  Using particle-scale models to accurately describe property enhancements and
phase transitions in macroscopic behavior is a major engineering challenge in
composite materials science. To address some of these challenges, we use the
graph theoretic property of rigidity to model me- chanical reinforcement in
composites with stiff rod-like particles. We develop an efficient algorithmic
approach called rigid graph compression (RGC) to describe the transition from
floppy to rigid in disordered fiber networks ('rod-hinge systems'), which form
the reinforcing phase in many composite systems. To establish RGC on a firm
theoretical foundation, we adapt rigidity matroid theory to identify primitive
topological network motifs that serve as rules for composing interacting rigid
par- ticles into larger rigid components. This approach is computationally
efficient and stable, because RGC requires only topological information about
rod interactions (encoded by a sparse unweighted network) rather than
geometrical details such as rod locations or pairwise distances (as required in
rigidity matroid theory). We conduct numerical experiments on simulated
two-dimensional rod-hinge systems to demonstrate that RGC closely approximates
the rigidity percolation threshold for such systems, through comparison with
the pebble game algorithm (which is exact in two dimensions). Importantly,
whereas the pebble game is derived from Laman's condition and is only valid in
two dimensions, the RGC approach naturally extends to higher dimensions.
