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Keller", "Author", TextJustification->1], Cell[TextData[{ "One may have difficulties to analyse economic models even with those \ containing a few number of equations. For large macroeconometric models, \ where the equations are often non-linear, a block triangular boolean 0-1 \ matrix will tell us how to organize the whole system of equations and will \ be helpful for recursive resolutions. Furthermore, the knowledge of the \ structural properties may show obvious but also unsuspected properties such \ that a dominant or dominated position, even within a set of interdependant \ variables. Then the degree of eccentricity of some component will be \ considered. This problematic may be approached by means of oriented graphs \ and their algorithms, as it was proposed earlier by us [", StyleBox["8", "Reference", FontWeight->"Plain", FontColor->RGBColor[1, 0, 0], FontVariations->{"Underline"->True, "CompatibilityType"->0}], StyleBox[", ", "Reference", FontColor->RGBColor[1, 0, 0]], StyleBox["9", "Reference", FontColor->RGBColor[1, 0, 0], FontVariations->{"Underline"->True}], "] in the 1970's.\nIn this paper, we are convinced of the major interest \ of using oriented digraphs (without loops) for analysing both theoretical \ and econometric models. In this paper, we are doing three main proposals for \ graph applications : the construction of a more performant graph \ representation using a largest circuit (after an exhaustive enumeration of \ circuits), the search after edge-disjoint cycles and the elaboration of a \ typology of the vertices . The application will be the dynamic \ Klein-Goldberger model [", StyleBox["3", "Reference", FontColor->RGBColor[1, 0, 0], FontVariations->{"Underline"->True}], StyleBox[", ", "Reference"], StyleBox["10", "Reference", FontColor->RGBColor[1, 0, 0], FontVariations->{"Underline"->True}], "] with 26 equations. The computations are done with ", StyleBox["Mathematica", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], StyleBox[" 5.1", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], " [", StyleBox["14", "Reference", FontColor->RGBColor[1, 0, 0], FontVariations->{"Underline"->True}], StyleBox[",", "Reference"], StyleBox["15", "Reference", FontColor->RGBColor[1, 0, 0], FontVariations->{"Underline"->True}], "] and specialized packages for graphs, such as ", StyleBox["DecisionAnalysis`Combinatorica ", "MB", FontColor->RGBColor[0, 0, 1]], StyleBox["[", "MB"], StyleBox["11", "Reference", FontColor->RGBColor[1, 0, 0], FontVariations->{"Underline"->True}], StyleBox["] and", "MB"], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox["DiscreteMath`GraphPlot ", "MB", FontColor->RGBColor[0, 0, 1]], StyleBox["which documentation can be found ", "MB"], StyleBox[" ", "MB", FontColor->RGBColor[0, 0, 1]], StyleBox["at", "MB"], StyleBox[" ", "MB", FontColor->RGBColor[0, 0, 1]], StyleBox["http:/library.wolfram.com/infocenter/", "MB", FontColor->RGBColor[1, 0, 0], FontVariations->{"Underline"->True}], StyleBox[". 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The Klein - Goldberger Macroeconometric Model " }], "Section", TextJustification->1, CellTags->"c:5"], Cell[TextData[{ "The Klein-Goldberger model for the United States [", StyleBox["3", "Reference", FontColor->RGBColor[1, 0, 0], FontVariations->{"Underline"->True}], StyleBox[",", "Reference"], StyleBox["10", "Reference", FontColor->RGBColor[1, 0, 0], FontVariations->{"Underline"->True}], "] (henceforth KG-model in the text) has been the essential reference in \ earlier econometric modelling which mainly purposes are forecasting and \ simulation of economic policies as well [", StyleBox["4", "Reference", FontColor->RGBColor[1, 0, 0], FontVariations->{"Underline"->True}], "]. Many analysis have been achieved with this small size keynesian model \ to discover its static and dynamic properties [", StyleBox["3", "Reference", FontColor->RGBColor[1, 0, 0], FontVariations->{"Underline"->True}], StyleBox[",", "Reference"], StyleBox["10", "Reference", FontColor->RGBColor[1, 0, 0], FontVariations->{"Underline"->True}], "]. In this section we will present the containts of the model and the \ organization of its equations." }], "Text", TextJustification->1], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], ". Presentation" }], "Subsection", TextJustification->1, CellTags->{"c:6", "b:0.1.0"}], Cell[TextData[{ " An economic model with n ", StyleBox["endogeneous", FontSlant->"Italic"], " (", Cell[BoxData[ \(X\_\(\(i\)\(,\)\(\ \)\)\)]], " i =1,n) and m ", StyleBox["exogeneous", FontSlant->"Italic"], " (", Cell[BoxData[ \(\(\(Z\_k, k = 0, m - 1\)\()\)\)\)]], "variables where ", Cell[BoxData[ \(Z\_0\)]], " is a constant, can be described by the following equations system " }], "Text", TextJustification->1], Cell[BoxData[ \(TraditionalForm\`X\_\(\(i\)\(\ \ \)\)\ \ = \ \(f\_i\)(X\_1, ... , X\_\(i - 1\), X\_\(i + 1\), ... , X\_n, \(Z\_\(\(0\)\(,\)\)\) Z\_1 ... , Z\_m), \ i, \(\(=\)\(1\)\), n, \)], "DisplayFormula", TextJustification->1], Cell["\<\ In the following list, we present the 26 endogeneous of the model. \ We indicate first the number of the equation that determines only one \ variable, the name of the variable and an approximative but more readible \ definition. 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To associate a graph to this model, the \ economist must say how to \"read\" the model : there may be different ways to \ associate the endogeneous variable to one equation of a model. A model may \ have different theoretical explanations. This problem is known under the \ graph-theoretic term ", StyleBox[" a m", FontVariations->{"CompatibilityType"->0}], StyleBox["aximal bipartite matching", "Text", FontVariations->{"CompatibilityType"->0}], StyleBox[". ", "SB", FontWeight->"Plain"], StyleBox[". A matching in the bipartite graph g =(X,Y,e) matches each \ vertex in X (the set of variables) to one in Y (the set of equations). ", "Text", FontWeight->"Plain"], StyleBox[" . The primitive ", "Text"], StyleBox["AddEdges[g,e]", "Text", FontColor->RGBColor[0, 0, 1]], StyleBox[" of the package ", "Text"], StyleBox[" ", "Text", FontColor->RGBColor[0, 1, 0]], StyleBox["Combinatorica ", "Text", FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], StyleBox["returns a graph g(X,Y,e) . An adequate test for that graph is \ given by", "Text"], StyleBox[" BipartiteQ[g]", "Text", FontColor->RGBColor[0, 0, 1]], StyleBox[". The ranked embedded graph is rendered by ", "Text"], StyleBox["RankedEmbedding[g,t]", "Text", FontColor->RGBColor[0, 0, 1]], StyleBox[", where parameter t numbers the vertices from 1 to 26 for the \ variables and from 27 to 52 for the equations. Finally, we may highlight \ edges that precise what association can be retained. The complete set of \ primitives is given by ", "Text"], StyleBox["g=AddEdges[EmptyGraph[26,Type\[Rule]Directed],{edges}]", FontColor->RGBColor[0, 0, 1]], ", where the term \"edges\" represents pairs like {x1,y1}, \ {x3,y1},(x2,y2),... such that equation y1 can determine variables x1 or x3 \ and where equation y2 determine x2, ", StyleBox["ShowGraph[SetGraphOptions[RankedEmbedding[g,t],{matching},options]\ ]", FontColor->RGBColor[0, 0, 1]], ", where matching represents the list of edges realizing the property of a \ matching in a bipartite graph. To load the package ", StyleBox["Combinatorica", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], " evaluate " }], "Text", TextJustification->1], Cell[BoxData[ \(\(\(<< DiscreteMath`Combinatorica`\)\(\ \)\)\)], "InputOnly"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], ". Block Triangularity" }], "Subsection", TextJustification->1, CellTags->{"c:6", "b:0.1.0"}], Cell[TextData[{ "The 26-vertex graph is displayed by the primitive ", StyleBox["g = AddEdges[EmptyGraph[26,Type->Directed],{edges}]]", FontColor->RGBColor[0, 0, 1]], " where the adjacency list of directed edges takes the form {{x,y},....} \ when x -> y. In the static version of the KG-model the graph is composed of \ 26 vertices and 68 edges. The adjacency 0-1 matrix of the graph g has ", Cell[BoxData[ \(26\^2\)]], " entries. It is achieved by using the the command ", StyleBox["ToAdjacencyMatrix[g]// TableForm", FontColor->RGBColor[0, 0, 1]], ". The adjacency matrix of the KG-model model shows a scarce matrix. The \ Figure 1. (right) shows how a block triangularity can be achieved. 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", "SB"], StyleBox["The block triangularity of the Klein-Goldberger model in static \ versio", FontWeight->"Bold"], "n. The figures illustrate the steps of the procedure. The figure (left) \ shows a boolean 0-1 scarce adjacency matrix . These tables have been obtained \ with help of the", StyleBox[" ", FontVariations->{"CompatibilityType"->0}], StyleBox["primitives of the ", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["DicreteMath`GraphPlot", "SB", FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True, "CompatibilityType"->0}], StyleBox[" package. In this package, the initial matrix is displayed by \ the primitive ", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["ArrayPlot[ind, ColorsRules->{0->White,1->Black}]", "SB", FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], StyleBox[". 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The figure \ (middle) shows a block triangular matrix with as many non zero entries on the \ diagonal as possible.Two primitives are used to make the matrix block \ triangular :", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[" MaximalBipartiteMatching[g]", "SB", FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], StyleBox[" to permute entries to the diagonal, and ", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["StrongComponents[mat]", "SB", FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], StyleBox[". 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(middle) are" }], "Text", TextJustification->1, CellTags->"b:0.0.1"], Cell[BoxData[ FormBox[ FrameBox[\(\(\ \)\(\(ind\ = \ MaximalBipartiteMatching[spar];\)\[IndentingNewLine] \ \ \ \ \ \ \ \ \(pind\ = \ SparseArray[{ind \[Rule] Table[1, {Length[ind]}]}, {7, 7}];\)\[IndentingNewLine] \ \ \ \ \ \ \ \ \ \ \ \ \ pind\ = \ Transpose[pind]; \ mat2\ = \ pind\ \ . \ spar; \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ figure2\ = \ ArrayPlot[mat2, ColorRules \[Rule] {0 \[Rule] White, 1 \[Rule] Black}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ TextStyle \[Rule] \ {FontSize\ \[Rule] 12, FontFamily -> "\"}]\)\)], TraditionalForm]], "InputOnly", Background->GrayLevel[0.900008], CellTags->"b:0.0.1"], Cell["The primitives of Figure 1. (right) are ", "Text", CellTags->"b:0.0.1"], Cell[BoxData[ FormBox[ FrameBox[ RowBox[{" ", RowBox[{ RowBox[{\(perm\ = \ StrongComponents[mat2]\), ";", " ", \(perm2\ = \ Flatten[perm]\), ";", " ", FormBox[\(\(\ \ \ \ \ \ \ \)\(func = mat2[\([perm2, perm2]\)]\)\), "TraditionalForm"], ";"}], "\[IndentingNewLine]", " ", FormBox[\(figure3\ = ArrayPlot[func, ColorRules \[Rule] {0 \[Rule] White, 1 \[Rule] Black}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotLabel -> "\<(c) block triangular 0-1 \ matrix\>", \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ TextStyle \[Rule] {FontSize\ \[Rule] 12, FontFamily -> "\"}]\), "TraditionalForm"]}]}]], TraditionalForm]], "InputOnly", Background->GrayLevel[0.900008], CellTags->"b:0.0.1"], Cell["\<\ The 26- equations of the model can be organized according to that \ structure and solved recursively.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ". Structural Analysis of the Klein - Goldberger Model " }], "Section", TextJustification->1, CellTags->"c:5"], Cell[TextData[{ "We will also use specialized packages for drawing graphs such as ", StyleBox["DecisionAnalysis`Combinatorica", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], " , ", StyleBox["DiscreteMath`GraphPlot", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], " and our own ", StyleBox["Mathlink", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], "-compatible external programs written in ", StyleBox["Fortran F77L", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], " \"", StyleBox["Baobab", FontColor->RGBColor[1, 0, 0]], "\" for the enumeration of circuits. The Section 2.2 deals with the \ connectivity of the graph .We will examine two versions of the model : a \ \"static\" one where the exogeneous (instantaeous and delayed) and the \ delayed endogeneous variables have been replaced by their observed values; \ the other is\" dynamic\" since the delayed endogeneous variables are taken \ into account." }], "Text", TextJustification->1], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], ". Graph Representation" }], "Subsection", TextJustification->1, CellTags->{"c:7", "b:0.1.1"}], Cell[TextData[{ "The 26-vertex graph of the KG-model is displayed by the primitive", StyleBox[" g = AddEdges[EmptyGraph[26,Type->Directed],{edges}]]", FontColor->RGBColor[0, 0, 1]], ". The whole graph is shown in the right Figure 2.(left) (section 2.2) . \ All vertices are organized around a circle. In this application, we use the \ primitives" }], "Text"], Cell[BoxData[ FormBox[ FrameBox[\(ShowGraph[g, \[IndentingNewLine]VertexNumber \[Rule] True, VertexColor \[Rule] Blue, TextStyle \[Rule] {FontSize\ \[Rule] 12, FontColor \[Rule] Red, FontFamily -> "\"}, \[IndentingNewLine]\ \ \ \ \ \ HeadCenter \[Rule] .9, HeadWidth \[Rule] .25, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ Background \[Rule] LightBlue, AspectRatio \[Rule] 1. ]\)], TraditionalForm]], "Input", Background->GrayLevel[0.900008]], Cell["", "Text"], Cell[TextData[{ StyleBox["Our proposal may be described as follows : firstly identify the \ largest strong connected components, secondly search for the (or one) \ largest circuit and place \n the vertices of the largest circuit on a \ polygon, thirdly put (inside the polygon) the remaining vertices of the \ component , introduce the remaining vertices outside and finally rely all \ the pairs of vertices by oriented edges (arrows) according to the \ information given by the graph. A polygonal curve is created by the \ primitives of ", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["Mathematica", "SB", FontWeight->"Plain", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True, "CompatibilityType"->0}], StyleBox[" ", "SB", FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], StyleBox["with ", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["data = Table[{Cos[n 2 Pi/20],Sin[n 2 Pi/20}],{n,0,20}]; \ Polygon[data]; InputForm[%]; Show[%,options] . ", "SB", FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], StyleBox["The arrows between vertices are ", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], " then drawn by using the primitives of the ", StyleBox["Standard Add-on", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], " package ", StyleBox[" Graphics`Arrow", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], StyleBox[" ", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], ", Thus, we have to use the function ", StyleBox["Arrow[{x,y},{u,v}]", FontColor->RGBColor[0, 0, 1]], " for P(x,y) ->Q(u,v). and ", StyleBox["Show[Graphics[{Arrow[{x,y},{u,v}] ,Arrow[{x,y},{r,s}],...}]] ", FontColor->RGBColor[0, 0, 1]], "to draw the entire graph. The Figure 2 (middle) shows a more readible \ graph. In this figure the graph is ", StyleBox[" restricted to the strong connected component.", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}] }], "Text", TextJustification->1, CellTags->"b:0.0.0"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], ". Strongly Connected Components" }], "Subsection", TextJustification->1, CellTags->{"c:7", "b:0.1.1"}], Cell[TextData[{ "\n The Figure 2. (left) shows the initial graph by ", StyleBox["C", FontColor->RGBColor[0, 0, 1]], StyleBox["ombinatorica", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], " , the largest strong connected component in Figure 2. (middle) and the \ reduced graph of the model in Figure 2 (right). We can soon deduce two main \ observations : the model is largely interdependant with in the graph a \ strong connected component composed of 20 vertices, the reduced graph \ emphazises the keynesian character of that model with a dominant investment, \ interest rates (short term and long term) that affect mainly the liquid \ assets held by agents together with the interdependant variables. 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Graph representations of the static KG-model. ", "SB"], StyleBox["The figure (left) has been drawn with help of the command ", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["ShowGraph[g ,options]", "SB", FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], StyleBox[" of the package ", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["Combinatorica", "SB", FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], StyleBox[". The figure (middle) shows the largest strong connected \ component. The presentation is that we preconize using the primitives of", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[" Mathematica", "SB", FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], StyleBox[" and those ", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[" of the ", FontVariations->{"CompatibilityType"->0}], StyleBox["Standard ", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True, "CompatibilityType"->0}], StyleBox["Add-on", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True, "CompatibilityType"->0}], StyleBox[" package", FontVariations->{"CompatibilityType"->0}], StyleBox[" Graphics`Arrow", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], StyleBox[" . The figure (right) represents the whole graph in a reduced \ form without circuits. In this figure, such that the strong connected \ component has been contracted to a single vertex. The structure is conformed \ to the keysesian theory of short term. The variable investment dominates , \ benefits of firms have an output position, and interest rates affect the \ liquid assets held by persons and businesses.", FontVariations->{"CompatibilityType"->0}] }], "NumberedFigureCaption", TextJustification->1, CellTags->"b:0.1.1"], Cell[TextData[{ "The contracted graph is obtained by using the command ", StyleBox["ShowGraphArray[{g,Contract[g,{list of vertices to \ delete}]},VertexNumber->True] .", FontColor->RGBColor[0, 0, 1]], "The primitive ", StyleBox["Contract ", FontColor->RGBColor[0, 0, 1]], "has some effects : this command maintains the embedding of the initial \ graph, deleting a vertex it deletes all incident edges, a new vertex is \ placed ar the midpoint of the contracted vertices, and the vertices are \ automatically renumbered. " }], "Text", TextJustification->1, CellTags->"b:0.1.1"], Cell["\<\ The Figure 3. show how the structure of the modified when we \ consider the delayed endogeneous variables ( the dynamic KG-model). Sure, \ the density of relations is increased in the initial graph. The largest \ strong connected component includes now variables that were \"external\" in \ the static version such as : investment, profit and liabilities. Interest \ rates are now dominant variables.\ \>", "Text", TextJustification->1, CellTags->"b:0.1.1"], Cell["\<\ The comparison between Figures 2. 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Graph representations of the dynamic KG model. ", "SB"], StyleBox["The figures for the dynamic version show some notable \ modifications. The strongly connected component (the largest) is more \ important and integrates investment and benefits of firms . Interest rates \ dominates in the figure (right).", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}] }], "NumberedFigureCaption", TextJustification->1, CellTags->"b:0.1.1"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], ". Lexicographically Ordered Circuits " }], "Subsection", TextJustification->1, CellTags->{"c:8", "b:0.1.2"}], Cell["\<\ The Figure 4. shows the repartition of the circuits for both static and \ dynamic versions of the KG-model with two presentations, a bart chart and a \ pie chart. 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{{335.125, 518.625}, {226.688, 96.5625}} -> {-6.88912, -1.85694, \ 0.0161385, 0.0161385}}], Cell[TextData[{ StyleBox["Figure 4. Lexicographic ordered circuits. ", "SB"], StyleBox["Our program", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[" \"Baobab\"", "SB", FontWeight->"Plain", FontColor->RGBColor[1, 0, 0], FontVariations->{"CompatibilityType"->0}], StyleBox[" gives an exhaustive list of circuits and classifies them \ according to an lexicographic order. We distinguish the static and the \ dynamic version of the KG-model. For each version, we produce a barchart \ using the command ", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["BarChart[data,options]", "SB", FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], StyleBox[" of the ", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["Standard Add-on", "SB", FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True, "CompatibilityType"->0}], StyleBox[" Package ", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["Graphics`Graphics", "SB", FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], StyleBox[" and ", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["Graphics`Legend.", "SB", FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], StyleBox[" We use the command in the following way : ", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["figure = BartChart[datalist, options]; ShowLegend[Show[figure],{{\ \"text\"}},options}]. ", "SB", FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], StyleBox["Pie charts are created by the simple command ", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox[" PieChart[ datalist,options]", "SB", FontWeight->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], StyleBox[" . The number of circuits in the dynamic version exceeds that \ one of the static version with 586 circuits (against 255 in the static \ KG-model) and with a largest length ot 18 vertices in dynamic version \ (against 15 in the static KG-model).\n ", "SB", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}] }], "NumberedFigureCaption", TextJustification->1, CellTags->"b:0.1.2"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], ". Edge-Disjoint Cycles" }], "Subsection", TextJustification->1, CellTags->{"c:8", "b:0.1.2"}], Cell[TextData[{ "\nWhen investigating all the circuits, we may be interested in the \ circuits that have no edge in common. The primitive ", StyleBox["ExtractCycles[g] ", FontColor->RGBColor[0, 0, 1]], "of the package", StyleBox[" Combinatorica ", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}], " determines these \"independant circuits\". In Figure 5. we have \ highlighted the extracted circuits with different colors." }], "Text", TextJustification->1], Cell[TextData[{ StyleBox["In the static KG-model", FontSlant->"Italic"], " (255 circuits) four \"independant\" circuits are observed . These \ circles has been colored differently. The figures with a lightblue \ background are those of the static KG-model and the other (right) that one of \ the dynamic KG-model. The Figure 5. (middle) shows an improvement in the \ presentation., when the position of one vertex is modified. The following \ results give the economic signification of such Figure 5. (middle). The color \ of the text corresponds to that of the circuits . The four \"independant\" \ circuits of the static version can be interpretated as follows with the \ number and the abreviation of the variables (see also the table of \ definitions in Section 1.1). The circuit ", StyleBox["{ 19, 5, 19} \[Congruent] {K, D, K}", FontWeight->"Bold", FontColor->RGBColor[0, 1, 0]], " represents real capital accumulation in the Figure 5 (middle) , the \ circuit ", StyleBox["{9, 16 6, 9} \[Congruent] {", FontWeight->"Bold", FontColor->RGBColor[1, 0.501961, 0]], Cell[BoxData[ \(Fi\)], FontWeight->"Bold", FontColor->RGBColor[1, 0.501961, 0]], StyleBox[", YTD, ", FontWeight->"Bold", FontColor->RGBColor[1, 0.501961, 0]], Cell[BoxData[ \(W\_1\)], FontWeight->"Bold", FontColor->RGBColor[1, 0.501961, 0]], StyleBox[",Fi}", FontWeight->"Bold", FontColor->RGBColor[1, 0.501961, 0]], " between the national product and importations, the circuit ", StyleBox["{3, 10, 17, 4, 3}\[Congruent] {", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(S\_p\)], FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" , ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(A\_1\)], FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[", P, ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(P\_c\)], FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[", ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(S\_p\)], FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["}", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " represents the repartition of the revenues, and the circuit ", StyleBox["{17, 1, 16, 5, 26, 17} \[Congruent] {P, C, YTD, D, Y, P}", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " is the circuit of the national revenue. " }], "Text", TextJustification->1, CellTags->"b:0.1.2"], Cell[TextData[{ " ", StyleBox["In the dynamic KG-model", FontSlant->"Italic"], " (586 circuits) we have eight \"independant\" circuits which \ interpretation is : ", StyleBox["{17, 10, 17} \[Congruent] {P, ", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(A\_1\)], FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[",P}", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " a circuit between farm and nonfarm incomes, the circuit ", StyleBox["{18,8,18} \[Congruent] {p,w,p}", FontWeight->"Bold", FontColor->RGBColor[1, 0.501961, 0]], StyleBox[" ", FontWeight->"Bold"], StyleBox["is", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontWeight->"Bold"], "the Phillips curve , the circuit ", StyleBox["{19, 2, 19} \[Congruent] {K, I, K}", FontWeight->"Bold", FontColor->RGBColor[0, 1, 0.25098]], " for investment, the circuit ", StyleBox["{20, 3,20} \[Congruent] {B, ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(S\_p\)], FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[",B}", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " between the corporate and savings, ", StyleBox["{6,1,16,6} \[Congruent] {", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(W\_1\)], FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[",C,YTD, ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(W\_1\)], FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["}", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], "a dark blue circuit for consumption, the circuit ", StyleBox["{10,2, 16, 7, 18,10} \[Congruent] {", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], Cell[BoxData[ \(A\_1\)], FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[", I,TTD, ", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], Cell[BoxData[ \(N\_w\)], FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[" , p, ", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], Cell[BoxData[ \(A\_1\)], FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox["}", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], " a purple circuit for investment demand , inflation and revenues, and the \ largest circuit of the revenue ", StyleBox["{4, 3, 10, 9, 16, 21, 26, 17,4} \[Congruent] {", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(P\_C\)], FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[", ", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(S\_P\)], FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[", ", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(A\_1\)], FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[", ", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ \(Fi\)], FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[", YTD, T, Y, P, ", 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