  Network-valued time series are currently a common form of network data.
However, the study of the aggregate behavior of network sequences generated
from network-valued stochastic processes is relatively rare. Most of the
existing research focuses on the simple setup where the networks are
independent (or conditionally independent) across time, and all edges are
updated synchronously at each time step. In this paper, we study the
concentration properties of the aggregated adjacency matrix and the
corresponding Laplacian matrix associated with network sequences generated from
lazy network-valued stochastic processes, where edges update asynchronously,
and each edge follows a lazy stochastic process for its updates independent of
the other edges. We demonstrate the usefulness of these concentration results
in proving consistency of standard estimators in community estimation and
changepoint estimation problems. We also conduct a simulation study to
demonstrate the effect of the laziness parameter, which controls the extent of
temporal correlation, on the accuracy of community and changepoint estimation.
