Observable-specific kinetic-domain hierarchies in Markovian population dynamics

Markovian population dynamics on a network are commonly formulated through a master equation for site populations and analyzed either in the site representation or in terms of independently evolving eigenmodes. Neither description directly resolves the relaxation of a selected observable into kinetically interpretable stages. We develop an observable-specific kinetic-domain (KD) representation that recasts this relaxation as a kinetic hierarchy. The resulting basis is algebraically equivalent to that obtained by Arnoldi orthogonalization of an observable-seeded Krylov sequence. This hierarchy describes the multidimensional population dynamics as a funnel-like sequence of kinetic stages with collective site-space structures and interstage transition rates. Generalized survival functions (GSFs) on nested kinetic subspaces then yield an exact stagewise decomposition of the time-dependent observable relaxation and, upon time integration, a characteristic passage time for each stage, thereby identifying the rate-determining stage and revealing the underlying relaxation mechanism. Applications to analytically solvable networks and extended two-dimensional lattices demonstrate how the KD/GSF framework resolves complex relaxation into interpretable collective stages and show that similar global relaxation time scales can conceal markedly different underlying sequences of kinetic steps. By extracting the collective kinetic pathway relevant to the selected observable, the KD representation complements site- and eigenmode-based descriptions and can enable dimensional reduction in large stochastic networks.