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

=== Assistant: Starting from the definition of ΔC\DeltaCΔC in that section specialized to the ℓ1\ell1ℓ1-regularized PPR objective 3 o… ===
Starting from the definition of Δ(C)\Delta(C)Δ(C) in that section (specialized to the ℓ1\ell_1ℓ1-regularized PPR objective (3) open-problem-fountoulakis22a

),

Δ(C)=1−α2μC ϕ(C) vol(C)dout din (αdv+ραvol(C)),\Delta(C)
=
\frac{1-\alpha}{2\mu_C}\,
\sqrt{\frac{\phi(C)\,\mathrm{vol}(C)}{d_{\mathrm{out}}\,d_{\mathrm{in}}}}\,
\left(\frac{\alpha}{\sqrt{d_v}}+\rho\alpha\sqrt{\mathrm{vol}(C)}\right),Δ(C)=2μC1−αdoutdinϕ(C)vol(C)(dvα+ραvol(C)),
you want to solve the inequality

Δ(C)≤(1−η)ραdout(η∈(0,1)).\Delta(C)\le (1-\eta)\rho\alpha\sqrt{d_{\mathrm{out}}}
\qquad(\eta\in(0,1)).Δ(C)≤(1−η)ραdout(η∈(0,1)).
===== Factor out α\alphaα inside the parentheses and cancel α>0\alpha>0α>0 on both sides: =====

1−α2μC ϕ(C) vol(C)dout din (1dv+ρvol(C))  ≤  (1−η)ρdout.\frac{1-\alpha}{2\mu_C}\,
\sqrt{\frac{\phi(C)\,\mathrm{vol}(C)}{d_{\mathrm{out}}\,d_{\mathrm{in}}}}\,
\left(\frac{1}{\sqrt{d_v}}+\rho\sqrt{\mathrm{vol}(C)}\right)
\;\le\;
(1-\eta)\rho\sqrt{d_{\mathrm{out}}}.2μC1−αdoutdinϕ(C)vol(C)(dv1+ρvol(C))≤(1−η)ρdout.
Rearrange to isolate the square root:

ϕ(C) vol(C)dout din  ≤  2μC1−α (1−η)ρdout1dv+ρvol(C).\sqrt{\frac{\phi(C)\,\mathrm{vol}(C)}{d_{\mathrm{out}}\,d_{\mathrm{in}}}}
\;\le\;
\frac{2\mu_C}{1-\alpha}\,
\frac{(1-\eta)\rho\sqrt{d_{\mathrm{out}}}}{\frac{1}{\sqrt{d_v}}+\rho\sqrt{\mathrm{vol}(C)}}.doutdinϕ(C)vol(C)≤1−α2μCdv1+ρvol(C)(1−η)ρdout.
Square both sides and solve for ϕ(C)\phi(C)ϕ(C):

ϕ(C)  ≤  ϕmax⁡(C):=4μC2(1−η)2ρ2 dout2 din(1−α)2 vol(C) (1dv+ρvol(C))2.\boxed{
\phi(C)\;\le\;\phi_{\max}(C)
:=
\frac{4\mu_C^2(1-\eta)^2\rho^2\,d_{\mathrm{out}}^2\,d_{\mathrm{in}}}
{(1-\alpha)^2\,\mathrm{vol}(C)\,\Bigl(\frac{1}{\sqrt{d_v}}+\rho\sqrt{\mathrm{vol}(C)}\Bigr)^2}.
}ϕ(C)≤ϕmax(C):=(1−α)2vol(C)(dv1+ρvol(C))24μC2(1−η)2ρ2dout2din.
That is the exact “how small should conductance be?” threshold implied by your condition.

==== The denominator contains (1dv+ρvol(C))2\Bigl(\frac{1}{\sqrt{d_v}}+\rho\sqrt{\mathrm{vol}(C)}\Bigr)^2(dv1+ρvol(C))2. Depending on which term dominates, you get simpler scaling: ====

===== If ρvol(C)≪1/dv\rho\sqrt{\mathrm{vol}(C)} \ll 1/\sqrt{d_v}ρvol(C)≪1/dv (equivalently vol(C)≪1/(ρ2dv)\mathrm{vol}(C)\ll 1/(\rho^2 d_v)vol(C)≪1/(ρ2dv)), then =====

(1dv+ρvol(C))2≈1dv,\Bigl(\tfrac{1}{\sqrt{d_v}}+\rho\sqrt{\mathrm{vol}(C)}\Bigr)^2 \approx \frac{1}{d_v},(dv1+ρvol(C))2≈dv1,
so

ϕmax⁡(C)  ≈  4μC2(1−η)2ρ2 dout2 din dv(1−α)2 vol(C).\phi_{\max}(C)\;\approx\;
\frac{4\mu_C^2(1-\eta)^2\rho^2\,d_{\mathrm{out}}^2\,d_{\mathrm{in}}\,d_v}{(1-\alpha)^2\,\mathrm{vol}(C)}.ϕmax(C)≈(1−α)2vol(C)4μC2(1−η)2ρ2dout2dindv.
===== If ρvol(C)≫1/dv\rho\sqrt{\mathrm{vol}(C)} \gg 1/\sqrt{d_v}ρvol(C)≫1/dv, then =====

(1dv+ρvol(C))2≈ρ2vol(C),\Bigl(\tfrac{1}{\sqrt{d_v}}+\rho\sqrt{\mathrm{vol}(C)}\Bigr)^2 \approx \rho^2\mathrm{vol}(C),(dv1+ρvol(C))2≈ρ2vol(C),
so

ϕmax⁡(C)  ≈  4μC2(1−η)2 dout2 din(1−α)2 vol(C)2.\phi_{\max}(C)\;\approx\;
\frac{4\mu_C^2(1-\eta)^2\,d_{\mathrm{out}}^2\,d_{\mathrm{in}}}{(1-\alpha)^2\,\mathrm{vol}(C)^2}.ϕmax(C)≈(1−α)2vol(C)24μC2(1−η)2dout2din.

==== In the corollary regime you were already using (typical local clustering assumptions): ====
* vol(C)≲V0/ρ\mathrm{vol}(C)\lesssim V_0/\rhovol(C)≲V0/ρ,
* din,dv≳d0d_{\mathrm{in}},d_v\gtrsim d_0din,dv≳d0 (bounded below),
* μC≳μ0\mu_C\gtrsim \mu_0μC≳μ0 (internally well-conditioned),
* dout≳c0/(α2ρ)d_{\mathrm{out}}\gtrsim c_0/(\alpha^2\rho)dout≳c0/(α2ρ) (outside degrees large enough so λi=ραdi\lambda_i=\rho\alpha\sqrt{d_i}λi=ραdi is Θ(ρ)\Theta(\sqrt{\rho})Θ(ρ)),

then ϕmax⁡(C)\phi_{\max}(C)ϕmax(C) simplifies to a constant multiple of ρ\rhoρ:

ϕmax⁡(C)  ≳  μ02(1−η)2 c02 d02α4(1−α)2 V0⏟constant depending on parameters⋅ρ,\phi_{\max}(C)\;\gtrsim\; \underbrace{
\frac{\mu_0^2(1-\eta)^2\,c_0^2\,d_0^2}{\alpha^4(1-\alpha)^2\,V_0}
}_{\text{constant depending on parameters}}
\cdot \rho,ϕmax(C)≳constant depending on parametersα4(1−α)2V0μ02(1−η)2c02d02⋅ρ,
(up to the same mild “ρ\rhoρ small enough so the seed term dominates” caveat you used earlier).

So in that common regime, a clean sufficient statement is:
* Conductance needs to be on the order of ρ\rhoρ (i.e. ϕ(C)≤c ρ\phi(C)\le c\,\rhoϕ(C)≤cρ for a small-enough constant ccc depending on α,μC,dout,din,vol(C),dv,η\alpha,\mu_C,d_{\mathrm{out}},d_{\mathrm{in}},\mathrm{vol}(C),d_v,\etaα,μC,dout,din,vol(C),dv,η).

If you want, I can also rewrite ϕmax⁡(C)\phi_{\max}(C)ϕmax(C) in terms of the cut size cut(C,Cˉ)\mathrm{cut}(C,\bar C)cut(C,Cˉ) instead of ϕ(C)\phi(C)ϕ(C), which sometimes reads more directly as a “boundary must be at most ___ edges” condition.