  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 sub-collections 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.
