  Persistent homology computes the multiscale topology of a data set by using a
sequence of discrete complexes. In this paper, we propose that persistent
homology may be a useful tool for studying the structure of the landscape of
string vacua. As a scaled-down version of the program, we use persistent
homology to characterize distributions of Type IIB flux vacua on moduli space
for three examples: the rigid Calabi-Yau, a hypersurface in weighted projective
space, and the symmetric six-torus $T^6=(T^2)^3$. These examples suggest that
persistence pairing and multiparameter persistence contain useful information
for characterization of the landscape in addition to the usual information
contained in standard persistent homology. We also study how restricting to
special vacua with phenomenologically interesting low-energy properties affects
the topology of a distribution.
