December 2023 Projective, sparse and learnable latent position network models
Neil A. Spencer, Cosma Rohilla Shalizi
Author Affiliations +
Ann. Statist. 51(6): 2506-2525 (December 2023). DOI: 10.1214/23-AOS2340


When modeling network data using a latent position model, it is typical to assume that the nodes’ positions are independently and identically distributed. However, this assumption implies the average node degree grows linearly with the number of nodes, which is inappropriate when the graph is thought to be sparse. We propose an alternative assumption—that the latent positions are generated according to a Poisson point process—and show that it is compatible with various levels of sparsity. Unlike other notions of sparse latent position models in the literature, our framework also defines a projective sequence of probability models, thus ensuring consistency of statistical inference across networks of different sizes. We establish conditions for consistent estimation of the latent positions, and compare our results to existing frameworks for modeling sparse networks.

Funding Statement

NAS received funding from the Natural Sciences and Engineering Research Council of Canada. CRS was supported by a grants from the National Science Foundation (DMS1418124) and the Institute for New Economic Thinking (INO1400020).


We are grateful to the members of the Carnegie Mellon Networkshop for feedback on our results and their presentation, and to conversations with Creagh Briercliffe, David Choi, Emily Fox, Alden Green, Peter Hoff, Jeannette Janssen, Dmitri Krioukov and Cristopher Moore. We would also like to thank the anonymous referees for their constructive comments and suggestions.


Download Citation

Neil A. Spencer. Cosma Rohilla Shalizi. "Projective, sparse and learnable latent position network models." Ann. Statist. 51 (6) 2506 - 2525, December 2023.


Received: 1 February 2020; Revised: 1 September 2023; Published: December 2023
First available in Project Euclid: 20 December 2023

MathSciNet: MR4682706
zbMATH: 07783624
Digital Object Identifier: 10.1214/23-AOS2340

Primary: 60G55 , 62M09 , 91D30

Keywords: consistent estimation , latent position network model , Latent space model , network models , network sparsity , projective family , Random geometric graph

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 6 • December 2023
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