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February 2020 Averages of unlabeled networks: Geometric characterization and asymptotic behavior
Eric D. Kolaczyk, Lizhen Lin, Steven Rosenberg, Jackson Walters, Jie Xu
Ann. Statist. 48(1): 514-538 (February 2020). DOI: 10.1214/19-AOS1820


It is becoming increasingly common to see large collections of network data objects, that is, data sets in which a network is viewed as a fundamental unit of observation. As a result, there is a pressing need to develop network-based analogues of even many of the most basic tools already standard for scalar and vector data. In this paper, our focus is on averages of unlabeled, undirected networks with edge weights. Specifically, we (i) characterize a certain notion of the space of all such networks, (ii) describe key topological and geometric properties of this space relevant to doing probability and statistics thereupon, and (iii) use these properties to establish the asymptotic behavior of a generalized notion of an empirical mean under sampling from a distribution supported on this space. Our results rely on a combination of tools from geometry, probability theory and statistical shape analysis. In particular, the lack of vertex labeling necessitates working with a quotient space modding out permutations of labels. This results in a nontrivial geometry for the space of unlabeled networks, which in turn is found to have important implications on the types of probabilistic and statistical results that may be obtained and the techniques needed to obtain them.


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Eric D. Kolaczyk. Lizhen Lin. Steven Rosenberg. Jackson Walters. Jie Xu. "Averages of unlabeled networks: Geometric characterization and asymptotic behavior." Ann. Statist. 48 (1) 514 - 538, February 2020.


Received: 1 October 2018; Published: February 2020
First available in Project Euclid: 17 February 2020

zbMATH: 07196549
MathSciNet: MR4065172
Digital Object Identifier: 10.1214/19-AOS1820

Primary: 62E20 , 62G20
Secondary: 53C20

Keywords: Fréchet mean , Fundamental domain , Graphs

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 1 • February 2020
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