"Spectrally gapped" random walks on networks: a Mean First Passage Time formula

We derive an approximate but explicit formula for the Mean First Passage Time of a random walker between source and target node directed weighted network. The does not require any matrix inversion, it takes as only input transition probabilities into node. It is derived from calculation average resolvent deformed ensemble sub-stochastic matrices $H=\langle H\rangle +\delta H$, with $\langle H\rangle$ rank-$1$ non-negative. accuracy depends on spectral gap reduced matrix, tested numerically several instances (weighted) networks away high sparsity regime, excellent agreement.