Identifying the Latent Space Geometry of Network Models through Analysis of Curvature

Shane Lubold, Arun Chandrasekhar and Tyler H. McCormick

No 28273, NBER Working Papers from National Bureau of Economic Research, Inc

Abstract: Statistically modeling networks, across numerous disciplines and contexts, is fundamentally challenging because of (often high-order) dependence between connections. A common approach assigns each person in the graph to a position on a low-dimensional manifold. Distance between individuals in this (latent) space is inversely proportional to the likelihood of forming a connection. The choice of the latent geometry (the manifold class, dimension, and curvature) has consequential impacts on the substantive conclusions of the model. More positive curvature in the manifold, for example, encourages more and tighter communities; negative curvature induces repulsion among nodes. Currently, however, the choice of the latent geometry is an a priori modeling assumption and there is limited guidance about how to make these choices in a data-driven way. In this work, we present a method to consistently estimate the manifold type, dimension, and curvature from an empirically relevant class of latent spaces: simply connected, complete Riemannian manifolds of constant curvature. Our core insight comes by representing the graph as a noisy distance matrix based on the ties between cliques. Leveraging results from statistical geometry, we develop hypothesis tests to determine whether the observed distances could plausibly be embedded isometrically in each of the candidate geometries. We explore the accuracy of our approach with simulations and then apply our approach to data-sets from economics and sociology as well as neuroscience.

JEL-codes: C01 C12 C4 C52 C6 D85 L14 (search for similar items in EconPapers)
Date: 2020-12
New Economics Papers: this item is included in nep-ecm, nep-net and nep-ore
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Published as Shane Lubold & Arun G Chandrasekhar & Tyler H McCormick, 2023. "Identifying the latent space geometry of network models through analysis of curvature," Journal of the Royal Statistical Society Series B: Statistical Methodology, vol 85(2), pages 240-292.

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