The interplay between ranking and communities in networks

Abstract

Community detection and hierarchy extraction are usually thought of as separate inference tasks on networks. Considering only one of the two when studying real-world data can be an oversimplification. In this work, we present a generative model based on an interplay between community and hierarchical structures. It assumes that each node has a preference in the interaction mechanism and nodes with the same preference are more likely to interact, while heterogeneous interactions are still allowed. The algorithmic implementation is efficient, as it exploits the sparsity of network datasets. We demonstrate our method on synthetic and real-world data and compare performance with two standard approaches for community detection and ranking extraction. We find that the algorithm accurately retrieves each node's preference in different scenarios and we show that it can distinguish small subsets of nodes that behave differently than the majority. As a consequence, the model can recognise whether a network has an overall preferred interaction mechanism. This is relevant in situations where there is no clear "a priori" information about what structure explains the observed network datasets well. Our model allows practitioners to learn this automatically from the data.

Ordered community detection in directed networks

“…The works that are perhaps closest to ours are the approaches from Letizia et al [26] and Iacovissi et al [27]. Letizia et al [26] considered a ranked SBM with uniform connection probabilities between groups depending only on whether the edge direction violates or not the hierarchy.…”

“…[26] is not based on a model likelihood, and hence cannot be used to evaluate statistical evidence. The method of Iacovissi et al [27] is based on a different idea, and combines the SBM with Springrank [23], such that a node can either have a group membership or a ranking, but not both simultaneously. Their model not only lacks degree correction, but its inference is performed in a parametric fashion: the number of groups in the SBM needs to be set a priori, and cannot be extracted from the data itself.…”

“…Furthermore, the inference procedure developed in Ref. [27] is based on a variational approximation, whereas our approach is based on MCMC using an exact likelihood.…”