  Geometric graph models of systems as diverse as proteins, robots, and
mechanical structures from DNA assemblies to architected materials point
towards a unified way to represent and control them in space and time. While
much work has been done in the context of characterizing the behavior of these
networks close to critical points associated with bond and rigidity
percolation, isostaticity, etc., much less is known about floppy,
under-constrained networks that are far more common in nature and technology.
Here we combine geometric rigidity and algebraic sparsity to provide a
framework for identifying the zero-energy floppy modes via a representation
that illuminates the underlying hierarchy and modularity of the network, and
thence the control of its nestedness and locality. Our framework allows us to
demonstrate a range of applications of this approach that include robotic
reaching tasks with motion primitives, and predicting the linear and nonlinear
response of elastic networks based solely on infinitesimal rigidity and
sparsity, which we test using physical experiments. Our approach is thus likely
to be of use broadly in dissecting the geometrical properties of floppy
networks using algebraic sparsity to optimize their function and performance.
