| Network | W = (wij)N×N i, j ∈ [1, N] |
| Refactoring |
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w↼1=descend (w11,w21,⋯,wN1)Tw↼2=descend (w12,w22,⋯,wN2)T⋯w↼N=descend (w1N,w2N,⋯,wNN)TW↼=(w↼1,w↼2,⋯,w↼N)=(w↼sj)N×Ns,j∈[1,N]
\matrix{ {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over w} }_1} = descend\,{{\left( {{w_{11}},{w_{21}}, \cdots ,{w_{N1}}} \right)}^T}} \cr {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over w} }_2} = descend\,{{\left( {{w_{12}},{w_{22}}, \cdots ,{w_{N2}}} \right)}^T}} \cr \cdots \cr {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over w} }_N} = descend\,{{\left( {{w_{1N}},{w_{2N}}, \cdots ,{w_{NN}}} \right)}^N}} \cr {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}}\over W} = \left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}}\over w} }_1},{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}}\over w} }_2}, \cdots ,{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}}\over w} }_N}} \right) = {{\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}}\over w} }_{sj}}} \right)}_{N \times N}}} \cr {s,j \in \left[ {1,N} \right]} \cr }
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w⇀1=descend (w11,w12,⋯,w1N)w⇀2=descend (w21,w22,⋯,w2N)⋯w⇀N=descend (wN1,wN2,⋯,wNN)W⇀←=(w⇀1w⇀2…w⇀N)=(w⇀it)N×Ni,t∈[ 1,N ]
\matrix{ {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over w} }_1} = descend\,\left( {{w_{11}},{w_{12}}, \cdots ,{w_{1N}}} \right)} \cr {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over w} }_2} = descend\,\left( {{w_{21}},{w_{22}}, \cdots ,{w_{2N}}} \right)} \cr \cdots \cr {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over w} }_N} = descend\,\left( {{w_{N1}},{w_{N2}}, \cdots ,{w_{NN}}} \right)} \cr {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over W} \overleftarrow {} = \left( {\matrix{ {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over w} }_1}} \cr {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over w} }_2}} \cr \ldots \cr {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over w} }_N}} \cr } } \right) = {{\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over w} }_{it}}} \right)}_{N \times N}}} \cr {i,t \in \left[ {1,N} \right]} \cr }
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| Conditions |
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∀a1,a2,⋯,aj∈[1,N]{∑s=1ajw↼sj∑s=1Nw↼sj≥1−ajN∑s=1aj−1w↼sj∑s=1Nw↼sj<1−aj−1N
\matrix{ {\forall {a_1},{a_2}, \cdots ,{a_j} \in \left[ {1,N} \right]} \cr {\left\{ {\matrix{ {{{\sum\nolimits_{s = 1}^{{a_j}} {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}}\over w} }_{sj}}} } \over {\sum\nolimits_{s = 1}^N {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}}\over w} }_{sj}}} }} \ge 1 - {{{a_j}} \over N}} \cr {{{\sum\nolimits_{s = 1}^{{a_j} - 1} {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}}\over w} }_{sj}}} } \over {\sum\nolimits_{s = 1}^N {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}}\over w} }_{sj}}} }} < 1 - {{{a_j} - 1} \over N}} \cr } } \right.} \cr }
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∀b1,b2,⋯,bj∈[1,N]{∑t=1biw⇀it∑t=1Nw⇀it≥1−biN∑t=1bi−1w⇀it∑t=1Nw⇀it<1−bi−1N
\matrix{ {\forall {b_1},{b_2}, \cdots ,{b_j} \in \left[ {1,N} \right]} \cr {\left\{ {\matrix{ {{{\sum\nolimits_{t = 1}^{{b_i}} {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over w} }_{it}}} } \over {\sum\nolimits_{t = 1}^N {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over w} }_{it}}} }} \ge 1 - {{{b_i}} \over N}} \cr {{{\sum\nolimits_{t = 1}^{{b_i} - 1} {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over w} }_{it}}} } \over {\sum\nolimits_{t = 1}^N {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over w} }_{it}}} }} < 1 - {{{b_i} - 1} \over N}} \cr } } \right.} \cr }
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| Definition |
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XIjB=ajNXIB=(XIjB)N×1
\matrix{ {XI_j^B = {{{a_j}} \over N}} \cr {X{I^B} = {{\left( {XI_j^B} \right)}_{N \times 1}}} \cr }
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XIiF=biNXIF=(XIiF)N×1
\matrix{ {XI_i^F = {{{b_i}} \over N}} \cr {X{I^F} = {{\left( {XI_i^F} \right)}_{N \times 1}}} \cr }
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| Pruning |
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w←ij={wij,wij=w←sj and s ≤aj0,otherwise
{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftarrow$}}\over w} _{ij}} = \left\{ {\matrix{ {{w_{ij}}} & , & {{w_{ij}} = {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}}\over w} }_{sj}}\,and\,s\, \le {a_j}} \cr 0 & , & {otherwise} \cr } } \right.
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w→ij={wij,wij=w⇀it and t ≤bi0,otherwise
{\vec w_{ij}} = \left\{ {\matrix{ {{w_{ij}}} & , & {{w_{ij}} = {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over w} }_{it}}\,and\,t\, \le {b_i}} \cr 0 & , & {otherwise} \cr } } \right.
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| Merging |
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w↔ij={wij,0w←ij≠0 or w→it ≠0,otherwise
{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over w} _{ij}} = \left\{ {\matrix{ {{w_{ij}}} & {,0} & {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftarrow$}}\over w} }_{ij}} \ne 0\,or\,{{\vec w}_{it}}\, \ne } \cr 0 & , & {otherwise} \cr } } \right.
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| Result |
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W↔=(w↔ij)N×N
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over W} = {\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over w} }_{ij}}} \right)_{N \times N}}
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