The Invariant Measure of Multiscale Markov Chains via Fast Arborescence Factorization

What the math gives to ML

The paper gives a constructive reduction theory for multiscale Markov chains: fast communicating states are grouped, slow behavior is represented by an effective chain, and stationary mass is expressed through weighted directed arborescences. Its most transferable engineering asset is the exact elimination rule for a hub state, which replaces two-hop hub-mediated transitions by a normalized rank-one update between retained states. A second opportunity is hierarchical stochastic routing, where rapidly mixing expert groups provide stable local specialization while a slower effective chain controls movement between groups. These ideas are most promising for relay-token attention, graph message passing, and MoE routing.