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1. #1

   (i) Track the perturbation of numerator and denominator separately.

   $$|{\Delta }_{\mathrm{cut}}|=|\sum _{i\in F}\sum _{j\notin F}(W_t^{\prime }(i,j)-W_t(i,j))|\le |F|\cdot |\bar{F}|\cdot \Vert \delta W{\Vert }_{\infty }.$$
2. #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. #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: