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

==== ### ====

Let C⊆VC\subseteq VC⊆V contain the seed node vvv (i.e. s=evs=e_vs=ev and v∈Cv\in Cv∈C). Define the complement Cˉ:=V∖C\bar C:=V\setminus CCˉ:=V∖C. Partition QQQ and bbb into blocks (C,Cˉ)(C,\bar C)(C,Cˉ).

Assume there exists a vector x^∈Rn\hat x\in\mathbb{R}^nx^∈Rn supported on CCC (i.e. x^Cˉ=0\hat x_{\bar C}=0x^Cˉ=0) such that:
# Stationarity on CCC with a known sign pattern. There is a sign vector σC∈{−1,+1}∣C∣\sigma_C\in\{-1,+1\}^{|C|}σC∈{−1,+1}∣C∣ such that QCCx^C−bC+λC∘σC=0.Q_{CC}\hat x_C - b_C + \lambda_C\circ \sigma_C = 0.QCCx^C−bC+λC∘σC=0. (In the PPR setting one often expects x^C≥0\hat x_C\ge 0x^C≥0, so σC=1\sigma_C=\mathbf{1}σC=1.)
# Strict dual feasibility off CCC (a uniform slack η∈(0,1)\eta\in(0,1)η∈(0,1)). For all i∈Cˉi\in\bar Ci∈Cˉ, ∣(QCˉCx^C−bCˉ)i∣≤(1−η)λi.|(Q_{\bar C C}\hat x_C - b_{\bar C})_i| \le (1-\eta)\lambda_i.∣(QCˉCx^C−bCˉ)i∣≤(1−η)λi.

Then:
* x^\hat xx^ is the unique global minimizer: x^=x⋆\hat x=x^\starx^=x⋆, hence \supp(x⋆)⊆C\supp(x^\star)\subseteq C\supp(x⋆)⊆C.
* The margin satisfies the explicit lower bound γ  ≥  ηmin⁡i∈Cˉλi  =  η ραmin⁡i∈Cˉdi.\gamma \;\ge\; \eta\min_{i\in\bar C}\lambda_i \;=\; \eta\,\rho\alpha\min_{i\in\bar C}\sqrt{d_i}.γ≥ηi∈Cˉminλi=ηραi∈Cˉmindi.

In particular, if the outside minimum degree satisfies

min⁡i∈Cˉdi  ≥  c0α2ρfor some c0>0,\min_{i\in\bar C} d_i \;\ge\; \frac{c_0}{\alpha^2\rho}
\quad\text{for some }c_0>0,i∈Cˉmindi≥α2ρc0for some c0>0,
then

γ  ≥  η ραc0α2ρ  =  ηc0 ρ  =  Ω(ρ).\gamma \;\ge\; \eta\,\rho\alpha\sqrt{\frac{c_0}{\alpha^2\rho}}
\;=\;\eta\sqrt{c_0}\,\sqrt{\rho}
\;=\;\Omega(\sqrt{\rho}).γ≥ηραα2ρc0=ηc0ρ=Ω(ρ).
Why this theorem is the right abstraction.
It’s the exact analogue of a standard Lasso “primal–dual witness” / strict dual feasibility condition: once you have strict feasibility on the complement, the support is stable and you get a quantitative margin.