Measure Theoretic Flow Centrality

We develop a measure-theoretic notion of Flow Centrality for stochastic routing and inference systems. The construction begins with occupancy events on Markov flows and lifts to tilted path ensembles whose log-partition equals a log spectral radius. Centrality scores arise as derivatives of this potential with respect to node/edge restrictions, delivering Action– Betweenness diagnostics that identify fragile channels. The same potential yields a convex rate function governing occupancy large deviations, which we use to define Murphy horizons for discovery schedules. We prove invariance and stability under quotienting (QFI), connect limiting cases to classical centralities (random-walk betweenness and current-flow centrality), and give algorithms based on sensitivity of the Perron root and matrix perturbation. The framework interoperates with FI/CRLB transport via Blackwell sufficiency, with ergodic drift guarantees, and with compile-time safety contracts that ensure measurable, support-respecting kernels. Claim-specific citations: random-walk and current-flow centralities (Newman 2005; Doyle and Snell 1984), spectral graph theory (Chung 1997), mixing/occupancy baselines (Levin et al. 2009; Kemeny and Snell 1976), large deviations via tilted spectra (Dembo and Zeitouni 1998), FI transport (Blackwell 1953; Amari and Nagaoka 2000; Cencov 1982), quotient invariants (Elliott 2025e), drift coverage (Meyn and Tweedie 2009; Elliott 2025c), and compile-time contracts (Elliott 2025d; Elliott 2025a).