Open Access
2024 Consistency of maximum likelihood for continuous-space network models I
Cosma Shalizi, Dena Asta
Author Affiliations +
Electron. J. Statist. 18(1): 335-354 (2024). DOI: 10.1214/23-EJS2169


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.

Funding Statement

Our work was supported by NSF grant DMS-1418124; DA also received support from NSF Graduate Research Fellowship under grant DGE-1252522, and CRS from NSF grant DMS-1207759.


We are grateful for valuable discussions with Carl Bergstrom, Elizabeth Casman, David Choi, Aaron Clauset, Steve Fienberg, Christopher Genovese, Aryeh Kontorovich, Dmitri Krioukov, Cris Moore, Alessandro Rinaldo, Mitch Small, Neil Spencer, Andrew Thomas, Larry Wasserman, and Chris Wiggins, and for feedback from seminar audiences at CMU, UCLA, UW-Seattle, and SFI.


Download Citation

Cosma Shalizi. Dena Asta. "Consistency of maximum likelihood for continuous-space network models I." Electron. J. Statist. 18 (1) 335 - 354, 2024.


Received: 1 June 2022; Published: 2024
First available in Project Euclid: 2 February 2024

arXiv: 1711.02123
Digital Object Identifier: 10.1214/23-EJS2169

Primary: 62G05 , 91D30

Keywords: consistency , graph embedding , latent space models , MLE , network models

Vol.18 • No. 1 • 2024
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