Example: Let cluster feature be

\[SSE(C_{j})=\sum_{i=1}^{n} w_{ij}^{p}dist(o_{i},c_{j})^2\]
\[SSE(o_{j})=\sum_{j=1}^{k} w_{ij}^{p}dist(o_{i},c_{j})^2\]




\[P(oC_{j})=w_{j}f_{j}(o)\]
\[P(oC)=\sum_{j=1}^{k}w_{j}f_{j}(o)\]
\[P(DC)=\prod_{i=1}^{n}P(o_{i}C)=\prod_{i=1}^{n}\sum_{j=1}^{k}w_{j}f_{j}(o_{i})\]
\[P(o_{i}\theta)=\sum_{j=1}^{k}w_{j}P_{j}(o_{i}\theta_{j} )\]
\[P(O\theta)=\prod_{i=1}^{n}\sum_{j=1}^{k}w_{j}P_{j}(o_{i}\theta_{j} )\]
\[P(o_{i}\theta_{j})=\frac{1}{\sqrt{2\pi}\sigma_{j} }e^{\frac{(o_{i}\mu_{j})^2 }{2\sigma^{2} }}\]
\[P(o_{i}\theta)=\sum_{j=1}^{k}\frac{1}{\sqrt{2\pi}\sigma_{j} }e^{\frac{(o_{i}\mu_{j})^2 }{2\sigma^{2} }}\]
\[P(O\theta)=\prod_{i=1}^{n}\sum_{j=1}^{k}\frac{1}{\sqrt{2\pi}\sigma_{j} }e^{\frac{(o_{i}\mu_{j})^2 }{2\sigma^{2} }}\]
\[ c_{j}=\frac{\sum_{eachpoint_{o}}w_{o,c_{j}}^{2}o }{\sum_{eachpoint_{o}}w_{o,c_{j}}^{2}} \]
\[c_{1}=(\frac{1^{2}\times 3+ 0^{2}\times4+ 0.48^{2}\times9+ 0.42^{2}\times14+ 0.41^{2}\times18+ 0.47^{2}\times21}{1^{2}+0^{2}+0.48^{2}+0.42^{2}+0.41^{2}+0.47^{2}},\frac{1^{2}\times3+ 0^{2}\times10+ 0.48^{2}\times6+ 0.42^{2}\times8+ 0.41^{2}\times11+ 0.47^{2}\times7}{1^{2}+0^{2}+0.48^{2}+0.42^{2}+0.41^{2}+0.47^{2}}) =(8.47,5.12) \]
\[P(\theta_{j} o_{i},\theta)=\frac{P(o_{i}\theta_{j})}{\sum_{l=1}^{k}P(o_{i} \theta_{l})}\]
\[\mu_{j}=\sum_{i=1}^{n}o_{i}\frac{P(\theta_{j} o_{i},\theta)}{\sum_{l=1}^{n}P(\theta_{j} o_{l},\theta)}=\frac{{\sum_{i=1}^{n}o_{i}P(\theta_{j} o_{i},\theta)}}{{\sum_{i=1}^{n}P(\theta_{j} o_{i},\theta)}}\]
\[\sigma_{j}=\sqrt{\frac{\sum_{i=1}^{n}P(\theta_{j} o_{i},\theta)(o_{i}\mu_ {j})^{2}}{\sum_{i=1}^{n}P(\theta_{j} o_{i},\theta)}}\]
\[dist(Ada,Bob)=dist(Bob,Cathy)=dist(Ada,Cathy)=\sqrt{2}\]
\[e_{iJ}=\frac{1}{J} \sum_{j\epsilon J}e_{ij}\]
\[e_{Ij}=\frac{1}{I} \sum_{i\epsilon I}e_{ij}\]
\[e_{IJ}=\frac{1}{IJ} \sum_{i\epsilon I,j\epsilon J}e_{ij}=\frac{1}{I} \sum_{i\epsilon I}e_{iJ}=\frac{1}{J} \sum_{j\epsilon J}e_{Ij}\]
\[H(I\times J)=\frac{1}{IJ} \sum_{i\epsilon I,j\epsilon J}(e_{ij}e_{iJ}e_{Ij}+e_{IJ})^{2}\]
\[d(i)=\frac{1}{J}\sum_{j\epsilon J}(e_{ij}e_{iJ}e_{Ij}+e_{IJ})^{2}\]
\[d(j)=\frac{1}{I}\sum_{i\epsilon I}(e_{ij}e_{iJ}e_{Ij}+e_{IJ})^{2}\]
\[pscore \begin{pmatrix} e_{i1j1} &e_{i1j2} \\ e_{i2j1}&e_{i2j2} \end{pmatrix} =(e_{i1j1}e_{i2j1})(e_{i1j2}e_{i2j2})\]
\[\frac{d_{xa}/d_{ya}}{d_{xb}/d_{yb}}<\delta \]
\[D_{ii}=\sum_{j=1}^{n}W_{ij}\]
\[W_{ij}=e^{\frac{dist(o_{i},o_{j})}{\sigma^{2}}}\]
\[A=D^{\frac{1}{2}}WD^{\frac{1}{2}}\]
\[s(u,v)=\frac{C}{I(u)I(v)}\sum_{x\epsilon I(u)}\sum_{y\epsilon I(v) }s(x,y) \]
\[ s_{0}(u,v)=\left\{\begin{matrix}0, if u\neq v \\ 1,if u=v \end{matrix}\right. \]
\[d(u,v)=\sum_{t:(u\rightarrow v)}P[t]l(t)\]
\[m(u,v)=\sum_{t:(u,v)\rightarrow (x,x)}P[t]l(t)\]
\[p(u,v)=\sum_{t:(u,v)\rightarrow (x,x)}P[t]C^{l(t)}\]
P[t] is the probability of the tour
φ=the size of the cut / min{S,T}
\[Q=\sum_{i=1}^{k}(\frac{l_{i}}{E}(\frac{d_{i}}{2E})^{2})\]
li: # edges between vertices in the ith cluster
di: the sum of the degrees of the vertices in the ith cluster
An Example Network
\[\sigma(v,w)=\frac{\tau (v)\bigcap \tau (w)}{\sqrt{\tau (v) \tau (w)}} \] 
\[ N_{\varepsilon }(v)=\left \{ w\epsilon \tau (v)\sigma (v,w)\geq \varepsilon \right \} \]
\[CORE_{\varepsilon,\mu }(v)\Leftrightarrow N_{\varepsilon } (v)\geq \mu \]
\[DirRECH_{\varepsilon,\mu }(v,w)\Leftrightarrow CORE_{\varepsilon,\mu }(v)\wedge w\epsilon N_{\varepsilon } (v)\]
\[CONNECT_{\varepsilon,\mu }(v,w)\Leftrightarrow \exists u\epsilon V:RECH_{\varepsilon,\mu }(u,v)\wedge RECH_{\varepsilon,\mu }(u,w)\]
\[\forall v,w\epsilon C:CONNECT_{\varepsilon ,\mu}(v,w)\]
\[\forall v,w\epsilon V:v\epsilon C\wedge REACH_{\varepsilon ,\mu}(v,w)\Rightarrow w\epsilon C\]