$w^{*}_{i}$

${T_{j}^{R}},j\in\{1,2\}$

$G=(V,E)$

$\displaystyle(k-k^{\prime})\times n_{L}\times|x-A|+n_{L}\times k^{\prime}% \times\big{(}|x-A|+|x-A-B|\big{)}\xrightarrow{\text{no edge weights are % changed, set }x=A}$

$E_{L}$

$\displaystyle(\underbrace{\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}% {2{n_{L}}}-i}_{\leq\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L% }}}})(\underbrace{{n^{2}_{L}}\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}(2j-{% \mathcal{R}})}_{\leq{n^{2}_{L}}\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}({% \mathcal{R}})})+j\underbrace{({n^{2}_{R}}(1-{\mathcal{R}})+{n_{L}}{n_{R}}({% \mathcal{L}}-2\times\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{% L}}})}_{=0})$

$x=\sum_{j=0}^{i}w_{j}$

$y=C+\epsilon_{2}$

$E$

$z$

$\overline{E_{m}}=E-E_{m}$

$\displaystyle\xrightarrow{0\leq x

$\epsilon_{1}+\epsilon_{2}\leq 1$

$C_{L}$

$(v,u)$

$w(e)$

$-L_{i}\times S_{RU}$

$V_{R}=\{w|(v_{2},w)\in E^{\prime}\},{\mathcal{R}}=|V_{R}|$

$v^{\prime}_{4}$

$S_{L}\xleftarrow{}\sum_{\forall e_{i}\in E_{L}}L_{i}$

$\alpha_{2}\geq B$

$w^{\prime}(e_{i})=w(e_{i})+\epsilon_{i}$

$e^{*}=(v_{n_{1}+1},v_{n_{1}+2})$

$i$

$j=0$

$w_{2}$

$\displaystyle=\mathcal{E}(M_{0})+\epsilon_{1}\times\sum_{e_{i}\in E_{L}}L_{i}% \times w^{*}\times(S_{L}-L_{i}-S_{R})+\epsilon_{2}\times\sum_{e_{i}\in E_{R}}R% _{i}\times w^{*}\times(S_{R}-R_{i}-S_{L})$

$S_{RU}\geq S_{LU}$

$c^{\prime}_{i}=c_{i}-\epsilon$

$n_{L}=n-(k+k^{\prime})$

$T_{i}^{R}$

$|\Delta E|=n_{L}(|x-A|+|x-A-B|)+n_{R}(|y-C|+|y-B-C|)+n_{L}n_{R}|x+y-A-B-C|=n_{% L}\alpha_{1}+n_{R}\alpha_{2}+n_{L}n_{R}|x+y-A-B-C|$

$\epsilon_{n_{1}}=w^{*}$

$\displaystyle\geq$

$|\Delta E|=\sum_{u\in V_{m},v\in\overline{V_{m}},\text{ or }u,v\in\overline{V_% {m}},u\neq v}\left|d_{G}(u,v)-d_{G^{\prime}}(u,v)\right|$

$n_{2}+1$

$7-21\leq 2=L_{1}$

$z=w^{*}$

$x=\sum_{j=0}^{i+1}w_{j}$

$M^{\prime}$

$\displaystyle\Delta=$

$u\in G_{1},v\in G_{2}$

$\displaystyle|\Delta E|^{\prime}=$

$x+y$

$u_{1}\in T_{1}^{L},u_{2}\in T_{1}^{R}$

$S_{L}=S_{LU}+S_{LM}$

$S_{R}=S_{RU}$

$\mathcal{E}_{R}$

$w:E\xrightarrow[]{}\mathbb{R}_{\geq 0}$

$c_{j}=\epsilon_{2}=0.5$

$0.1$

$e^{*}=(v_{2},v_{3})$

${\mathcal{L}}\choose 2$

$\mathcal{E}(M^{\prime\prime})=\mathcal{E}(M^{\prime})+\Delta_{2}({\text{MARK\_% LEFT}})=\underbrace{\mathcal{E}(M^{\prime})+(c_{1})\times X}_{=\mathcal{E}(M^{% \prime})+\Delta_{1}({\text{MARK\_LEFT}})=\mathcal{E}(M)}+(1-c_{1})\times X<% \mathcal{E}(M)$

$|w^{*}|-\left|w^{*}-\sum_{k=n_{1}+1}^{j-2}\epsilon_{k}\right|$

$V_{R}=\{v_{5},v_{6}\}$

$u\in V_{L}$

$L_{i}$

$0

$\overline{V_{m}}=V-\{v_{1},v_{2}\}$

$(n-2k)w^{*}_{k}$

$u_{1}\in T_{1}^{R},u_{2}\in T_{2}^{R}$

$\alpha_{1}\geq B$

$e$

$L_{1}\times L_{2}\times 2w^{*}$

$\mathcal{E}(M_{L}^{*})=\underbrace{\mathcal{E}(M_{0})}_{\text{The error % associated with the empty marking}}+\underbrace{\sum_{e_{i}\in E_{L}}L_{i}% \times w^{*}\times(S_{L}-L_{i}-S_{R})}_{\text{The sum of all }\Delta({\text{% MARK\_LEFT}})\text{'s that transform }M_{0}\text{ into }M_{L}^{*}}$

$\mathcal{E}=\mathcal{E}_{L}+\mathcal{E}_{R}+\mathcal{E}_{LR}$

$\epsilon_{i}\neq 0$

$\epsilon\times w^{*}$

$\pi:\mathcal{G}\xrightarrow{}\mathbb{R}$

$j$

$y=B+C$

$S_{L}-S_{R}>L_{i}$

$21-7\leq 20=R_{1}$

$\mathcal{E}(.)$

$e^{*}_{k}$

${\mathcal{L}}={\mathcal{R}}=2$

$\displaystyle(k-k^{\prime})\times n_{R}\times|y-B-C|+n_{R}\times k^{\prime}% \times\big{(}|y-C|+|y-B-C|\big{)}$

$w(e_{i})+\epsilon_{i}$

$w^{\prime}(e_{3})=w_{3}+w^{*}$

$c_{i}=\epsilon_{1}$

$0\leq\epsilon_{2}\leq w^{*}$

$x=A+B$

$T_{i}={t^{1}_{1},\dots,t^{1}_{k}}$

$X=\{x_{1},x_{2},...,x_{n}\}$

$T_{i}\sim p_{\theta}(t^{1}_{1\dots k}\mid x_{1}\dots,x_{n})$

$t^{i}_{j}*\sim p^{vote}_{\theta}(t^{i}_{j}*|B)$

$X$

$n-1$

$V(p_{\theta},T_{i})(t^{i}_{j})=1[t^{i}_{j}=t^{i}_{j}*]$

$n_{i}\in N$

$t^{n}_{1}\sim t^{n}_{k}$

$G=F(X)$

$X=\{x_{1},x_{2},...,x_{m}\}$

$t^{i-1}_{j}$

$x_{i}$

$t^{i}_{j}$

$F$

$G=(N,E)$

$\{t^{i}_{1},\dots,t^{i}_{k}\}\sim p_{\theta}(t^{i}_{1\dots k}\mid x_{1\dots n}% ,t^{i-1}_{j})$