  We perform an extensive analysis of how sampling impacts the estimate of
several relevant network measures.
  In particular, we focus on how a sampling strategy optimized to recover a
particular spectral centrality measure impacts other topological quantities.
Our goal is on one hand to extend the analysis of the behavior of TCEC
[Ruggeri2019], a theoretically-grounded sampling method for eigenvector
centrality estimation.
  On the other hand, to demonstrate more broadly how sampling can impact the
estimation of relevant network properties like centrality measures different
than the one aimed at optimizing, community structure and node attribute
distribution.
  Finally, we adapt the theoretical framework behind TCEC for the case of
PageRank centrality and propose a sampling algorithm aimed at optimizing its
estimation. We show that, while the theoretical derivation can be suitably
adapted to cover this case, the resulting algorithm suffers of a high
computational complexity that requires further approximations compared to the
eigenvector centrality case.
