The Heavy-tailed Frog Model

What the math gives to ML

The paper identifies a sharp propagation law for activation cascades driven by translation-invariant heavy-tailed jumps: when the tail exponent satisfies $0<\alpha<d$, a distance-$L$ region can be activated in only $(\log L)^{\Delta+o(1)}$ steps, with $\Delta^{-1}=\log_2(2d/(d+\alpha))$. The transferable asset is not the frog process itself but its multiscale mechanism: a sparse collection of power-law long-range links can repeatedly expand the communication radius, replacing local-depth growth by polylogarithmic-depth growth. A practical neural-network translation is a stochastic sparse attention or message-passing layer whose communication edges are sampled from a truncated power-law distance kernel and refreshed across layers; the classical local baseline and the heavy-tail phase transition provide clear ablations.