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We are grateful to Francesc Comellas for letting us \ know that the mentioned functions were not working properly in version 1.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Packages", "Section"], Cell[BoxData[ \(\(\(\(Needs["\"];\)\ (*\ For\ the\ analysis\ of\ \(\(graphs\)\(.\)\)\ *) \[IndentingNewLine] \(Needs["\"];\)\ (*\ For\ the\ \(\(Histograms\)\(.\)\)\ *) \[IndentingNewLine] \(Needs["\"];\)\)\(\ \)\( (*\ For\ Frequencies, \ used\ in\ the\ function\ \(\(ReachabilityCloseness\)\(.\)\)\ *) \ \)\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Functions", "Section"], Cell[BoxData[{ \(\(Off[General::spell];\)\ \[IndentingNewLine]\[IndentingNewLine] (*\ This\ function\ returns\ the\ number\ of\ reciprocated\ links\ in\ a\ \ simple\ \(\(network\)\(.\)\)\ *) \), "\[IndentingNewLine]", \(\(ReciprocatedEdges[g_Graph] := Module[{adjMatrix = ToAdjacencyMatrix[g]}, Count[adjMatrix + Transpose[adjMatrix], 2, {2}]];\)\[IndentingNewLine]\[IndentingNewLine] (*\ A\ clique\ is\ a\ subset\ of\ the\ vertices\ such\ that\ every\ pair\ \ of\ vertices\ in\ the\ subset\ is\ connected\ by\ an\ edge . \ This\ function\ returns\ a\ clique\ of\ maximum\ size\ allowing\ \ for\ paths\ of\ any\ length\ no\ greater\ than\ n, \ i . e . \ an\ n - clique\ *) \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(MaximumNClique[g_Graph, n_Integer] := MaximumClique[ RemoveSelfLoops[ RemoveMultipleEdges[ GraphPower[g, n]]]];\)\[IndentingNewLine]\[IndentingNewLine]\ \[IndentingNewLine] (*\ Returns\ true\ if\ c\ is\ an\ n - clique, \ i . e . \ allowing\ for\ paths\ of\ any\ length\ no\ greater\ than\ n . \ False\ \(\(otherwise\)\(.\)\)\ *) \[IndentingNewLine]\), "\ \[IndentingNewLine]", \(\(NCliqueQ[g_Graph, n_Integer, c_List] := CliqueQ[RemoveSelfLoops[RemoveMultipleEdges[GraphPower[g, n]]], c];\)\[IndentingNewLine]\[IndentingNewLine] (*\ An\ n - clan\ is\ an\ n - clique\ where\ all\ ties\ among\ members\ of\ the\ n - clique\ occur\ through\ members\ of\ the\ n - clique . \ This\ function\ returns\ a\ list\ of\ all\ the\ n - clans\ of\ maximum\ size, \ i . e . \ allowing\ for\ paths\ of\ any\ length\ no\ greater\ than\ \ \(\(n\)\(.\)\)\ *) \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(MaximumNClans[g_Graph, n_Integer] := Module[{listOfDegrees, allSuitableNodes = Range[V[g]], clans = {}}, \[IndentingNewLine]\[IndentingNewLine] (*\ A\ clan\ is, \ in\ particular, \ a\ clique, \ so\ find\ out\ the\ maximum\ possible\ size\ of\ the\ \ \(\(clan\)\(.\)\)\ *) \[IndentingNewLine]maxSizeOfClan = Length[MaximumNClique[g, n]]; \[IndentingNewLine]\[IndentingNewLine] (*\ There\ is\ no\ need\ to\ consider\ the\ nodes\ of\ degree\ \ maxSizeOfClan - 2\ or\ less\ \((in\ the\ powered\ graph)\)\ as\ they\ cannot\ \ be\ part\ of\ an\ n - clique\ or\ an\ n - \(\(clan\)\(.\)\)\ \ *) \[IndentingNewLine]listOfDegrees = Degrees[RemoveSelfLoops[ RemoveMultipleEdges[ GraphPower[g, n]]]]; \[IndentingNewLine]allSuitableNodes = Select[allSuitableNodes, \ \((listOfDegrees[\([#]\)] \ \[GreaterEqual] \ maxSizeOfClan - 1)\) &]; \[IndentingNewLine]\[IndentingNewLine]While[ Length[clans] \[Equal] 0\ && \ maxSizeOfClan \[GreaterEqual] \ 1, \[IndentingNewLine]possibleClans = Subsets[allSuitableNodes, {maxSizeOfClan}]; \ \[IndentingNewLine]clans = Select[possibleClans, CompleteQ[ RemoveSelfLoops[ RemoveMultipleEdges[ GraphPower[InduceSubgraph[g, #], n]]]] &]; \[IndentingNewLine]\(maxSizeOfClan--\);\ \[IndentingNewLine]]; \[IndentingNewLine]\[IndentingNewLine]clans\ \[IndentingNewLine]];\)\[IndentingNewLine]\[IndentingNewLine]\ \[IndentingNewLine] (*\ Returns\ true\ if\ c\ is\ an\ n - clan, \ i . e . \ allowing\ for\ paths\ of\ any\ length\ no\ greater\ than\ n . \ False\ \(\(otherwise\)\(.\)\)\ *) \[IndentingNewLine]\), "\ \[IndentingNewLine]", \(\(NClanQ[g_Graph, n_Integer, c_List] := CompleteQ[ RemoveSelfLoops[ RemoveMultipleEdges[ GraphPower[InduceSubgraph[g, c], n]]]];\)\[IndentingNewLine]\[IndentingNewLine] (*\ The\ clustering\ coefficient\ of\ a\ node\ is\ the\ ratio\ of\ existing\ \ links\ connecting\ the\ node' s\ neighbours\ to\ each\ other, to\ the\ maximum\ possible\ number\ of\ such\ links . \ For\ nodes\ with\ fewer\ than\ two\ neighbours\ the\ clustering\ \ coefficient\ is\ \(\(undefined\)\(.\)\)*) \[IndentingNewLine]\), "\ \[IndentingNewLine]", \(\(Clustering[v_Integer, g_Graph] := Module[\[IndentingNewLine]{nbrs, subGraph}, \[IndentingNewLine]\[IndentingNewLine]nbrs = Complement[Neighborhood[g, v, 1], {v}]; \[IndentingNewLine]If[ Length[nbrs] < 2, Return["\"]]; \ \[IndentingNewLine]\[IndentingNewLine]subGraph = InduceSubgraph[g, nbrs]; \[IndentingNewLine]\[IndentingNewLine]If[ UndirectedQ[g], \[IndentingNewLine]Return[ 2*M[subGraph]/\((V[ subGraph]*\((V[subGraph] - 1)\))\)], \[IndentingNewLine]Return[ M[subGraph]/\((V[ subGraph]*\((V[subGraph] - 1)\))\)]\[IndentingNewLine]];\[IndentingNewLine]];\)\ \[IndentingNewLine]\n (*\ An\ n - clan\ is\ an\ n - clique\ where\ all\ ties\ among\ members\ of\ the\ n - clique\ occur\ through\ members\ of\ the\ n - clique . \ This\ function\ returns\ a\ list\ of\ all\ the\ n - clans\ of\ maximum\ size, \ i . e . \ allowing\ for\ paths\ of\ any\ length\ no\ greater\ than\ \ \(\(n\)\(.\)\)\ *) \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(MaximumNClans[g_Graph, n_Integer] := Module[{listOfDegrees, allSuitableNodes = Range[V[g]], clans = {}}, \[IndentingNewLine]\[IndentingNewLine] (*\ A\ clan\ is, \ in\ particular, \ a\ clique, \ so\ find\ out\ the\ maximum\ possible\ size\ of\ the\ \ \(\(clan\)\(.\)\)\ *) \[IndentingNewLine]maxSizeOfClan = Length[MaximumNClique[g, n]]; \[IndentingNewLine]\[IndentingNewLine] (*\ There\ is\ no\ need\ to\ consider\ the\ nodes\ of\ degree\ \ maxSizeOfClan - 2\ or\ less\ \((in\ the\ powered\ graph)\)\ as\ they\ cannot\ \ be\ part\ of\ an\ n - clique\ or\ an\ n - \(\(clan\)\(.\)\)\ \ *) \[IndentingNewLine]listOfDegrees = Degrees[RemoveSelfLoops[ RemoveMultipleEdges[ GraphPower[g, n]]]]; \[IndentingNewLine]allSuitableNodes = Select[allSuitableNodes, \ \((listOfDegrees[\([#]\)] \ \[GreaterEqual] \ maxSizeOfClan - 1)\) &]; \[IndentingNewLine]\[IndentingNewLine]While[ Length[clans] \[Equal] 0\ && \ maxSizeOfClan \[GreaterEqual] \ 1, \[IndentingNewLine]possibleClans = Subsets[allSuitableNodes, {maxSizeOfClan}]; \ \[IndentingNewLine]clans = Select[possibleClans, CompleteQ[ RemoveSelfLoops[ RemoveMultipleEdges[ GraphPower[InduceSubgraph[g, #], n]]]] &]; \[IndentingNewLine]\(maxSizeOfClan--\);\ \[IndentingNewLine]]; \[IndentingNewLine]\[IndentingNewLine]clans\ \[IndentingNewLine]];\)\[IndentingNewLine]\[IndentingNewLine] (*\ Returns\ true\ if\ c\ is\ an\ n - clan, \ i . e . \ allowing\ for\ paths\ of\ any\ length\ no\ greater\ than\ n . \ False\ \(\(otherwise\)\(.\)\)\ *) \[IndentingNewLine]\), "\ \[IndentingNewLine]", \(\(NClanQ[g_Graph, n_Integer, c_List] := CompleteQ[ RemoveSelfLoops[ RemoveMultipleEdges[ GraphPower[InduceSubgraph[g, c], n]]]];\)\[IndentingNewLine]\[IndentingNewLine]\ \[IndentingNewLine] (*\ The\ clustering\ coefficient\ of\ a\ node\ is\ the\ ratio\ of\ existing\ \ links\ connecting\ the\ node' s\ neighbours\ to\ each\ other, to\ the\ maximum\ possible\ number\ of\ such\ links . \ For\ nodes\ with\ fewer\ than\ two\ neighbours\ the\ clustering\ \ coefficient\ is\ \(\(undefined\)\(.\)\)*) \[IndentingNewLine]\), "\ \[IndentingNewLine]", \(\(Clustering[v_Integer, g_Graph] := Module[\[IndentingNewLine]{nbrs, subGraph}, \[IndentingNewLine]\[IndentingNewLine]nbrs = Complement[Neighborhood[g, v, 1], {v}]; \[IndentingNewLine]If[ Length[nbrs] < 2, Return["\"]]; \ \[IndentingNewLine]\[IndentingNewLine]subGraph = InduceSubgraph[g, nbrs]; \[IndentingNewLine]\[IndentingNewLine]If[ UndirectedQ[g], \[IndentingNewLine]Return[ 2*M[subGraph]/\((V[ subGraph]*\((V[subGraph] - 1)\))\)], \[IndentingNewLine]Return[ M[subGraph]/\((V[ subGraph]*\((V[subGraph] - 1)\))\)]\[IndentingNewLine]];\[IndentingNewLine]];\)\ \[IndentingNewLine]\[IndentingNewLine] (*\ This\ function\ returns\ a\ list\ of\ 2 - dimensional\ vectors . \ The\ first\ component\ of\ each\ vector\ is\ a\ number\ nbrs\ \ denoting\ a\ number\ of\ neighbours . \ The\ second\ component\ is\ the\ average\ number\ of\ edges\ \ between\ the\ neighbours\ of\ every\ node\ that\ has\ nbrs\ neighbours . \ It\ works\ for\ both\ directed\ and\ undirected\ \ \(\(graphs\)\(.\)\)\ *) \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(Clustering[g_Graph] := Module[\[IndentingNewLine]{listOfNeighbourhoods, listOfNeighbourhoodsGreaterThan1, listOfSubgraphs, myClusteringList, nOfNbrs, listOfDiffOutDegress, totalNumEdges}, \ \[IndentingNewLine]\[IndentingNewLine]listOfNeighbourhoods = Map[Complement[Neighborhood[g, #, 1], {#}] &, Range[V[g]]]; \ \[IndentingNewLine]listOfNeighbourhoodsGreaterThan1 = Select[listOfNeighbourhoods, \((Length[#] > 1)\) &]; \[IndentingNewLine]listOfSubgraphs = Map[InduceSubgraph[g, #] &, listOfNeighbourhoodsGreaterThan1]; \[IndentingNewLine]\ \[IndentingNewLine] (*\ myClusteringList\ is\ a\ list\ of\ two\ elements\ for\ each\ node\ \ with\ 2\ or\ more\ neighbours . \ The\ first\ element\ of\ the\ list\ is\ the\ number\ of\ \ neighbours, \ and\ the\ second\ element\ is\ the\ number\ of\ edges\ between\ \ the\ neighbours . \ It\ works\ for\ both\ directed\ and\ undirected\ \(\(graphs\)\ \(.\)\)\ *) \[IndentingNewLine]myClusteringList = Map[{V[#], M[#]} &, listOfSubgraphs]; \[IndentingNewLine]\[IndentingNewLine]nOfNbrs \ = Length[listOfNeighbourhoodsGreaterThan1]; \ \[IndentingNewLine]listOfDiffOutDegress = Union[myClusteringList[\([All, 1]\)]]; \[IndentingNewLine]totalNumEdges = Table[0, {Length[ listOfDiffOutDegress]}]; \n\[IndentingNewLine]For[i = 1, i \[LessEqual] nOfNbrs, \(i++\), \ \[IndentingNewLine]\(totalNumEdges[\([\(Position[listOfDiffOutDegress, myClusteringList[\([i, 1]\)], 1, 1]\)[\([1, 1]\)]]\)] += myClusteringList[\([i, 2]\)];\)\[IndentingNewLine]]; \n\[IndentingNewLine]If[ UndirectedQ[g], \[IndentingNewLine]Inner[List, listOfDiffOutDegress, \ 2*totalNumEdges/\((listOfDiffOutDegress*\((listOfDiffOutDegress \ - 1)\)*Map[Count[myClusteringList[\([All, 1]\)], #] &, listOfDiffOutDegress])\), List], \[IndentingNewLine]Inner[List, listOfDiffOutDegress, \ totalNumEdges/\((listOfDiffOutDegress*\((listOfDiffOutDegress - 1)\)*Map[Count[myClusteringList[\([All, 1]\)], #] &, listOfDiffOutDegress])\), List], \[IndentingNewLine]]\[IndentingNewLine]];\)\ \[IndentingNewLine]\[IndentingNewLine] (*\ The\ closeness\ centrality\ of\ a\ node\ using\ geodesic\ distance\ is\ \ the\ reciprocal\ of\ the\ sum\ of\ geodesic\ distances\ from\ the\ node\ to\ \ all\ other\ vertices\ in\ the\ graph . \ This\ function\ returns\ the\ closeness\ centrality\ using\ \ geodesic\ distance\ of\ every\ node\ in\ the\ \(\(graph\)\(.\)\)\ *) \ \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(GeodesicCloseness[g_Graph] := 1/Total[AllPairsShortestPath[ g]];\)\[IndentingNewLine]\[IndentingNewLine]\[IndentingNewLine] \ (*\ This\ function\ returns\ a\ list\ with\ the\ reachability\ closeness\ of\ \ each\ node\ in\ the\ graph . \ The\ reachability\ closeness\ of\ a\ node\ is\ calculated\ by\ \ summing\ up\ the\ proportion\ of\ other\ nodes\ reached\ \((for\ the\ first\ \ time)\)\ at\ a\ given\ distance, appropriately\ weighted . \ In\ this\ case\ the\ weights\ used\ are\ 1/ n\ where\ n\ denotes\ the\ distance\ between\ the\ nodes \((e . g . \ 1\ for\ nodes\ at\ distance\ 1, \ \.bd\ for\ nodes\ at\ \ distance\ 2 \[Ellipsis])\)\ *) \[IndentingNewLine]\), "\[IndentingNewLine]", \ \(\(ReachabilityCloseness[g_Graph] := Module[{freqLists}, \[IndentingNewLine]freqLists = Map[Rest[Frequencies[#]] &, AllPairsShortestPath[ g]]; \[IndentingNewLine]Map[\((Total[#[\([All, 1]\)]*\((1/#[\([All, 2]\)])\)])\) &, freqLists]/\((V[g] - 1)\)\[IndentingNewLine]];\)\[IndentingNewLine]\ \[IndentingNewLine] (*\ This\ function\ returns\ a\ matrix\ indicating\ the\ number\ of\ \ shortest\ paths\ between\ any\ two\ nodes\ in\ the\ \(\(network\)\(.\)\)*) \ \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(Options[ NumberOfShortestPaths] = {CheckGraphIsConnected \[Rule] True};\)\), "\[IndentingNewLine]", \(\(NumberOfShortestPaths[g_Graph, opts___?OptionQ] := Module[\[IndentingNewLine]{distancesMatrix\ , adjMatrix\ , powersOfAdjMatrix\ }, \ \[IndentingNewLine]\[IndentingNewLine]checkForConnected = CheckGraphIsConnected /. Flatten[{opts, Options[NumberOfShortestPaths]}]; \[IndentingNewLine] (*\ Replace\ \(( /. )\)\ returns\ the\ result\ from\ using\ the\ \ first\ rule\ that\ \(\(applies\)\(.\)\)\ *) \[IndentingNewLine]If[ checkForConnected, \[IndentingNewLine]\(If[\((UndirectedQ[ g]\ && \ \(! ConnectedQ[ g]\))\)\ || \ \((\(! UndirectedQ[ g]\)\ && \ \(! ConnectedQ[g, Strong]\))\), \ \[IndentingNewLine]Print["\"]; \ Return[];\[IndentingNewLine]];\)\[IndentingNewLine]]; \ \[IndentingNewLine]\[IndentingNewLine]distancesMatrix\ = \ AllPairsShortestPath[g]; \[IndentingNewLine]If[ Not[checkForConnected], distancesMatrix = Replace[distancesMatrix, \[Infinity] \[Rule] 0, {2}]]; \[IndentingNewLine] (*\ A\ distance\ of\ 0\ will\ give\ 0\ shortest\ paths\ between\ \ those\ \(\(nodes\)\(.\)\)\ *) \[IndentingNewLine]\[IndentingNewLine]adjMatrix\ \ = \ ToAdjacencyMatrix[g]; \[IndentingNewLine]powersOfAdjMatrix\ = \ NestList[\((# . adjMatrix)\) &, adjMatrix, Max[distancesMatrix] - 1]; \[IndentingNewLine]\[IndentingNewLine]PrependTo[ powersOfAdjMatrix, Table[0, {V[g]}, {V[g]}]]; \[IndentingNewLine] (*\ We\ prepend\ the\ matrix\ that\ pairs\ showing\ a\ distance\ of\ \ 0\ will\ read . \ These\ pairs\ are\ those\ that\ are\ not\ connected, \ and\ the\ pairs\ consisting\ of\ a\ node\ with\ \(\(itself\)\(.\)\ \)\ *) \[IndentingNewLine]MapIndexed[\((powersOfAdjMatrix[\([#1, Apply[Sequence, #2]]\)])\) &, distancesMatrix + Table[1, {V[g]}, {V[ g]}], {2}]\[IndentingNewLine]];\)\[IndentingNewLine]\ \[IndentingNewLine]\[IndentingNewLine] (*\ This\ function\ returns\ the\ betweeness\ of\ node\ v . \ The\ betweenness\ of\ a\ node\ v\ is\ calculated\ by\ computing, \ for\ every\ pair\ of\ other\ nodes\ s, t\ in\ the\ network\ \((s \[NotEqual] v \[NotEqual] t)\), \ the\ fraction\ of\ all\ the\ shortest\ paths\ between\ s\ and\ t\ that\ \ pass\ through\ v . \ These\ \((n\ \[Dash]\ 1)\) \((n\ \[Dash]\ 2)\)\ fractions\ are\ \ then\ summed\ up . \ The\ idea\ in\ this\ algorithm\ is\ that\ the\ number\ of\ shortest\ \ paths\ between\ s\ and\ s\ that\ do\ not\ pass\ through\ a\ node\ v\ are\ \ the\ number\ of\ paths\ between\ s\ and\ t\ OF\ LENGTH\ EQUAL\ TO\ THE\ \ DISTANCE\ BETWEEN\ s\ AND\ t\ IN\ THE\ ORIGINAL\ GRAPH, \ in\ a\ graph\ where\ node\ v\ has\ been\ \(\(deleted\)\(.\)\)\ *) \ \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(Betweenness[v_Integer, g_Graph] := Module[{distancesMatrix, adjMatrix, powersOfAdjMatrix, nShortestPaths, adjMatrixWithoutVertex, powersOfAdjMatrixWithoutVertex, nShortestPathsNotThroughV}, \ \[IndentingNewLine]\[IndentingNewLine]If[\((UndirectedQ[ g]\ && \ \(! ConnectedQ[ g]\))\)\ || \ \((\(! UndirectedQ[ g]\)\ && \ \(! ConnectedQ[g, Strong]\))\), \ \[IndentingNewLine]Print["\"]; \ Return[];\[IndentingNewLine]]; \[IndentingNewLine]\ \[IndentingNewLine]distancesMatrix\ = \ AllPairsShortestPath[ g]; \[IndentingNewLine]\[IndentingNewLine] (*\ Calculate\ the\ number\ of\ shortest\ paths\ between\ any\ two\ \ nodes\ in\ the\ graph\ *) \[IndentingNewLine]adjMatrix\ = \ ToAdjacencyMatrix[g]; \[IndentingNewLine]powersOfAdjMatrix\ = \ NestList[\((# . adjMatrix)\) &, adjMatrix, Max[distancesMatrix] - 1]; \[IndentingNewLine]PrependTo[ powersOfAdjMatrix, Table[0, {V[g]}, {V[g]}]]; (*\ We\ prepend\ the\ matrix\ that\ pairs\ showing\ a\ distance\ of\ \ 0\ will\ read . \ These\ pairs\ are\ those\ that\ are\ not\ connected, \ and\ the\ pairs\ consisting\ of\ a\ node\ with\ \(\(itself\)\(.\)\ \)\ *) \[IndentingNewLine]nShortestPaths = MapIndexed[\((powersOfAdjMatrix[\([#1, Apply[Sequence, #2]]\)])\) &, distancesMatrix + Table[1, {V[g]}, {V[ g]}], {2}]; \[IndentingNewLine]\[IndentingNewLine] (*\ Calculate\ the\ number\ of\ paths\ between\ i\ and\ j\ OF\ LENGTH\ \ EQUAL\ TO\ THE\ DISTANCE\ BETWEEN\ i\ AND\ j\ IN\ THE\ ORIGINAL\ GRAPH, \ in\ a\ graph\ where\ node\ v\ has\ been\ deleted . \ This\ is\ the\ number\ of\ shortest\ paths\ between\ i\ and\ \ j\ that\ do\ not\ pass\ through\ node\ v\ in\ the\ original\ \(\(graph\)\(.\)\ \)\ *) \[IndentingNewLine]adjMatrixWithoutVertex\ = \ Drop[adjMatrix, {v}, {v}]; \ \[IndentingNewLine]powersOfAdjMatrixWithoutVertex\ = \ NestList[\((# . adjMatrixWithoutVertex)\) &, adjMatrixWithoutVertex, Max[distancesMatrix] - 1]; \[IndentingNewLine]PrependTo[ powersOfAdjMatrixWithoutVertex, Table[0, {V[g]}, {V[g]}]]; (*\ We\ prepend\ the\ matrix\ that\ pairs\ showing\ a\ distance\ of\ \ 0\ will\ read . \ These\ pairs\ are\ those\ that\ are\ not\ connected, \ and\ the\ pairs\ consisting\ of\ a\ node\ with\ \(\(itself\)\(.\)\ \)\ *) \[IndentingNewLine]nShortestPathsNotThroughV = MapIndexed[\((powersOfAdjMatrixWithoutVertex[\([#1, Apply[Sequence, #2]]\)])\) &, Drop[distancesMatrix, {v}, {v}] + Table[1, {V[g] - 1}, {V[g] - 1}], {2}]; \[IndentingNewLine]\[IndentingNewLine] (*\ Return\ the\ betweenness\ *) \[IndentingNewLine]Total[\((Drop[ nShortestPaths, {v}, {v}] - nShortestPathsNotThroughV)\)/\((Drop[ nShortestPaths, {v}, {v}] - IdentityMatrix[V[g] - 1])\), 2]\[IndentingNewLine]];\)\[IndentingNewLine]\[IndentingNewLine]\ \[IndentingNewLine] (*\ This\ function\ returns\ a\ list\ with\ the\ betweeness\ of\ every\ \ node\ in\ the\ network . \ The\ betweenness\ of\ a\ node\ v\ is\ calculated\ by\ computing, \ for\ every\ pair\ of\ other\ nodes\ s, t\ in\ the\ network\ \((s \[NotEqual] v \[NotEqual] t)\), \ the\ fraction\ of\ all\ the\ shortest\ paths\ between\ s\ and\ t\ that\ \ pass\ through\ v . \ These\ \((n\ \[Dash]\ 1)\) \((n\ \[Dash]\ 2)\)\ fractions\ are\ \ then\ summed\ up . \ The\ idea\ in\ this\ algorithm\ is\ that\ the\ number\ of\ shortest\ \ paths\ between\ s\ and\ s\ that\ do\ not\ pass\ through\ a\ node\ v\ are\ \ the\ number\ of\ paths\ between\ s\ and\ t\ OF\ LENGTH\ EQUAL\ TO\ THE\ \ DISTANCE\ BETWEEN\ s\ AND\ t\ IN\ THE\ ORIGINAL\ GRAPH, \ in\ a\ graph\ where\ node\ v\ has\ been\ \(\(deleted\)\(.\)\)\ *) \ \[IndentingNewLine]\), "\[IndentingNewLine]", \(\(Betweenness[g_Graph] := Module[{nNodes, distancesMatrix, adjMatrix, powersOfAdjMatrix, nShortestPaths, v, adjMatrixWithoutVertex, powersOfAdjMatrixWithoutVertex, nShortestPathsNotThroughV, betweennessList}, \ \[IndentingNewLine]\[IndentingNewLine]If[\((UndirectedQ[ g]\ && \ \(! ConnectedQ[ g]\))\)\ || \ \((\(! UndirectedQ[ g]\)\ && \ \(! ConnectedQ[g, Strong]\))\), \ \[IndentingNewLine]Print["\"]; \ Return[];\[IndentingNewLine]]; \[IndentingNewLine]\ \[IndentingNewLine]distancesMatrix\ = \ AllPairsShortestPath[g]; \[IndentingNewLine]nNodes = V[g]; \[IndentingNewLine]\[IndentingNewLine] (*\ Calculate\ the\ number\ of\ shortest\ paths\ between\ any\ two\ \ nodes\ in\ the\ graph\ *) \[IndentingNewLine]adjMatrix\ = \ ToAdjacencyMatrix[g]; \[IndentingNewLine]powersOfAdjMatrix\ = \ NestList[\((# . adjMatrix)\) &, adjMatrix, Max[distancesMatrix] - 1]; \[IndentingNewLine]PrependTo[ powersOfAdjMatrix, Table[0, {nNodes}, {nNodes}]]; (*\ We\ prepend\ the\ matrix\ that\ pairs\ showing\ a\ distance\ of\ \ 0\ will\ read . \ These\ pairs\ are\ those\ that\ are\ not\ connected, \ and\ the\ pairs\ consisting\ of\ a\ node\ with\ \(\(itself\)\(.\)\ \)\ *) \[IndentingNewLine]nShortestPaths = MapIndexed[\((powersOfAdjMatrix[\([#1, Apply[Sequence, #2]]\)])\) &, distancesMatrix + Table[1, {nNodes}, {nNodes}], {2}]; \[IndentingNewLine]\ \[IndentingNewLine]betweennessList = {}; \[IndentingNewLine] (*\ Calculate\ the\ number\ of\ paths\ between\ i\ and\ j\ OF\ LENGTH\ \ EQUAL\ TO\ THE\ DISTANCE\ BETWEEN\ i\ AND\ j\ IN\ THE\ ORIGINAL\ GRAPH, \ in\ a\ graph\ where\ node\ v\ has\ been\ deleted . \ This\ is\ the\ number\ of\ shortest\ paths\ between\ i\ and\ \ j\ that\ do\ not\ pass\ through\ node\ v\ in\ the\ original\ \(\(graph\)\(.\)\ \)\ *) \[IndentingNewLine]For[v = 1, v \[LessEqual] nNodes, \(v++\), \[IndentingNewLine]adjMatrixWithoutVertex\ = \ \ Drop[adjMatrix, {v}, {v}]; powersOfAdjMatrixWithoutVertex\ = \ NestList[\((# . adjMatrixWithoutVertex)\) &, adjMatrixWithoutVertex, Max[distancesMatrix] - 1]; \[IndentingNewLine]PrependTo[ powersOfAdjMatrixWithoutVertex, Table[0, {nNodes}, {nNodes}]]; (*\ We\ prepend\ the\ matrix\ that\ pairs\ showing\ a\ distance\ of\ \ 0\ will\ read . \ These\ pairs\ are\ those\ that\ are\ not\ connected, \ and\ the\ pairs\ consisting\ of\ a\ node\ with\ \(\(itself\)\(.\ \)\)\ *) \[IndentingNewLine]nShortestPathsNotThroughV = MapIndexed[\((powersOfAdjMatrixWithoutVertex[\([#1, Apply[Sequence, #2]]\)])\) &, Drop[distancesMatrix, {v}, {v}] + Table[1, {nNodes - 1}, {nNodes - 1}], {2}]; \ \[IndentingNewLine]\[IndentingNewLine]AppendTo[betweennessList, Total[\((Drop[nShortestPaths, {v}, {v}] - nShortestPathsNotThroughV)\)/\((Drop[ nShortestPaths, {v}, {v}] - IdentityMatrix[V[g] - 1])\), 2]];\[IndentingNewLine]]; \[IndentingNewLine]\ \[IndentingNewLine] (*\ Return\ the\ betweenness\ *) \[IndentingNewLine]betweennessList\ \[IndentingNewLine]];\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Basics Concepts and Terminology", "Section"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\(myBinaryTree = CompleteBinaryTree[50];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(g1 = ShowGraph[myBinaryTree, \ DisplayFunction \[Rule] Identity];\)\), "\n", \(\(g2 = ShowGraph[SpringEmbedding[myBinaryTree, 200, 0.05], \ DisplayFunction \[Rule] Identity];\)\), "\n", 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.95455 L s .30519 .93599 m .30519 .95455 L s .37941 .93599 m .37941 .95455 L s .60204 .93599 m .62059 .93599 L s .60204 .93599 m .60204 .95455 L s .62059 .93599 m .63915 .93599 L s .62059 .93599 m .62059 .95455 L s .63915 .93599 m .6577 .93599 L s .63915 .93599 m .63915 .95455 L s .6577 .93599 m .6577 .95455 L s .71336 .93599 m .73191 .93599 L s .71336 .93599 m .71336 .95455 L s .73191 .93599 m .73191 .95455 L s .78757 .93599 m .78757 .95455 L s .82468 .93599 m .84323 .93599 L s .82468 .93599 m .82468 .95455 L s .84323 .93599 m .86178 .93599 L s .84323 .93599 m .84323 .95455 L s .86178 .93599 m .86178 .95455 L s .89889 .93599 m .91744 .93599 L s .89889 .93599 m .89889 .95455 L s .10111 .95455 m .11967 .95455 L s .21243 .95455 m .23098 .95455 L s .28664 .95455 m .30519 .95455 L s .36085 .95455 m .37941 .95455 L s .41651 .95455 m .43506 .95455 L s .43506 .95455 m .45362 .95455 L s .45362 .95455 m .47217 .95455 L s .47217 .95455 m .49072 .95455 L s .54638 .95455 m .56494 .95455 L s .56494 .95455 m .58349 .95455 L s .58349 .95455 m .60204 .95455 L s .60204 .95455 m .62059 .95455 L s .62059 .95455 m .63915 .95455 L s .63915 .95455 m .6577 .95455 L s .6577 .95455 m .67625 .95455 L s .71336 .95455 m .73191 .95455 L s .73191 .95455 m .75046 .95455 L s .78757 .95455 m .80612 .95455 L s .80612 .95455 m .82468 .95455 L s .82468 .95455 m .84323 .95455 L s .84323 .95455 m .86178 .95455 L s .86178 .95455 m .88033 .95455 L s .88033 .95455 m .89889 .95455 L s 0 w .04545 .04545 Mdot .11967 .04545 Mdot .15677 .04545 Mdot .19388 .04545 Mdot .21243 .04545 Mdot .23098 .04545 Mdot .24954 .04545 Mdot .28664 .04545 Mdot .32375 .04545 Mdot .3423 .04545 Mdot .36085 .04545 Mdot .37941 .04545 Mdot .56494 .04545 Mdot .69481 .04545 Mdot .71336 .04545 Mdot .73191 .04545 Mdot .75046 .04545 Mdot .76902 .04545 Mdot .78757 .04545 Mdot .82468 .04545 Mdot .88033 .04545 Mdot .93599 .04545 Mdot .04545 .06401 Mdot .06401 .06401 Mdot .10111 .06401 Mdot .15677 .06401 Mdot .21243 .06401 Mdot .23098 .06401 Mdot .24954 .06401 Mdot .28664 .06401 Mdot .3423 .06401 Mdot .37941 .06401 Mdot .41651 .06401 Mdot .43506 .06401 Mdot .58349 .06401 Mdot .62059 .06401 Mdot .67625 .06401 Mdot .69481 .06401 Mdot .80612 .06401 Mdot .82468 .06401 Mdot .84323 .06401 Mdot .88033 .06401 Mdot .04545 .08256 Mdot .10111 .08256 Mdot .11967 .08256 Mdot .13822 .08256 Mdot .19388 .08256 Mdot .21243 .08256 Mdot .23098 .08256 Mdot .24954 .08256 Mdot .30519 .08256 Mdot .39796 .08256 Mdot .45362 .08256 Mdot .47217 .08256 Mdot .54638 .08256 Mdot .58349 .08256 Mdot .63915 .08256 Mdot .6577 .08256 Mdot .69481 .08256 Mdot .84323 .08256 Mdot .86178 .08256 Mdot .91744 .08256 Mdot .93599 .08256 Mdot .95455 .08256 Mdot .04545 .10111 Mdot .06401 .10111 Mdot .08256 .10111 Mdot .19388 .10111 Mdot .21243 .10111 Mdot .23098 .10111 Mdot .26809 .10111 Mdot .30519 .10111 Mdot .32375 .10111 Mdot .37941 .10111 Mdot .39796 .10111 Mdot .43506 .10111 Mdot .45362 .10111 Mdot .50928 .10111 Mdot .56494 .10111 Mdot .58349 .10111 Mdot .73191 .10111 Mdot .78757 .10111 Mdot .80612 .10111 Mdot .82468 .10111 Mdot .89889 .10111 Mdot .93599 .10111 Mdot .95455 .10111 Mdot .04545 .11967 Mdot .11967 .11967 Mdot .15677 .11967 Mdot .21243 .11967 Mdot .24954 .11967 Mdot .26809 .11967 Mdot .28664 .11967 Mdot .30519 .11967 Mdot .3423 .11967 Mdot .36085 .11967 Mdot .41651 .11967 Mdot .47217 .11967 Mdot .50928 .11967 Mdot .56494 .11967 Mdot .58349 .11967 Mdot .63915 .11967 Mdot .69481 .11967 Mdot .75046 .11967 Mdot .80612 .11967 Mdot .82468 .11967 Mdot .86178 .11967 Mdot .89889 .11967 Mdot .04545 .13822 Mdot .11967 .13822 Mdot .19388 .13822 Mdot .28664 .13822 Mdot .36085 .13822 Mdot .39796 .13822 Mdot .43506 .13822 Mdot .45362 .13822 Mdot .49072 .13822 Mdot .50928 .13822 Mdot .52783 .13822 Mdot .56494 .13822 Mdot .60204 .13822 Mdot .62059 .13822 Mdot .67625 .13822 Mdot .75046 .13822 Mdot .76902 .13822 Mdot .82468 .13822 Mdot .91744 .13822 Mdot .93599 .13822 Mdot .95455 .13822 Mdot .11967 .15677 Mdot .15677 .15677 Mdot .21243 .15677 Mdot .24954 .15677 Mdot .28664 .15677 Mdot .30519 .15677 Mdot .37941 .15677 Mdot .39796 .15677 Mdot .41651 .15677 Mdot .49072 .15677 Mdot .50928 .15677 Mdot .54638 .15677 Mdot .56494 .15677 Mdot .60204 .15677 Mdot .62059 .15677 Mdot .63915 .15677 Mdot .69481 .15677 Mdot .73191 .15677 Mdot .75046 .15677 Mdot .76902 .15677 Mdot .80612 .15677 Mdot .86178 .15677 Mdot .88033 .15677 Mdot .89889 .15677 Mdot .93599 .15677 Mdot .04545 .17532 Mdot .08256 .17532 Mdot .15677 .17532 Mdot .21243 .17532 Mdot .26809 .17532 Mdot .30519 .17532 Mdot .41651 .17532 Mdot .47217 .17532 Mdot .49072 .17532 Mdot .52783 .17532 Mdot .56494 .17532 Mdot .62059 .17532 Mdot .69481 .17532 Mdot .71336 .17532 Mdot .73191 .17532 Mdot .76902 .17532 Mdot .78757 .17532 Mdot .84323 .17532 Mdot .88033 .17532 Mdot .89889 .17532 Mdot .93599 .17532 Mdot .04545 .19388 Mdot .06401 .19388 Mdot .08256 .19388 Mdot .10111 .19388 Mdot .11967 .19388 Mdot .17532 .19388 Mdot .26809 .19388 Mdot .30519 .19388 Mdot .32375 .19388 Mdot .3423 .19388 Mdot .45362 .19388 Mdot .47217 .19388 Mdot .50928 .19388 Mdot .6577 .19388 Mdot .67625 .19388 Mdot .75046 .19388 Mdot .78757 .19388 Mdot .84323 .19388 Mdot .86178 .19388 Mdot .88033 .19388 Mdot .93599 .19388 Mdot .06401 .21243 Mdot .08256 .21243 Mdot .11967 .21243 Mdot .19388 .21243 Mdot .21243 .21243 Mdot .23098 .21243 Mdot .24954 .21243 Mdot .28664 .21243 Mdot .30519 .21243 Mdot .3423 .21243 Mdot .39796 .21243 Mdot .41651 .21243 Mdot .47217 .21243 Mdot .50928 .21243 Mdot .52783 .21243 Mdot .54638 .21243 Mdot .56494 .21243 Mdot .60204 .21243 Mdot .62059 .21243 Mdot .63915 .21243 Mdot .6577 .21243 Mdot .67625 .21243 Mdot .71336 .21243 Mdot .75046 .21243 Mdot .78757 .21243 Mdot .80612 .21243 Mdot .82468 .21243 Mdot .86178 .21243 Mdot .88033 .21243 Mdot .91744 .21243 Mdot .95455 .21243 Mdot .04545 .23098 Mdot .13822 .23098 Mdot .21243 .23098 Mdot .23098 .23098 Mdot .24954 .23098 Mdot .26809 .23098 Mdot .30519 .23098 Mdot .37941 .23098 Mdot .39796 .23098 Mdot .43506 .23098 Mdot .58349 .23098 Mdot .62059 .23098 Mdot .63915 .23098 Mdot .6577 .23098 Mdot .67625 .23098 Mdot .69481 .23098 Mdot .76902 .23098 Mdot .78757 .23098 Mdot .93599 .23098 Mdot .04545 .24954 Mdot .10111 .24954 Mdot .11967 .24954 Mdot .13822 .24954 Mdot .15677 .24954 Mdot .26809 .24954 Mdot .28664 .24954 Mdot .30519 .24954 Mdot .32375 .24954 Mdot .3423 .24954 Mdot .36085 .24954 Mdot .37941 .24954 Mdot .54638 .24954 Mdot .56494 .24954 Mdot .62059 .24954 Mdot .63915 .24954 Mdot .67625 .24954 Mdot .73191 .24954 Mdot .75046 .24954 Mdot .76902 .24954 Mdot .82468 .24954 Mdot .88033 .24954 Mdot .95455 .24954 Mdot .08256 .26809 Mdot .17532 .26809 Mdot .23098 .26809 Mdot .28664 .26809 Mdot .30519 .26809 Mdot .3423 .26809 Mdot .37941 .26809 Mdot .41651 .26809 Mdot .49072 .26809 Mdot .50928 .26809 Mdot .58349 .26809 Mdot .60204 .26809 Mdot .62059 .26809 Mdot .63915 .26809 Mdot .6577 .26809 Mdot .71336 .26809 Mdot .75046 .26809 Mdot .76902 .26809 Mdot .80612 .26809 Mdot .88033 .26809 Mdot .89889 .26809 Mdot .11967 .28664 Mdot .13822 .28664 Mdot .15677 .28664 Mdot .21243 .28664 Mdot .26809 .28664 Mdot .28664 .28664 Mdot .30519 .28664 Mdot .32375 .28664 Mdot .3423 .28664 Mdot .36085 .28664 Mdot .37941 .28664 Mdot .39796 .28664 Mdot .41651 .28664 Mdot .43506 .28664 Mdot .50928 .28664 Mdot .52783 .28664 Mdot .54638 .28664 Mdot .56494 .28664 Mdot .58349 .28664 Mdot .60204 .28664 Mdot .63915 .28664 Mdot .76902 .28664 Mdot .88033 .28664 Mdot .89889 .28664 Mdot .91744 .28664 Mdot .04545 .30519 Mdot .06401 .30519 Mdot .08256 .30519 Mdot .10111 .30519 Mdot .15677 .30519 Mdot .17532 .30519 Mdot .19388 .30519 Mdot .24954 .30519 Mdot .26809 .30519 Mdot .28664 .30519 Mdot .30519 .30519 Mdot .3423 .30519 Mdot .36085 .30519 Mdot .37941 .30519 Mdot .39796 .30519 Mdot .41651 .30519 Mdot .47217 .30519 Mdot .49072 .30519 Mdot .50928 .30519 Mdot .56494 .30519 Mdot .58349 .30519 Mdot .60204 .30519 Mdot .63915 .30519 Mdot .67625 .30519 Mdot .71336 .30519 Mdot .73191 .30519 Mdot .76902 .30519 Mdot .82468 .30519 Mdot .84323 .30519 Mdot .93599 .30519 Mdot .06401 .32375 Mdot .11967 .32375 Mdot .13822 .32375 Mdot .24954 .32375 Mdot .28664 .32375 Mdot .37941 .32375 Mdot .39796 .32375 Mdot .41651 .32375 Mdot .43506 .32375 Mdot .47217 .32375 Mdot .52783 .32375 Mdot .58349 .32375 Mdot .60204 .32375 Mdot .62059 .32375 Mdot .63915 .32375 Mdot .6577 .32375 Mdot .67625 .32375 Mdot .69481 .32375 Mdot .75046 .32375 Mdot .76902 .32375 Mdot .80612 .32375 Mdot .88033 .32375 Mdot .95455 .32375 Mdot .06401 .3423 Mdot .08256 .3423 Mdot .13822 .3423 Mdot .15677 .3423 Mdot .17532 .3423 Mdot .19388 .3423 Mdot .30519 .3423 Mdot .3423 .3423 Mdot .36085 .3423 Mdot .39796 .3423 Mdot .41651 .3423 Mdot .47217 .3423 Mdot .52783 .3423 Mdot .56494 .3423 Mdot .6577 .3423 Mdot .67625 .3423 Mdot .73191 .3423 Mdot .75046 .3423 Mdot .76902 .3423 Mdot .86178 .3423 Mdot .93599 .3423 Mdot .04545 .36085 Mdot .06401 .36085 Mdot .08256 .36085 Mdot .11967 .36085 Mdot .13822 .36085 Mdot .17532 .36085 Mdot .28664 .36085 Mdot .30519 .36085 Mdot .3423 .36085 Mdot .37941 .36085 Mdot .43506 .36085 Mdot .45362 .36085 Mdot .47217 .36085 Mdot .49072 .36085 Mdot .54638 .36085 Mdot .60204 .36085 Mdot .6577 .36085 Mdot .67625 .36085 Mdot .69481 .36085 Mdot .76902 .36085 Mdot .86178 .36085 Mdot .89889 .36085 Mdot .95455 .36085 Mdot .06401 .37941 Mdot .08256 .37941 Mdot .10111 .37941 Mdot .11967 .37941 Mdot .15677 .37941 Mdot .19388 .37941 Mdot .21243 .37941 Mdot .23098 .37941 Mdot .26809 .37941 Mdot .28664 .37941 Mdot .30519 .37941 Mdot .39796 .37941 Mdot .43506 .37941 Mdot .49072 .37941 Mdot .52783 .37941 Mdot .54638 .37941 Mdot .56494 .37941 Mdot .58349 .37941 Mdot .60204 .37941 Mdot .63915 .37941 Mdot .67625 .37941 Mdot .75046 .37941 Mdot .76902 .37941 Mdot .78757 .37941 Mdot .82468 .37941 Mdot .84323 .37941 Mdot .86178 .37941 Mdot .88033 .37941 Mdot .04545 .39796 Mdot .06401 .39796 Mdot .08256 .39796 Mdot .10111 .39796 Mdot .13822 .39796 Mdot .15677 .39796 Mdot .23098 .39796 Mdot .28664 .39796 Mdot .30519 .39796 Mdot .32375 .39796 Mdot .3423 .39796 Mdot .37941 .39796 Mdot .43506 .39796 Mdot .45362 .39796 Mdot .49072 .39796 Mdot .50928 .39796 Mdot .52783 .39796 Mdot .54638 .39796 Mdot .63915 .39796 Mdot .6577 .39796 Mdot .67625 .39796 Mdot .69481 .39796 Mdot .78757 .39796 Mdot .84323 .39796 Mdot .89889 .39796 Mdot .04545 .41651 Mdot .10111 .41651 Mdot .15677 .41651 Mdot .19388 .41651 Mdot .24954 .41651 Mdot .28664 .41651 Mdot .30519 .41651 Mdot .32375 .41651 Mdot .37941 .41651 Mdot .39796 .41651 Mdot .41651 .41651 Mdot .43506 .41651 Mdot .45362 .41651 Mdot .49072 .41651 Mdot .54638 .41651 Mdot .60204 .41651 Mdot .6577 .41651 Mdot .67625 .41651 Mdot .69481 .41651 Mdot .71336 .41651 Mdot .84323 .41651 Mdot .86178 .41651 Mdot .88033 .41651 Mdot .89889 .41651 Mdot .91744 .41651 Mdot .93599 .41651 Mdot .95455 .41651 Mdot .06401 .43506 Mdot .10111 .43506 Mdot .11967 .43506 Mdot .15677 .43506 Mdot .17532 .43506 Mdot .19388 .43506 Mdot .21243 .43506 Mdot .24954 .43506 Mdot .26809 .43506 Mdot .30519 .43506 Mdot .32375 .43506 Mdot .36085 .43506 Mdot .37941 .43506 Mdot .41651 .43506 Mdot .43506 .43506 Mdot .50928 .43506 Mdot .54638 .43506 Mdot .56494 .43506 Mdot .58349 .43506 Mdot .63915 .43506 Mdot .67625 .43506 Mdot .69481 .43506 Mdot .71336 .43506 Mdot .75046 .43506 Mdot .76902 .43506 Mdot .78757 .43506 Mdot .80612 .43506 Mdot .82468 .43506 Mdot .86178 .43506 Mdot .88033 .43506 Mdot .89889 .43506 Mdot .91744 .43506 Mdot .95455 .43506 Mdot .04545 .45362 Mdot .10111 .45362 Mdot .11967 .45362 Mdot .19388 .45362 Mdot .24954 .45362 Mdot .26809 .45362 Mdot .30519 .45362 Mdot .32375 .45362 Mdot .36085 .45362 Mdot .47217 .45362 Mdot .50928 .45362 Mdot .52783 .45362 Mdot .54638 .45362 Mdot .56494 .45362 Mdot .58349 .45362 Mdot .60204 .45362 Mdot .63915 .45362 Mdot .6577 .45362 Mdot .71336 .45362 Mdot .73191 .45362 Mdot .75046 .45362 Mdot .78757 .45362 Mdot .80612 .45362 Mdot .82468 .45362 Mdot .88033 .45362 Mdot .89889 .45362 Mdot .93599 .45362 Mdot .04545 .47217 Mdot .06401 .47217 Mdot .08256 .47217 Mdot .10111 .47217 Mdot .11967 .47217 Mdot .13822 .47217 Mdot .19388 .47217 Mdot .24954 .47217 Mdot .26809 .47217 Mdot .30519 .47217 Mdot .3423 .47217 Mdot .36085 .47217 Mdot .39796 .47217 Mdot .45362 .47217 Mdot .47217 .47217 Mdot .50928 .47217 Mdot .52783 .47217 Mdot .58349 .47217 Mdot .63915 .47217 Mdot .6577 .47217 Mdot .71336 .47217 Mdot .73191 .47217 Mdot .75046 .47217 Mdot .86178 .47217 Mdot .91744 .47217 Mdot .93599 .47217 Mdot .08256 .49072 Mdot .13822 .49072 Mdot .17532 .49072 Mdot .21243 .49072 Mdot .23098 .49072 Mdot .24954 .49072 Mdot .26809 .49072 Mdot .30519 .49072 Mdot .3423 .49072 Mdot .36085 .49072 Mdot .37941 .49072 Mdot .39796 .49072 Mdot .41651 .49072 Mdot .47217 .49072 Mdot .49072 .49072 Mdot .50928 .49072 Mdot .54638 .49072 Mdot .71336 .49072 Mdot .73191 .49072 Mdot .78757 .49072 Mdot .84323 .49072 Mdot .86178 .49072 Mdot .88033 .49072 Mdot .91744 .49072 Mdot .93599 .49072 Mdot .95455 .49072 Mdot .04545 .50928 Mdot .06401 .50928 Mdot .11967 .50928 Mdot .13822 .50928 Mdot .15677 .50928 Mdot .24954 .50928 Mdot .36085 .50928 Mdot .37941 .50928 Mdot .43506 .50928 Mdot .47217 .50928 Mdot .49072 .50928 Mdot .52783 .50928 Mdot .56494 .50928 Mdot .58349 .50928 Mdot .60204 .50928 Mdot .63915 .50928 Mdot .6577 .50928 Mdot .67625 .50928 Mdot .73191 .50928 Mdot .76902 .50928 Mdot .78757 .50928 Mdot .82468 .50928 Mdot .86178 .50928 Mdot .91744 .50928 Mdot .95455 .50928 Mdot .11967 .52783 Mdot .19388 .52783 Mdot .24954 .52783 Mdot .30519 .52783 Mdot .32375 .52783 Mdot .3423 .52783 Mdot .36085 .52783 Mdot .39796 .52783 Mdot .43506 .52783 Mdot .52783 .52783 Mdot .54638 .52783 Mdot .56494 .52783 Mdot .62059 .52783 Mdot .63915 .52783 Mdot .6577 .52783 Mdot .67625 .52783 Mdot .71336 .52783 Mdot .73191 .52783 Mdot .76902 .52783 Mdot .80612 .52783 Mdot .84323 .52783 Mdot .86178 .52783 Mdot .88033 .52783 Mdot .89889 .52783 Mdot .95455 .52783 Mdot .08256 .54638 Mdot .10111 .54638 Mdot .11967 .54638 Mdot .15677 .54638 Mdot .19388 .54638 Mdot .23098 .54638 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.01472 L .52113 .01472 L .51643 .01472 L s 1 0 0 r .52113 .01472 m .52113 .01472 L .52582 .01472 L .52582 .01472 L F 0 g .52113 .01472 m .52113 .01472 L .52582 .01472 L .52582 .01472 L .52113 .01472 L s 1 0 0 r .52582 .01472 m .52582 .01472 L .53052 .01472 L .53052 .01472 L F 0 g .52582 .01472 m .52582 .01472 L .53052 .01472 L .53052 .01472 L .52582 .01472 L s 1 0 0 r .53052 .01472 m .53052 .01472 L .53521 .01472 L .53521 .01472 L F 0 g .53052 .01472 m .53052 .01472 L .53521 .01472 L .53521 .01472 L .53052 .01472 L s 1 0 0 r .53521 .01472 m .53521 .01472 L .53991 .01472 L .53991 .01472 L F 0 g .53521 .01472 m .53521 .01472 L .53991 .01472 L .53991 .01472 L .53521 .01472 L s 1 0 0 r .53991 .01472 m .53991 .01472 L .5446 .01472 L .5446 .01472 L F 0 g .53991 .01472 m .53991 .01472 L .5446 .01472 L .5446 .01472 L .53991 .01472 L s 1 0 0 r .5446 .01472 m .5446 .01472 L .5493 .01472 L .5493 .01472 L F 0 g .5446 .01472 m .5446 .01472 L .5493 .01472 L .5493 .01472 L .5446 .01472 L s 1 0 0 r .5493 .01472 m .5493 .01472 L .55399 .01472 L .55399 .01472 L F 0 g .5493 .01472 m .5493 .01472 L .55399 .01472 L .55399 .01472 L .5493 .01472 L s 1 0 0 r .55399 .01472 m .55399 .01472 L .55869 .01472 L .55869 .01472 L F 0 g .55399 .01472 m .55399 .01472 L .55869 .01472 L .55869 .01472 L .55399 .01472 L s 1 0 0 r .55869 .01472 m .55869 .01472 L .56338 .01472 L .56338 .01472 L F 0 g .55869 .01472 m .55869 .01472 L .56338 .01472 L .56338 .01472 L .55869 .01472 L s 1 0 0 r .56338 .01472 m .56338 .01472 L .56808 .01472 L .56808 .01472 L F 0 g .56338 .01472 m .56338 .01472 L .56808 .01472 L .56808 .01472 L .56338 .01472 L s 1 0 0 r .56808 .01472 m .56808 .01472 L .57277 .01472 L .57277 .01472 L F 0 g .56808 .01472 m .56808 .01472 L .57277 .01472 L .57277 .01472 L .56808 .01472 L s 1 0 0 r .57277 .01472 m .57277 .01472 L .57746 .01472 L .57746 .01472 L F 0 g .57277 .01472 m .57277 .01472 L .57746 .01472 L .57746 .01472 L .57277 .01472 L s 1 0 0 r .57746 .01472 m .57746 .01472 L .58216 .01472 L .58216 .01472 L F 0 g .57746 .01472 m .57746 .01472 L .58216 .01472 L .58216 .01472 L .57746 .01472 L s 1 0 0 r .58216 .01472 m .58216 .01472 L .58685 .01472 L .58685 .01472 L F 0 g .58216 .01472 m .58216 .01472 L .58685 .01472 L .58685 .01472 L .58216 .01472 L s 1 0 0 r .58685 .01472 m .58685 .01472 L .59155 .01472 L .59155 .01472 L F 0 g .58685 .01472 m .58685 .01472 L .59155 .01472 L .59155 .01472 L .58685 .01472 L s 1 0 0 r .59155 .01472 m .59155 .01472 L .59624 .01472 L .59624 .01472 L F 0 g .59155 .01472 m .59155 .01472 L .59624 .01472 L .59624 .01472 L .59155 .01472 L s 1 0 0 r .59624 .01472 m .59624 .01472 L .60094 .01472 L .60094 .01472 L F 0 g .59624 .01472 m .59624 .01472 L .60094 .01472 L .60094 .01472 L .59624 .01472 L s 1 0 0 r .60094 .01472 m .60094 .01472 L .60563 .01472 L .60563 .01472 L F 0 g .60094 .01472 m .60094 .01472 L .60563 .01472 L .60563 .01472 L .60094 .01472 L s 1 0 0 r .60563 .01472 m .60563 .01472 L .61033 .01472 L .61033 .01472 L F 0 g .60563 .01472 m .60563 .01472 L .61033 .01472 L .61033 .01472 L .60563 .01472 L s 1 0 0 r .61033 .01472 m .61033 .01472 L .61502 .01472 L .61502 .01472 L F 0 g .61033 .01472 m .61033 .01472 L .61502 .01472 L .61502 .01472 L .61033 .01472 L s 1 0 0 r .61502 .01472 m .61502 .01472 L .61972 .01472 L .61972 .01472 L F 0 g .61502 .01472 m .61502 .01472 L .61972 .01472 L .61972 .01472 L .61502 .01472 L s 1 0 0 r .61972 .01472 m .61972 .01472 L .62441 .01472 L .62441 .01472 L F 0 g .61972 .01472 m .61972 .01472 L .62441 .01472 L .62441 .01472 L .61972 .01472 L s 1 0 0 r .62441 .01472 m .62441 .01472 L .62911 .01472 L .62911 .01472 L F 0 g .62441 .01472 m .62441 .01472 L .62911 .01472 L .62911 .01472 L .62441 .01472 L s 1 0 0 r .62911 .01472 m .62911 .01472 L .6338 .01472 L .6338 .01472 L F 0 g .62911 .01472 m .62911 .01472 L .6338 .01472 L .6338 .01472 L .62911 .01472 L s 1 0 0 r .6338 .01472 m .6338 .01472 L .6385 .01472 L .6385 .01472 L F 0 g .6338 .01472 m .6338 .01472 L .6385 .01472 L .6385 .01472 L .6338 .01472 L s 1 0 0 r .6385 .01472 m .6385 .01472 L .64319 .01472 L .64319 .01472 L F 0 g .6385 .01472 m .6385 .01472 L .64319 .01472 L .64319 .01472 L .6385 .01472 L s 1 0 0 r .64319 .01472 m .64319 .01472 L .64789 .01472 L .64789 .01472 L F 0 g .64319 .01472 m .64319 .01472 L .64789 .01472 L .64789 .01472 L .64319 .01472 L s 1 0 0 r .64789 .01472 m .64789 .01472 L .65258 .01472 L .65258 .01472 L F 0 g .64789 .01472 m .64789 .01472 L .65258 .01472 L .65258 .01472 L .64789 .01472 L s 1 0 0 r .65258 .01472 m .65258 .01472 L .65728 .01472 L .65728 .01472 L F 0 g .65258 .01472 m .65258 .01472 L .65728 .01472 L .65728 .01472 L .65258 .01472 L s 1 0 0 r .65728 .01472 m .65728 .02246 L .66197 .02246 L .66197 .01472 L F 0 g .65728 .01472 m .65728 .02246 L .66197 .02246 L .66197 .01472 L .65728 .01472 L s 1 0 0 r .66197 .01472 m .66197 .01472 L .66667 .01472 L .66667 .01472 L F 0 g .66197 .01472 m .66197 .01472 L .66667 .01472 L .66667 .01472 L .66197 .01472 L s 1 0 0 r .66667 .01472 m .66667 .01472 L .67136 .01472 L .67136 .01472 L F 0 g .66667 .01472 m .66667 .01472 L .67136 .01472 L .67136 .01472 L .66667 .01472 L s 1 0 0 r .67136 .01472 m .67136 .01472 L .67606 .01472 L .67606 .01472 L F 0 g .67136 .01472 m .67136 .01472 L .67606 .01472 L .67606 .01472 L .67136 .01472 L s 1 0 0 r .67606 .01472 m .67606 .01472 L .68075 .01472 L .68075 .01472 L F 0 g .67606 .01472 m .67606 .01472 L .68075 .01472 L .68075 .01472 L .67606 .01472 L s 1 0 0 r .68075 .01472 m .68075 .01472 L .68545 .01472 L .68545 .01472 L F 0 g .68075 .01472 m .68075 .01472 L .68545 .01472 L .68545 .01472 L .68075 .01472 L s 1 0 0 r .68545 .01472 m .68545 .01472 L .69014 .01472 L .69014 .01472 L F 0 g .68545 .01472 m .68545 .01472 L .69014 .01472 L .69014 .01472 L .68545 .01472 L s 1 0 0 r .69014 .01472 m .69014 .01472 L .69484 .01472 L .69484 .01472 L F 0 g .69014 .01472 m .69014 .01472 L .69484 .01472 L .69484 .01472 L .69014 .01472 L s 1 0 0 r .69484 .01472 m .69484 .01472 L .69953 .01472 L .69953 .01472 L F 0 g .69484 .01472 m .69484 .01472 L .69953 .01472 L .69953 .01472 L .69484 .01472 L s 1 0 0 r .69953 .01472 m .69953 .01472 L .70423 .01472 L .70423 .01472 L F 0 g .69953 .01472 m .69953 .01472 L .70423 .01472 L .70423 .01472 L .69953 .01472 L s 1 0 0 r .70423 .01472 m .70423 .01472 L .70892 .01472 L .70892 .01472 L F 0 g .70423 .01472 m .70423 .01472 L .70892 .01472 L .70892 .01472 L .70423 .01472 L s 1 0 0 r .70892 .01472 m .70892 .01472 L .71362 .01472 L .71362 .01472 L F 0 g .70892 .01472 m .70892 .01472 L .71362 .01472 L .71362 .01472 L .70892 .01472 L s 1 0 0 r .71362 .01472 m .71362 .01472 L .71831 .01472 L .71831 .01472 L F 0 g .71362 .01472 m .71362 .01472 L .71831 .01472 L .71831 .01472 L .71362 .01472 L s 1 0 0 r .71831 .01472 m .71831 .01472 L .723 .01472 L .723 .01472 L F 0 g .71831 .01472 m .71831 .01472 L .723 .01472 L .723 .01472 L .71831 .01472 L s 1 0 0 r .723 .01472 m .723 .01472 L .7277 .01472 L .7277 .01472 L F 0 g .723 .01472 m .723 .01472 L .7277 .01472 L .7277 .01472 L .723 .01472 L s 1 0 0 r .7277 .01472 m .7277 .01472 L .73239 .01472 L .73239 .01472 L F 0 g .7277 .01472 m .7277 .01472 L .73239 .01472 L .73239 .01472 L .7277 .01472 L s 1 0 0 r .73239 .01472 m .73239 .01472 L .73709 .01472 L .73709 .01472 L F 0 g .73239 .01472 m .73239 .01472 L .73709 .01472 L .73709 .01472 L .73239 .01472 L s 1 0 0 r .73709 .01472 m .73709 .01472 L .74178 .01472 L .74178 .01472 L F 0 g .73709 .01472 m .73709 .01472 L .74178 .01472 L .74178 .01472 L .73709 .01472 L s 1 0 0 r .74178 .01472 m .74178 .01472 L .74648 .01472 L .74648 .01472 L F 0 g .74178 .01472 m .74178 .01472 L .74648 .01472 L .74648 .01472 L .74178 .01472 L s 1 0 0 r .74648 .01472 m .74648 .01472 L .75117 .01472 L .75117 .01472 L F 0 g .74648 .01472 m .74648 .01472 L .75117 .01472 L .75117 .01472 L .74648 .01472 L s 1 0 0 r .75117 .01472 m .75117 .01472 L .75587 .01472 L .75587 .01472 L F 0 g .75117 .01472 m .75117 .01472 L .75587 .01472 L .75587 .01472 L .75117 .01472 L s 1 0 0 r .75587 .01472 m .75587 .01472 L .76056 .01472 L .76056 .01472 L F 0 g .75587 .01472 m .75587 .01472 L .76056 .01472 L .76056 .01472 L .75587 .01472 L s 1 0 0 r .76056 .01472 m .76056 .01472 L .76526 .01472 L .76526 .01472 L F 0 g .76056 .01472 m .76056 .01472 L .76526 .01472 L .76526 .01472 L .76056 .01472 L s 1 0 0 r .76526 .01472 m .76526 .01472 L .76995 .01472 L .76995 .01472 L F 0 g .76526 .01472 m .76526 .01472 L .76995 .01472 L .76995 .01472 L .76526 .01472 L s 1 0 0 r .76995 .01472 m .76995 .01472 L .77465 .01472 L .77465 .01472 L F 0 g .76995 .01472 m .76995 .01472 L .77465 .01472 L .77465 .01472 L .76995 .01472 L s 1 0 0 r .77465 .01472 m .77465 .01472 L .77934 .01472 L .77934 .01472 L F 0 g .77465 .01472 m .77465 .01472 L .77934 .01472 L .77934 .01472 L .77465 .01472 L s 1 0 0 r .77934 .01472 m .77934 .01472 L .78404 .01472 L .78404 .01472 L F 0 g .77934 .01472 m .77934 .01472 L .78404 .01472 L .78404 .01472 L .77934 .01472 L s 1 0 0 r .78404 .01472 m .78404 .01472 L .78873 .01472 L .78873 .01472 L F 0 g .78404 .01472 m .78404 .01472 L .78873 .01472 L .78873 .01472 L .78404 .01472 L s 1 0 0 r .78873 .01472 m .78873 .01472 L .79343 .01472 L .79343 .01472 L F 0 g .78873 .01472 m .78873 .01472 L .79343 .01472 L .79343 .01472 L .78873 .01472 L s 1 0 0 r .79343 .01472 m .79343 .01472 L .79812 .01472 L .79812 .01472 L F 0 g .79343 .01472 m .79343 .01472 L .79812 .01472 L .79812 .01472 L .79343 .01472 L s 1 0 0 r .79812 .01472 m .79812 .01472 L .80282 .01472 L .80282 .01472 L F 0 g .79812 .01472 m .79812 .01472 L .80282 .01472 L .80282 .01472 L .79812 .01472 L s 1 0 0 r .80282 .01472 m .80282 .01472 L .80751 .01472 L .80751 .01472 L F 0 g .80282 .01472 m .80282 .01472 L .80751 .01472 L .80751 .01472 L .80282 .01472 L s 1 0 0 r .80751 .01472 m .80751 .01472 L .81221 .01472 L .81221 .01472 L F 0 g .80751 .01472 m .80751 .01472 L .81221 .01472 L .81221 .01472 L .80751 .01472 L s 1 0 0 r .81221 .01472 m .81221 .01472 L .8169 .01472 L .8169 .01472 L F 0 g .81221 .01472 m .81221 .01472 L .8169 .01472 L .8169 .01472 L .81221 .01472 L s 1 0 0 r .8169 .01472 m .8169 .01472 L .8216 .01472 L .8216 .01472 L F 0 g .8169 .01472 m .8169 .01472 L .8216 .01472 L .8216 .01472 L .8169 .01472 L s 1 0 0 r .8216 .01472 m .8216 .01472 L .82629 .01472 L .82629 .01472 L F 0 g .8216 .01472 m .8216 .01472 L .82629 .01472 L .82629 .01472 L .8216 .01472 L s 1 0 0 r .82629 .01472 m .82629 .01472 L .83099 .01472 L .83099 .01472 L F 0 g .82629 .01472 m .82629 .01472 L .83099 .01472 L .83099 .01472 L .82629 .01472 L s 1 0 0 r .83099 .01472 m .83099 .01472 L .83568 .01472 L .83568 .01472 L F 0 g .83099 .01472 m .83099 .01472 L .83568 .01472 L .83568 .01472 L .83099 .01472 L s 1 0 0 r .83568 .01472 m .83568 .01472 L .84038 .01472 L .84038 .01472 L F 0 g .83568 .01472 m .83568 .01472 L .84038 .01472 L .84038 .01472 L .83568 .01472 L s 1 0 0 r .84038 .01472 m .84038 .01472 L .84507 .01472 L .84507 .01472 L F 0 g .84038 .01472 m .84038 .01472 L .84507 .01472 L .84507 .01472 L .84038 .01472 L s 1 0 0 r .84507 .01472 m .84507 .01472 L .84977 .01472 L .84977 .01472 L F 0 g .84507 .01472 m .84507 .01472 L .84977 .01472 L .84977 .01472 L .84507 .01472 L s 1 0 0 r .84977 .01472 m .84977 .01472 L .85446 .01472 L .85446 .01472 L F 0 g .84977 .01472 m .84977 .01472 L .85446 .01472 L .85446 .01472 L .84977 .01472 L s 1 0 0 r .85446 .01472 m .85446 .01472 L .85915 .01472 L .85915 .01472 L F 0 g .85446 .01472 m .85446 .01472 L .85915 .01472 L .85915 .01472 L .85446 .01472 L s 1 0 0 r .85915 .01472 m .85915 .01472 L .86385 .01472 L .86385 .01472 L F 0 g .85915 .01472 m .85915 .01472 L .86385 .01472 L .86385 .01472 L .85915 .01472 L s 1 0 0 r .86385 .01472 m .86385 .01472 L .86854 .01472 L .86854 .01472 L F 0 g .86385 .01472 m .86385 .01472 L .86854 .01472 L .86854 .01472 L .86385 .01472 L s 1 0 0 r .86854 .01472 m .86854 .01472 L .87324 .01472 L .87324 .01472 L F 0 g .86854 .01472 m .86854 .01472 L .87324 .01472 L .87324 .01472 L .86854 .01472 L s 1 0 0 r .87324 .01472 m .87324 .01472 L .87793 .01472 L .87793 .01472 L F 0 g .87324 .01472 m .87324 .01472 L .87793 .01472 L .87793 .01472 L .87324 .01472 L s 1 0 0 r .87793 .01472 m .87793 .01472 L .88263 .01472 L .88263 .01472 L F 0 g .87793 .01472 m .87793 .01472 L .88263 .01472 L .88263 .01472 L .87793 .01472 L s 1 0 0 r .88263 .01472 m .88263 .01472 L .88732 .01472 L .88732 .01472 L F 0 g .88263 .01472 m .88263 .01472 L .88732 .01472 L .88732 .01472 L .88263 .01472 L s 1 0 0 r .88732 .01472 m .88732 .01472 L .89202 .01472 L .89202 .01472 L F 0 g .88732 .01472 m .88732 .01472 L .89202 .01472 L .89202 .01472 L .88732 .01472 L s 1 0 0 r .89202 .01472 m .89202 .01472 L .89671 .01472 L .89671 .01472 L F 0 g .89202 .01472 m .89202 .01472 L .89671 .01472 L .89671 .01472 L .89202 .01472 L s 1 0 0 r .89671 .01472 m .89671 .01472 L .90141 .01472 L .90141 .01472 L F 0 g .89671 .01472 m .89671 .01472 L .90141 .01472 L .90141 .01472 L .89671 .01472 L s 1 0 0 r .90141 .01472 m .90141 .01472 L .9061 .01472 L .9061 .01472 L F 0 g .90141 .01472 m .90141 .01472 L .9061 .01472 L .9061 .01472 L .90141 .01472 L s 1 0 0 r .9061 .01472 m .9061 .01472 L .9108 .01472 L .9108 .01472 L F 0 g .9061 .01472 m .9061 .01472 L .9108 .01472 L .9108 .01472 L .9061 .01472 L s 1 0 0 r .9108 .01472 m .9108 .01472 L .91549 .01472 L .91549 .01472 L F 0 g .9108 .01472 m .9108 .01472 L .91549 .01472 L .91549 .01472 L .9108 .01472 L s 1 0 0 r .91549 .01472 m .91549 .01472 L .92019 .01472 L .92019 .01472 L F 0 g .91549 .01472 m .91549 .01472 L .92019 .01472 L .92019 .01472 L .91549 .01472 L s 1 0 0 r .92019 .01472 m .92019 .01472 L .92488 .01472 L .92488 .01472 L F 0 g .92019 .01472 m .92019 .01472 L .92488 .01472 L .92488 .01472 L .92019 .01472 L s 1 0 0 r .92488 .01472 m .92488 .01472 L .92958 .01472 L .92958 .01472 L F 0 g .92488 .01472 m .92488 .01472 L .92958 .01472 L .92958 .01472 L .92488 .01472 L s 1 0 0 r .92958 .01472 m .92958 .01472 L .93427 .01472 L .93427 .01472 L F 0 g .92958 .01472 m .92958 .01472 L .93427 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