Entanglement Dynamics in Katz-Weighted Graph States
We investigate the entanglement dynamics of quantum states defined on graphs with non-local Ising interactions governed by the Katz kernel of the underlying network. The interaction pattern is physically motivated by a gapped fermionic mediator propagating on the same graph, whose perturbative elimination yields an effective Katz-weighted Ising Hamiltonian. Using the Entanglement Distance, we derive an exact analytical expression for the entanglement generated from an initially separable state and apply it to representative deterministic graph families. We then characterize the dynamics in different propagation regimes. In the weak-Katz regime, the dynamics admits a systematic motif expansion with triangles entering at first order order and four-cycles, local degree structure, and overlappin triangles appearing at second order. In the strong-propagation regime, the interaction is instead dominated by the principal adjacency mode and by the localization properties of its eigenvector. For Erdős--Rényi graphs, the weak-propagation expansion can be averaged analytically, revealing a locally tree-like contribution in the sparse regime and saturation of the Entanglement Distance density in the dense regime. Our results connect entanglement dynamics with both the walk-based and spectral structure of complex networks.