Competing Localizations on Disordered Non-Hermitian Random Graph Lattice

Phase transitions in one-dimensional lattice systems are well established and have been extensively studied within both Hermitian and non-Hermitian frameworks. In this work, we extend this understanding to a more general setting by investigating localization and delocalization transitions and the behavior of the non-Hermitian skin effect (NHSE) using a tight-binding model on a generalized random graph lattice. Our model incorporates three key parameters, asymmetric hopping $\Delta$, on-site disorder $W$, and a random long-range coupling $p$ that together define the underlying random graph structure. By varying $p$, $\Delta$, and the disorder strength, we explore the interplay between topology, randomness, and non-Hermiticity in determining localization properties. Our results show a strong competition between skin effect driven and Anderson driven localizations across parameter regimes. Notably, even in the presence of strong disorder, skin effect driven localization coexists with Anderson-driven localization. We further discuss the relevance of these results to machine-learning architectures and information propagation in complex networks and other real-world problems.