  This paper surveys various distance measures for networks and graphs that
were introduced in persistent homology. The scope of the paper is limited to
network distances that were actually used in brain networks but the methods can
be easily adapted to any weighted graph in other fields. The network version of
Gromov-Hausdorff, bottleneck, kernel distances are introduced. We also
introduce a recently developed KS-test like distance based on monotonic
topology features such as the zeroth Betti number. Numerous toy examples and
the result of applying many different distances to the brain networks of
different clinical status and populations are given.
