April 2021 Network representation using graph root distributions
Jing Lei
Author Affiliations +
Ann. Statist. 49(2): 745-768 (April 2021). DOI: 10.1214/20-AOS1976


Exchangeable random graphs serve as an important probabilistic framework for the statistical analysis of network data. In this work, we develop an alternative parameterization for a large class of exchangeable random graphs, where the nodes are independent random vectors in a linear space equipped with an indefinite inner product, and the edge probability between two nodes equals the inner product of the corresponding node vectors. Therefore, the distribution of exchangeable random graphs in this subclass can be represented by a node sampling distribution on this linear space, which we call the graph root distribution. We study existence and identifiability of such representations, the topological relationship between the graph root distribution and the exchangeable random graph sampling distribution and estimation of graph root distributions.

Funding Statement

Jing Lei’s research was partially supported by NSF Grants DMS-1553884 and DMS-2015492.


The author would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the quality of this paper.


Download Citation

Jing Lei. "Network representation using graph root distributions." Ann. Statist. 49 (2) 745 - 768, April 2021. https://doi.org/10.1214/20-AOS1976


Received: 1 March 2019; Revised: 1 November 2019; Published: April 2021
First available in Project Euclid: 2 April 2021

Digital Object Identifier: 10.1214/20-AOS1976

Primary: 62E10
Secondary: 62G05 , 62M15

Keywords: exchangeable random graph , Kreǐn space , network data , spectral embedding

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 2 • April 2021
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