Recurrent segmentation meets block models in temporal networks

A popular approach to model interactions is to represent them as a network with nodes being the agents and the interactions being the edges. Interactions are often timestamped, which leads to having timestamped edges. Many real-world temporal networks have a recurrent or possibly cyclic behaviour. In this paper, our main interest is to model recurrent activity in such temporal networks. As a starting point we use stochastic block model, a popular choice for modelling static networks, where nodes are split into R groups. We extend the block model to temporal networks by modelling the edges with a Poisson process. We make the parameters of the process dependent on time by segmenting the time line into K segments. We require that only H <= K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H \le K$$\end{document} different set of parameters can be used. If H<K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H < K$$\end{document}, then several, not necessarily consecutive, segments must share their parameters, modelling repeating behaviour. We propose two variants where a group membership of a node is fixed over the course of entire time line and group memberships are allowed to vary from segment to segment. We prove that searching for optimal groups and segmentation in both variants is NP-hard. Consequently, we split the problem into 3 subproblems where we optimize groups, model parameters, and segmentation in turn while keeping the remaining structures fixed. We propose an iterative algorithm that requires OKHm+Rn+R2H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O} \left( KHm + Rn + R<^>2\,H\right)$$\end{document} time per iteration, where n and m are the number of nodes and edges in the network. We demonstrate experimentally that the number of required iterations is typically low, the algorithm is able to discover the ground truth from synthetic datasets, and show that certain real-world networks exhibit recurrent behaviour as the likelihood does not deteriorate when H is lowered.