Inference of hierarchical core-periphery structure in temporal networks

Networks can have various types of mesoscale structures. One such structure is core--periphery structure, which consists of sets of densely-connected core nodes and sets of sparsely-connected peripheral nodes. Core nodes are connected densely to other core nodes and are potentially connected densely to peripheral nodes, which are connected sparsely to other nodes. There has been much research on core--periphery structure in time-independent networks, but few core--periphery detection methods have been developed for time-dependent (i.e., ``temporal") networks. Using a multilayer-network representation of temporal networks and an inference approach that employs stochastic block models, we generalize a recent method for detecting hierarchical core--periphery structure \cite{Polanco23} from time-independent networks to temporal networks. In contrast to ``onion-like'' nested core--periphery structures (where each node is assigned to a group according to how deeply it is nested in a network's core), hierarchical core--periphery structures encompass networks with nested structures, tree-like structures (where any two groups must either be disjoint or have one group as a strict subset of the other group), and general non-nested mesoscale structures (where the group assignments of nodes do not have to be nested in any way). To perform statistical inference and thereby identify hierarchical core--periphery structure, we use a Markov-chain Monte Carlo (MCMC) approach. We illustrate our method for detecting hierarchical core--periphery structure in one synthetic temporal network and two real-world temporal networks, and we discuss the structures that we identify in these networks.