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].

Diameter Dependence of Water Filling in Lithographically Segmented Isolated Carbon Nanotubes.

Although the structure and properties of water under conditions of extreme confinement are fundamentally important for a variety of applications, they remain poorly understood, especially for dimensions less than 2 nm. This problem is confounded by the difficulty in controlling surface roughness and dimensionality in fabricated nanochannels, contributing to a dearth of experimental platforms capable of carrying out the necessary precision measurements. In this work, we utilize an experimental platform based on the interior of lithographically segmented, isolated single-walled carbon nanotubes to study water under extreme nanoscale confinement. This platform generates multiple copies of nanotubes with identical chirality, of diameters from 0.8 to 2.5 nm and lengths spanning 6 to 160 µm, that can be studied individually in real time before and after opening, exposure to water, and subsequent water filling. We demonstrate that, under controlled conditions, the diameter-dependent blue shift of the Raman radial breathing mode (RBM) between 1 and 8 cm-1 measures an increase in the interior mechanical modulus associated with liquid water filling, with no response from exterior water exposure. The observed RBM shift with filling demonstrates a non-monotonic trend with diameter, supporting the assignment of a minimum of 1.81 ± 0.09 cm-1 at 0.93 ± 0.08 nm with a nearly linear increase at larger diameters. We find that a simple hard-sphere model of water in the confined nanotube interior describes key features of the diameter-dependent modulus change of the carbon nanotube and supports previous observations in the literature. Longer segments of 160 µm show partial filling from their ends, consistent with pore clogging. These devices provide an opportunity to study fluid behavior under extreme confinement with high precision and repeatability.