  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.
