correlate with observed interventions (e.g., vaccination rates and mobility restrictions). However, this results in a non-zero "steady-state" of case counts, implying current mitigation strategies cannot eradicate COVID-19. Hence, we suggest making implicit policies a function of cumulative cases, resulting in proportional-integral-derivative control with higher potential to eliminate COVID-19.

Simplicial closure and higher-order link prediction.

Networks provide a powerful formalism for modeling complex systems by using a model of pairwise interactions. But much of the structure within these systems involves interactions that take place among more than two nodes at once-for example, communication within a group rather than person to person, collaboration among a team rather than a pair of coauthors, or biological interaction between a set of molecules rather than just two. Such higher-order interactions are ubiquitous, but their empirical study has received limited attention, and little is known about possible organizational principles of such structures. Here we study the temporal evolution of 19 datasets with explicit accounting for higher-order interactions. We show that there is a rich variety of structure in our datasets but datasets from the same system types have consistent patterns of higher-order structure. Furthermore, we find that tie strength and edge density are competing positive indicators of higher-order organization, and these trends are consistent across interactions involving differing numbers of nodes. To systematically further the study of theories for such higher-order structures, we propose higher-order link prediction as a benchmark problem to assess models and algorithms that predict higher-order structure. We find a fundamental difference from traditional pairwise link prediction, with a greater role for local rather than long-range information in predicting the appearance of new interactions.