  Complex networks with directed, local interactions are ubiquitous in nature,
and often occur with probabilistic connections due to both intrinsic
stochasticity and disordered environments. Sparse non-Hermitian random matrices
arise naturally in this context, and are key to describing statistical
properties of the non-equilibrium dynamics that emerges from interactions
within the network structure. Here, we study one-dimensional (1d) spatial
structures and focus on sparse non-Hermitian random matrices in the spirit of
tight-binding models in solid state physics. We first investigate two-point
eigenvalue correlations in the complex plane for sparse non-Hermitian random
matrices using methods developed for the statistical mechanics of inhomogeneous
2d interacting particles. We find that eigenvalue repulsion in the complex
plane directly correlates with eigenvector delocalization. In addition, for 1d
chains and rings with both disordered nearest neighbor connections and
self-interactions, the self-interaction disorder tends to de-correlate
eigenvalues and localize eigenvectors more than simple hopping disorder.
However, remarkable resistance to eigenvector localization by disorder is
provided by large cycles, such as those embodied in 1d periodic boundary
conditions under strong directional bias. The directional bias also spatially
separates the left and right eigenvectors, leading to interesting dynamics in
excitation and response. These phenomena have important implications for
asymmetric random networks and highlight a need for mathematical tools to
describe and understand them analytically.
