Sparse and Smooth: improved guarantees for Spectral Clustering in the Dynamic Stochastic Block Model
This work provides incremental improvements for community detection in dynamic networks, benefiting researchers in network analysis and machine learning.
The paper tackles the problem of spectral clustering in the dynamic stochastic block model by showing that increased smoothness allows for sparser networks while ensuring consistent community recovery, achieving improved error bounds and extending guarantees to normalized Laplacian.
In this paper, we analyse 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 more regular 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. We also extend these guarantees to the normalized Laplacian, and 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.