  We present mathematical models based on persistent homology for analyzing
force distributions in particulate systems. We define three distinct chain
complexes: digital, position, and interaction, motivated by different
capabilities of collecting experimental or numerical data, e.g. digital images,
location of the particles, and normal forces between particles, respectively.
We describe how algebraic topology, in particular, homology allows one to
obtain algebraic representations of the geometry captured by these complexes.
To each complexes we define an associated force network from which persistent
homology is computed. Using numerical data obtained from molecular dynamics
simulations of a system of particles being slowly compressed we demonstrate how
persistent homology can be used to compare the geometries of the force
distributions in different granular systems. We also discuss the properties of
force networks as a function of the underlying complexes, and hence, as a
function of the type of experimental or numerical data provided.
