A stochastic block model for interaction lengths
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A stochastic block model for interaction lengths Riccardo Rastelli1
· Michael Fop1
Received: 15 January 2019 / Revised: 6 May 2020 / Accepted: 19 May 2020 / Published online: 18 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We propose a new stochastic block model that focuses on the analysis of interaction lengths in dynamic networks. The model does not rely on a discretization of the time dimension and may be used to analyze networks that evolve continuously over time. The framework relies on a clustering structure on the nodes, whereby two nodes belonging to the same latent group tend to create interactions and non-interactions of similar lengths. We introduce a variational expectation–maximization algorithm to perform inference, and adapt a widely used clustering criterion to perform model choice. Finally, we validate our methodology using simulated data experiments and showing two illustrative applications concerning face-to-face interaction data and a bike sharing network. Keywords Interaction lengths · Stochastic block model · Variational inference · Integrated completed likelihood · Social network analysis Mathematics Subject Classification 62H30 · 91D30
1 Introduction In recent years, a number of network models have been introduced in the literature to study how binary interactions between entities evolve over time. One common approach relies on the discretization of the time dimension: once an appropriate time grid is specified, the continuous data are essentially transformed into a collection of static network snapshots. This approach has facilitated the extension of many static network models to a dynamic framework. For example, the Stochastic Block Model (SBM) of Wang and Wong (1987) has been recently adapted to the dynamic case by Yang et al. (2011) and Matias and Miele (2017). In the same fashion, extensions of the Latent Position Model (LPM) of Hoff et al. (2002) have been proposed by Sarkar
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Riccardo Rastelli [email protected] School of Mathematics and Statistics, University College Dublin, Dublin, Ireland
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and Moore (2005) and Sewell and Chen (2015), among others. The model of Hanneke et al. (2010) extends instead the well known Exponential Random Graph Model of Holland and Leinhardt (1981). However, the approach based on the discretization of the time dimension has been recently criticized, mainly due to the non-negligible effects that the data transformation may have on the results (Corneli et al. 2017; Matias et al. 2018). In fact, the discretization process always involves a certain level of arbitrariness, either due to the data being collected at specific given times, or because of a postcollection transformation. In truth, in the vast majority of data analysis applications, the interactions evolve over time in a continuous fashion. Dynamic binary interactions could either be instantaneous or protracted over a time interval. An example of the first situation is the well-known email network Enron, and this framework
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