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A very popular class of models for networks posits that each node is represented by a point in a continuous latent space, and that the probability of an edge between nodes is a decreasing function of the distance between them in this latent space. We study the embedding problem for these models, of recovering the latent positions from the observed graph. Assuming certain natural symmetry and smoothness properties, we establish the uniform convergence of the log-likelihood of latent positions as the number of nodes grows. A consequence is that the maximum likelihood embedding converges on the true positions in a certain information-theoretic sense. Extensions of these results, to recovering distributions in the latent space, and so distributions over arbitrarily large graphs, will be treated in the sequel.
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Joint latent space models for network data with high-dimensional node variables
Summary Network latent space models assume that each node is associated with an unobserved latent position in a Euclidean space, and such latent variables determine the probability of two nodes connecting with each other. In many applications, nodes in the network are often observed along with high-dimensional node variables, and these node variables provide important information for understanding the network structure. However, classical network latent space models have several limitations in incorporating node variables. In this paper, we propose a joint latent space model where we assume that the latent variables not only explain the network structure, but are also informative for the multivariate node variables. We develop a projected gradient descent algorithm that estimates the latent positions using a criterion incorporating both network structure and node variables. We establish theoretical properties of the estimators and provide insights into how incorporating high-dimensional node variables could improve the estimation accuracy of the latent positions. We demonstrate the improvement in latent variable estimation and the improvements in associated downstream tasks, such as missing value imputation for node variables, by simulation studies and an application to a Facebook data example.
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- Award ID(s):
- 1846747
- PAR ID:
- 10316654
- Date Published:
- Journal Name:
- Biometrika
- ISSN:
- 0006-3444
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/biomet/asab063
