∑Notes

World

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#1
(i) Track the perturbation of numerator and denominator separately.
$|\Delta_{\mathrm{cut}}| = \Bigl|\sum_{i\in F}\sum_{j\notin F}(W'_t(i,j)-W_t(i,j))\Bigr| \le |F|\cdot|\bar F|\cdot\|\delta W\|_\infty.$
#2
$$ |\Delta{\mathrm{vol}}| = \Bigl|\sum{i\in F}\sum{j\in V}(W't(i,j)-Wt(i,j))\Bigr| \le |F|\cdot|V|\cdot\|\delta W\|\infty.
$The fractional perturbation of $\phi_t(F)=\mathrm{cut}/\mathrm{vol}$:$
#3
The fractional perturbation of : $$ \phi't(F)-\phit(F) = \frac{\mathrm{cut}'\cdot\mathrm{vol}-\mathrm{cut}\cdot\mathrm{vol}'}{\mathrm{vol}'\cdot\mathrm{vol}}.
$The numerator is bounded by $\|\delta W\|_\infty\cdot|V|^2\cdot C\cdot\mathrm{vol}_t(F)$. The denominator $\ge\mathrm{vol}_t(F)^2/2$ for small perturbations. Hence $|\phi'-\phi|\le C_1\cdot\|\delta W\|_\infty\cdot|V|^2/\mathrm{vol}_t(F)$, where:$