Spectral clustering in the dynamic stochastic block model

Abstract

In the present paper, we studied a Dynamic Stochastic Block Model (DSBM) under the assumptions that the connection probabilities, as functions of time, are smooth and that at most s nodes can switch their class memberships between two consecutive time points. We estimate the edge probability tensor by a kernel-type procedure and extract the group memberships of the nodes by spectral clustering. The procedure is computationally viable, adaptive to the unknown smoothness of the functional connection probabilities, to the rate s of membership switching and to the unknown number of clusters. In addition, it is accompanied by non-asymptotic guarantees for the precision of estimation and clustering.

A Dynamic Network Perspective on the Latent Group Structure of Cryptocurrencies

“…For the first issue, we follow Zhang et al (2018) by setting W t = X L τ,t X, which measures the correlation between covariates along the graph. For the second issue, we follow Pensky and Zhang (2019) by constructing the estimator of S t with a discrete kernel to bring in historical network information. Klochkov et al (2019) present a similar idea.…”

Hierarchical Change Point Detection on Dynamic Networks

“…Among the work on dynamic stochastic block models, Xu (2015) proposed a stochastic block transition model using a hidden Markov-type approach; Xu and Hero (2014) proposed to track dynamic stochastic block models using Gaussian approximation and an extended Kalman filter algorithm; Matias and Miele (2017) integrated a Markov chain determined group labels evolving process; Pensky and Zhang (2017) exploited kernel-based smoothing techniques dealing with the evolving block structures; Bhattacharyya and Chatterjee (2017) focused on time-varying stochastic block model and variants thereof with time-independent community labels, applied spectral clustering on an averaged version of adjacency matrices, and achieved consistent community detection. Wang et al (2014) used two types of scan statistics investigating change point detection on time-varying stochastic block model sequences, emphasizing testing connectivity matrices changes.…”

Estimation of Dynamic Networks for High-Dimensional Nonstationary Time Series

“…This assumption is quite restrictive in practice and hardly plausible for many real-world applications, such as gene regulatory networks, social networks, and stocking market, where the underlying data generating mechanisms are often dynamic. On the other hand, dynamic random networks have been extensively studied from the perspective of large random graphs, such as community detection and edge probability estimation for dynamic stochastic block models (DSBMs) [ 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 ]. Such approaches do not model the sampling distributions of the error (or noise), since the “true” networks are connected with random edges sampled from certain probability models, such as the Erdős–Rényi graphs [ 31 ] and random geometric graphs [ 32 ].…”