  Computation of persistent homology of simplicial representations such as the
Rips and the C\v{e}ch complexes do not efficiently scale to large point clouds.
It is, therefore, meaningful to devise approximate representations and evaluate
the trade-off between their efficiency and effectiveness. The lazy witness
complex economically defines such a representation using only a few selected
points, called landmarks.
  Topological data analysis traditionally considers a point cloud in a
Euclidean space. In many situations, however, data is available in the form of
a weighted graph. A graph along with the geodesic distance defines a metric
space. This metric space of a graph is amenable to topological data analysis.
  We discuss the computation of persistent homologies on a weighted graph. We
present a lazy witness complex approach leveraging the notion of $\epsilon$-net
that we adapt to weighted graphs and their geodesic distance to select
landmarks. We show that the value of the $\epsilon$ parameter of the
$\epsilon$-net provides control on the trade-off between choice and number of
landmarks and the quality of the approximate simplicial representation.
  We present three algorithms for constructing an $\epsilon$-net of a graph. We
comparatively and empirically evaluate the efficiency and effectiveness of the
choice of landmarks that they induce for the topological data analysis of
different real-world graphs.
