Stochastic shielding and edge importance for Markov chains with timescale separation
Fig 7
Factors contributing to edge importance reversal.
The relative importance due to the hidden edges, η = R12/(R12 + R23), was calculated for an ensemble of 3-state chains (100,000 samples, see text for details). Relative edge importance is inverted when η > 0.5. Left column shows η plotted versus stationary occupancy probability of node 3 (π3, panel A), node 2 (π2, C), and the ratio of nodes 2 to 3 (π2/π3, E). The corresponding plot for π1 appears similar to that for π3 (not shown). Edge importance can be inverted for any values of π1 and π3, but requires π2 ≲ 1/6. Right column shows η plotted versus timescale separation (ν = λ3/λ2, B), relative fraction of flux generated by the hidden edges (ΔJ = (J12 − J23)/(J12 + J23), D), and ratio of relaxation times for isolated 2-state systems corresponding to the hidden versus observable transitions (τ12/τ23, F). Edge importance reversal requires timescale separation (|λ3| ≳ 15|λ2| or ν ≳ 15), larger mean flux along the observable edges than the hidden edges (J23 > J12), and faster relaxation along the visible edges than along the hidden edges (τ12 > τ23). None of these conditions alone are sufficient. However, panel G shows η versus the two factors F1 and F2 in the exact expression for η, black ‘+’ line is F1 ⋅ F2 (see Eq 22).