June 2024 Learning common structures in a collection of networks. An application to food webs
Saint-Clair Chabert-Liddell, Pierre Barbillon, Sophie Donnet
Author Affiliations +
Ann. Appl. Stat. 18(2): 1213-1235 (June 2024). DOI: 10.1214/23-AOAS1831


Let a collection of networks represent interactions within several (social or ecological) systems. We pursue two objectives: identifying similarities in the topological structures that are held in common between the networks and clustering the collection into subcollections of structurally homogeneous networks. We tackle these two questions with a probabilistic model-based approach. We propose an extension of the stochastic block model (SBM) adapted to the joint modeling of a collection of networks. The networks in the collection are assumed to be independent realizations of SBMs. The common connectivity structure is imposed through the equality of some parameters.

The model parameters are estimated with a variational expectation-maximization (EM) algorithm. We derive an ad hoc penalized likelihood criterion to select the number of blocks and to assess the adequacy of the consensus found between the structures of the different networks. This same criterion can also be used to cluster networks on the basis of their connectivity structure. It thus provides a partition of the collection into subsets of structurally homogeneous networks.

The relevance of our proposition is assessed on two collections of ecological networks. First, an application to three stream food webs reveals the homogeneity of their structures and the correspondence between groups of species in different ecosystems playing equivalent ecological roles. Moreover, the joint analysis allows a finer analysis of the structure of smaller networks. Second, we cluster 67 food webs according to their connectivity structures and demonstrate that five mesoscale structures are sufficient to describe this collection.

Funding Statement

This work was supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH. This work was partially supported by the grant ANR-18-CE02-0010-01 of the French National Research Agency ANR (project EcoNet).


The authors would like to thank Stéphane Robin for his helpful advice.


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Saint-Clair Chabert-Liddell. Pierre Barbillon. Sophie Donnet. "Learning common structures in a collection of networks. An application to food webs." Ann. Appl. Stat. 18 (2) 1213 - 1235, June 2024. https://doi.org/10.1214/23-AOAS1831


Received: 1 June 2022; Revised: 1 September 2023; Published: June 2024
First available in Project Euclid: 5 April 2024

Digital Object Identifier: 10.1214/23-AOAS1831

Keywords: clustering , ecology , latent variable models , networks , Stochastic block model

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.18 • No. 2 • June 2024
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