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

===== Let C⊆VC\subseteq VC⊆V contain the seed vvv, and suppose: =====

(A) Internal connectivity / conditioning on CCC.
The principal submatrix QCCQ_{CC}QCC satisfies

λmin⁡(QCC)≥μC,\lambda_{\min}(Q_{CC}) \ge \mu_C,λmin(QCC)≥μC,
for some μC>0\mu_C>0μC>0.
A natural way to ensure this is to assume CCC has a positive Dirichlet spectral gap for the normalized Laplacian (or, more informally, CCC is an internal expander); then μC\mu_CμC is bounded below by α+1−α2λmin⁡(LCD)\alpha+\tfrac{1-\alpha}{2}\lambda_{\min}(L^D_C)α+21−αλmin(LCD).

(B) Low external conductance and “dispersed boundary.”
Let cut(C,Cˉ)\mathrm{cut}(C,\bar C)cut(C,Cˉ) be the number of edges crossing (C,Cˉ)(C,\bar C)(C,Cˉ). Assume

ϕ(C)=cut(C,Cˉ)vol(C)≤coutρandmax⁡i∈Cˉdeg⁡C(i)≤b,\phi(C)=\frac{\mathrm{cut}(C,\bar C)}{\mathrm{vol}(C)} \le c_{\mathrm{out}}\rho
\quad\text{and}\quad
\max_{i\in\bar C}\deg_C(i)\le b,ϕ(C)=vol(C)cut(C,Cˉ)≤coutρandi∈CˉmaxdegC(i)≤b,
where deg⁡C(i)\deg_C(i)degC(i) is the number of neighbors of iii inside CCC, and bbb is a modest constant.
(Heuristically: not only is the cut small, it’s not concentrated onto a single outside low-degree node.)

(C) Degree lower bounds.

dj≥dC,min⁡  ∀j∈C,di≥dout  ∀i∈Cˉ,dout≥c0α2ρ.d_j \ge d_{C,\min}\ \ \forall j\in C,
\qquad
d_i \ge d_{\mathrm{out}}\ \ \forall i\in\bar C,
\qquad
d_{\mathrm{out}} \ge \frac{c_0}{\alpha^2\rho}.dj≥dC,min  ∀j∈C,di≥dout  ∀i∈Cˉ,dout≥α2ρc0.
(D) Choose ρ\rhoρ so the restricted solution is nonnegative (and not at a breakpoint).
Define the candidate supported on CCC (assuming positivity)

x^C:=QCC−1(bC−λC),x^Cˉ:=0,\hat x_C := Q_{CC}^{-1}(b_C-\lambda_C),
\qquad \hat x_{\bar C}:=0,x^C:=QCC−1(bC−λC),x^Cˉ:=0,
and assume x^C≥0\hat x_C\ge 0x^C≥0 componentwise, and that the following “boundary domination” inequality holds for some η∈(0,1)\eta\in(0,1)η∈(0,1):

1−α2bdoutdC,min⁡⏟(boundary coupling)⋅∥x^C∥2⏟(signal on C) ≤ (1−η)ραdout⏟min⁡i∈Cˉλi.\underbrace{\frac{1-\alpha}{2}\sqrt{\frac{b}{d_{\mathrm{out}}d_{C,\min}}}}_{\text{(boundary coupling)}}
\cdot
\underbrace{\|\hat x_C\|_2}_{\text{(signal on }C\text{)}}
\ \le\
(1-\eta)\underbrace{\rho\alpha\sqrt{d_{\mathrm{out}}}}_{\min_{i\in\bar C}\lambda_i}.(boundary coupling)21−αdoutdC,minb⋅(signal on C)∥x^C∥2 ≤ (1−η)mini∈Cˉλiραdout.
Then the strict dual feasibility condition holds, so x^=x⋆\hat x=x^\starx^=x⋆ and

γ≥η ραdout≥ηc0ρ.\gamma \ge \eta\,\rho\alpha\sqrt{d_{\mathrm{out}}} \ge \eta\sqrt{c_0}\sqrt{\rho}.γ≥ηραdout≥ηc0ρ.
Interpretation.
* Low conductance keeps the interaction across (C,Cˉ)(C,\bar C)(C,Cˉ) limited.
* Internal connectivity keeps QCCQ_{CC}QCC well-conditioned, preventing ∥x^C∥\|\hat x_C\|∥x^C∥ from blowing up.
* High degree outside ensures the thresholds λi=ραdi\lambda_i=\rho\alpha\sqrt{d_i}λi=ραdi are at least Θ(ρ)\Theta(\sqrt{\rho})Θ(ρ) (this is necessary for γ\gammaγ to be Θ(ρ)\Theta(\sqrt{\rho})Θ(ρ), because γ≤min⁡i∈I⋆λi\gamma\le \min_{i\in I^\star}\lambda_iγ≤mini∈I⋆λi).
* Dispersed boundary prevents a single outside vertex from having a large coupling QiCQ_{iC}QiC.