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. For example, social network activity may be heightened during certain hours of day. 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 this 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. To enforce the recurring activity we require that only $H < K$ different set of parameters can be used, that is, several, not necessarily consecutive, segments must share their parameters. We prove that the searching for optimal blocks and segmentation is an NP-hard problem. Consequently, we split the problem into 3 subproblems where we optimize blocks, model parameters, and segmentation in turn while keeping the remaining structures fixed. We propose an iterative algorithm that requires $O(KHm + Rn + R^2H)$ 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.

翻译：一种常见的交互建模方法是将交互表示为网络，其中节点代表智能体，边表示交互。交互通常带有时间戳，从而产生带时间戳的边。许多真实世界的时序网络具有递归或可能的循环行为。例如，社交网络活动可能在一天中的特定时段增强。本文主要关注建模此类时序网络中的递归活动。我们以随机块模型作为起点，该模型是建模静态网络的常用选择，其中节点被分为$R$组。我们将该模型扩展至时序网络，通过泊松过程对边进行建模。为使过程参数随时间变化，我们将时间线分割为$K$个段。为强制实现递归活动，我们要求仅能使用$H < K$组不同的参数，即多个（不必相邻的）段必须共享其参数。我们证明寻找最优块与分割是NP难问题。因此，我们将问题分解为三个子问题，分别优化块、模型参数和分割，同时保持其余结构固定。我们提出一种迭代算法，每轮迭代需要$O(KHm + Rn + R^2H)$时间，其中$n$和$m$分别为网络中的节点数和边数。实验表明，所需迭代次数通常较低，该算法能从合成数据集中发现真实结构，且当$H$降低时似然度并未恶化，表明某些真实网络确实表现出递归行为。