Editing Openai/693e3ce6-229c-8008-97dc-ab720cb1f95a (section)

==== Pick typical PageRank teleportation, say: ====
* α=0.15\alpha = 0.15α=0.15,
* star size m=100m = 100m=100.

Then

ρ0=1−α2m=0.85200=0.00425.\rho_0 = \frac{1-\alpha}{2m}=\frac{0.85}{200}=0.00425.ρ0=2m1−α=2000.85=0.00425.
Take ρ=0.004251\rho = 0.004251ρ=0.004251 (so δ=10−6\delta = 10^{-6}δ=10−6). Then

γ=2α1+αδ=0.31.15⋅10−6≈2.61×10−7.\gamma = \frac{2\alpha}{1+\alpha}\delta
= \frac{0.3}{1.15}\cdot 10^{-6}
\approx 2.61\times 10^{-7}.γ=1+α2αδ=1.150.3⋅10−6≈2.61×10−7.
But

ρ≈0.004251≈6.52×10−2.\sqrt{\rho} \approx \sqrt{0.004251} \approx 6.52\times 10^{-2}.ρ≈0.004251≈6.52×10−2.
So

γ≈2.6⋅10−7  ≪  6.5⋅10−2  =  ρ.\gamma \approx 2.6\cdot 10^{-7} \;\ll\; 6.5\cdot 10^{-2} \;=\; \sqrt{\rho}.γ≈2.6⋅10−7≪6.5⋅10−2=ρ.
This is a clean “bad” case: connected graph, ∣S⋆∣=1≤1/ρ|S^\star|=1 \le 1/\rho∣S⋆∣=1≤1/ρ, yet γ\gammaγ is orders of magnitude smaller than ρ\sqrt{\rho}ρ because ρ\rhoρ is chosen near a support-change breakpoint.

===== Step size ttt, momentum β\betaβ, etc. do not change γ\gammaγ: γ\gammaγ is a property of the problem instance (graph + seed + α,ρ\alpha,\rhoα,ρ). What the algorithm parameters can do is affect how long you linger in the transient, but they don’t alter the KKT slack at the true minimizer. =====

If you want, I can also give a path graph construction where ρ0\rho_0ρ0 is small even with moderate mmm, but the star example is the simplest explicit one where you can compute γ(ρ)\gamma(\rho)γ(ρ) in closed form.