Sparse and smooth: Improved guarantees for spectral clustering in the dynamic stochastic block model

Abstract

In this paper, we analyze classical variants of the Spectral Clustering (SC) algorithm in the Dynamic Stochastic Block Model (DSBM). Existing results show that, in the relatively sparse case where the expected degree grows logarithmically with the number of nodes, guarantees in the static case can be extended to the dynamic case and yield improved error bounds when the DSBM is sufficiently smooth in time, that is, the communities do not change too much between two time steps. We improve over these results by drawing a new link between the sparsity and the smoothness of the DSBM: the smoother the DSBM is, the more sparse it can be, while still guaranteeing consistent recovery. In particular, a mild condition on the smoothness allows to treat the sparse case with bounded degree. These guarantees are valid for the SC applied to the adjacency matrix or the normalized Laplacian. As a by-product of our analysis, we obtain to our knowledge the best spectral concentration bound available for the normalized Laplacian of matrices with independent Bernoulli entries.

Reliable prediction in the Markov stochastic block model

“…when the maximum expected node degree is of order log(n) or higher), the proportion of misclassified nodes tends to 0 with a probability that goes to 1 when the number of nodes n increases using spectral clustering. This result inspired the recent paper [37] which considers a Dynamic Stochastic Block Model where the communities can change with time. They provide direct connection between the density of the graph and its smoothness (which measures how much the graph changes with time).…”

Interpretable Clustering on Dynamic Graphs with Recurrent Graph Neural Networks

“…Traditional spectral clustering algorithms use the spectrum of the graph adjacency matrix to generate a compact representation of the graph connectivity (Lei, Rinaldo et al 2015;Qin and Rohe 2013). A line of work on static clustering algorithms uses the stochastic block model for graph connectivity (Abbe 2017), which more recent works have extended to a dynamic stochastic block model (Keriven and Vaiter 2020;Pensky, Zhang et al 2019). While these works do not distinguish between clusters with different transition probabilities, earlier models incorporate such heterogeneity (Xu 2015).…”

“…Accounting for historical node connections with a single decay rate parameter offers the advantage of interpretability: the decay rate quantifies the emphasis put on historically formed edges, which can be tuned for specific datasets. Yet while prior works have examined the optimal decay rate for stylized network models, they use a single decay rate for all edges (Keriven and Vaiter 2020). In practice, the optimal rate will likely vary, e.g., with higher decay rates for clusters with higher membership turnover where historical information might reflect outdated cluster memberships, making it less useful.…”

“…• A theoretical analysis of the optimal decay rates for spectral clustering algorithms applied to the dynamic stochastic block model, a common model of graph clustering dynamics (Keriven and Vaiter 2020).…”