Function Repository Resource:

WeakPathGraphs

Create path graphs between two vertices in a graph where the paths are constructed without consideration of the direction of the edges

Contributed by: Seth J. Chandler

ResourceFunction["WeakPathGraphs"][g,start,end]

takes a graph g and finds all subgraphs for paths of any length between start and end in the undirected version of g.

ResourceFunction["WeakPathGraphs"][g,start,end,kspec]

returns only paths satisfying the length specified by kspec as in FindPath.

ResourceFunction["WeakPathGraphs"][g,start,end,kspec,n]

finds at most n paths.

Details and Options

The documentation for FindPath provides detailed information on how to use the kspec parameter.

One can use all options for Graph in ResourceFunction["WeakPathGraphs"]. Options given to ResourceFunction["WeakPathGraphs"] will take precedence over the option settings from the underlying graph.

An issue in creating weak path graphs is inheritance of options from the underlying graph where the user has not specified option settings. Sometimes it will be desirable to preserve those options, sometimes it will be undesirable, and sometimes preservation of the options will prevent rendering of the ResourceFunction["WeakPathGraphs"].

The "Inherit" and "Disinherit" options give you additional control over this option inheritance. The "Inherit" option lets you specify what options from the underlying graph you wish to keep.

The following settings are supported for "Inherit" or "Disinherit":

Automatic

(default) inherit or disinherit all settings that do not prevent rendering in the output

All

inherit or disinherit all settings

None

inherit or disinherit no settings

{opt₁,opt₂,…}

list of Graph option names to inherit or disinherit

The "Disinherit" option overrides option inheritance decisions made by the "Inherit" option. Thus, if one wanted to use All or Automatic to preserve most of the options but not VertexShapeFunction, one might use "Inherit"→Automatic, "Disinherit"→{VertexShapeFunction} as option settings.

Note, however: any options set directly from within ResourceFunction["WeakPathGraphs"] take precedence over any option settings in the underlying graph.

Examples

Basic Examples (2)

Find a weak path graph for a simple directed graph:

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Take a graph and find all paths between "sodium" and "sbp" assuming the graph was undirected, but preserve the direction of the original edges in the resulting "weak path graph":

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Scope (6)

WeakPathGraphs works with undirected graphs:

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$ResourceFunction["WeakPathGraphs"][\!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {Null, SparseArray[ Automatic, {12, 12}, 0, {1, {{0, 2, 5, 7, 10, 14, 17, 20, 24, 27, 29, 32, 34}, {{ 2}, {4}, {1}, {3}, {5}, {2}, {6}, {1}, {5}, {7}, {2}, { 4}, {6}, {8}, {3}, {5}, {9}, {4}, {8}, {10}, {5}, {7}, { 9}, {11}, {6}, {8}, {12}, {7}, {11}, {8}, {10}, {12}, { 9}, {11}}}, Pattern}]}, {GraphLayout -> {"GridEmbedding", "Dimension" -> {3, 4}}, VertexShapeFunction -> {2 -> "Name", 7 -> "Name"}}]], Typeset`boxes, Typeset`boxes$s2d = GraphicsGroupBox[{{ Directive[ Opacity[0.7], Hue[0.6, 0.7, 0.5]], LineBox[{{ DynamicLocation["VertexID$1", Automatic, Center], DynamicLocation["VertexID$2", Automatic, Center]}, { DynamicLocation[ "VertexID$1", Automatic, Center], DynamicLocation["VertexID$4", Automatic, Center]}, { DynamicLocation[ "VertexID$2", Automatic, Center], DynamicLocation["VertexID$3", Automatic, Center]}, { DynamicLocation[ "VertexID$2", Automatic, Center], DynamicLocation["VertexID$5", Automatic, Center]}, { DynamicLocation[ "VertexID$3", Automatic, Center], DynamicLocation["VertexID$6", Automatic, Center]}, { DynamicLocation[ "VertexID$4", Automatic, Center], DynamicLocation["VertexID$5", Automatic, Center]}, { DynamicLocation[ "VertexID$4", Automatic, Center], DynamicLocation["VertexID$7", Automatic, Center]}, { DynamicLocation[ "VertexID$5", Automatic, Center], DynamicLocation["VertexID$6", Automatic, Center]}, { DynamicLocation[ "VertexID$5", Automatic, Center], DynamicLocation["VertexID$8", Automatic, Center]}, { DynamicLocation[ "VertexID$6", Automatic, Center], DynamicLocation["VertexID$9", Automatic, Center]}, { DynamicLocation[ "VertexID$7", Automatic, Center], DynamicLocation["VertexID$8", Automatic, Center]}, { DynamicLocation[ "VertexID$7", Automatic, Center], DynamicLocation["VertexID$10", Automatic, Center]}, { DynamicLocation[ "VertexID$8", Automatic, Center], DynamicLocation["VertexID$9", Automatic, Center]}, { DynamicLocation[ "VertexID$8", Automatic, Center], DynamicLocation["VertexID$11", Automatic, Center]}, { DynamicLocation[ "VertexID$9", Automatic, Center], DynamicLocation["VertexID$12", Automatic, Center]}, { DynamicLocation[ "VertexID$10", Automatic, Center], DynamicLocation["VertexID$11", Automatic, Center]}, { DynamicLocation[ "VertexID$11", Automatic, Center], DynamicLocation["VertexID$12", Automatic, Center]}}]}, { Directive[ Hue[0.6, 0.2, 0.8], EdgeForm[ Directive[ GrayLevel[0], Opacity[0.7]]]], TagBox[ DiskBox[{1., 1.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$1"], TagBox[ InsetBox[ BoxData[ FormBox[ PaneBox["2", Alignment -> Center, ImageMargins -> 2], TraditionalForm]], {1., 2.}, BaseStyle -> "Graphics"], "DynamicName", BoxID -> "VertexID$2"], TagBox[ DiskBox[{1., 3.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$3"], TagBox[ DiskBox[{2., 1.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$4"], TagBox[ DiskBox[{2., 2.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$5"], TagBox[ DiskBox[{2., 3.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$6"], TagBox[ InsetBox[ BoxData[ FormBox[ PaneBox["7", Alignment -> Center, ImageMargins -> 2], TraditionalForm]], {3., 1.}, BaseStyle -> "Graphics"], "DynamicName", BoxID -> "VertexID$7"], TagBox[ DiskBox[{3., 2.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$8"], TagBox[ DiskBox[{3., 3.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$9"], TagBox[ DiskBox[{4., 1.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$10"], TagBox[ DiskBox[{4., 2.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$11"], TagBox[ DiskBox[{4., 3.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$12"]}}], $CellContext`flag}, TagBox[ DynamicBox[GraphComputation`NetworkGraphicsBox[ 3, Typeset`graph, Typeset`boxes, $CellContext`flag], {CachedValue :> Typeset`boxes, SingleEvaluation -> True, SynchronousUpdating -> False, TrackedSymbols :> {$CellContext`flag}}, ImageSizeCache->{{-1., 89.24835191419872}, {-35.275820010886434`, 32.}}], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False, UnsavedVariables:>{$CellContext`flag}]], DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FrameTicks->None]\), 2, 7, ImageSize -> 100, VertexLabels -> "Name"]$

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Same idea but with different options to improve rendering:

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Directed graphs:

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$ResourceFunction["WeakPathGraphs"][\!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, { SparseArray[ Automatic, {12, 12}, 0, {1, {{0, 2, 4, 5, 7, 9, 10, 12, 14, 15, 16, 17, 17}, {{ 2}, {4}, {3}, {5}, {6}, {5}, {7}, {6}, {8}, {9}, {8}, { 10}, {9}, {11}, {12}, {11}, {12}}}, Pattern}], Null}, {EdgeStyle -> { Arrowheads[0.08]}, GraphLayout -> {"GridEmbedding", "Dimension" -> {3, 4}}, VertexShapeFunction -> {2 -> "Name", 12 -> "Name"}}]], Typeset`boxes, Typeset`boxes$s2d = GraphicsGroupBox[{{ Arrowheads[0.028937842778793414`], Directive[ Opacity[0.7], Hue[0.6, 0.7, 0.5]], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$1", Automatic, Center], DynamicLocation["VertexID$2", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$1", Automatic, Center], DynamicLocation["VertexID$4", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$2", Automatic, Center], DynamicLocation["VertexID$3", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$2", Automatic, Center], DynamicLocation["VertexID$5", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$3", Automatic, Center], DynamicLocation["VertexID$6", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$4", Automatic, Center], DynamicLocation["VertexID$5", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$4", Automatic, Center], DynamicLocation["VertexID$7", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$5", Automatic, Center], DynamicLocation["VertexID$6", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$5", Automatic, Center], DynamicLocation["VertexID$8", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$6", Automatic, Center], DynamicLocation["VertexID$9", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$7", Automatic, Center], DynamicLocation["VertexID$8", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$7", Automatic, Center], DynamicLocation["VertexID$10", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$8", Automatic, Center], DynamicLocation["VertexID$9", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$8", Automatic, Center], DynamicLocation["VertexID$11", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$9", Automatic, Center], DynamicLocation["VertexID$12", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$10", Automatic, Center], DynamicLocation["VertexID$11", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False], StyleBox[ ArrowBox[ LineBox[{ DynamicLocation["VertexID$11", Automatic, Center], DynamicLocation["VertexID$12", Automatic, Center]}]], Arrowheads[0.08], StripOnInput -> False]}, { Directive[ Hue[0.6, 0.2, 0.8], EdgeForm[ Directive[ GrayLevel[0], Opacity[0.7]]]], TagBox[ DiskBox[{1., 1.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$1"], TagBox[ InsetBox[ BoxData[ FormBox[ PaneBox["2", Alignment -> Center, ImageMargins -> 2], TraditionalForm]], {1., 2.}, BaseStyle -> "Graphics"], "DynamicName", BoxID -> "VertexID$2"], TagBox[ DiskBox[{1., 3.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$3"], TagBox[ DiskBox[{2., 1.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$4"], TagBox[ DiskBox[{2., 2.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$5"], TagBox[ DiskBox[{2., 3.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$6"], TagBox[ DiskBox[{3., 1.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$7"], TagBox[ DiskBox[{3., 2.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$8"], TagBox[ DiskBox[{3., 3.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$9"], TagBox[ DiskBox[{4., 1.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$10"], TagBox[ DiskBox[{4., 2.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$11"], TagBox[ InsetBox[ BoxData[ FormBox[ PaneBox["12", Alignment -> Center, ImageMargins -> 2], TraditionalForm]], {4., 3.}, BaseStyle -> "Graphics"], "DynamicName", BoxID -> "VertexID$12"]}}], $CellContext`flag}, TagBox[ DynamicBox[GraphComputation`NetworkGraphicsBox[ 3, Typeset`graph, Typeset`boxes, $CellContext`flag], {CachedValue :> Typeset`boxes, SingleEvaluation -> True, SynchronousUpdating -> False, TrackedSymbols :> {$CellContext`flag}}, ImageSizeCache->{{-1.015368312707917, 91.01536831270788}, {-35.1958952588157, 28.44432240452428}}], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False, UnsavedVariables:>{$CellContext`flag}]], DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FrameTicks->None]\), 2, 12, {1, 6}, 5, ImageSize -> 200, VertexSize -> 0.7, VertexLabels -> Placed[Automatic, Center], "Disinherit" -> VertexShapeFunction]$

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Multigraphs:

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$ResourceFunction["WeakPathGraphs"][\!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {Null, {{1, 2}, {1, 4}, {2, 3}, {2, 5}, {3, 6}, {4, 5}, { 4, 7}, {5, 6}, {5, 8}, {6, 9}, {7, 8}, {7, 10}, {8, 9}, {8, 11}, {9, 12}, {10, 11}, {11, 12}, {5, 8}}}, {PerformanceGoal -> "Q", VertexCoordinates -> {{1., 1.}, {1., 2.}, {1., 3.}, {2., 1.}, {2., 2.}, {2., 3.}, {3., 1.}, {3., 2.}, {3., 3.}, {4., 1.}, {4., 2.}, {4., 3.}}, VertexShapeFunction -> {2 -> "Name", 7 -> "Name"}}]], Typeset`boxes, Typeset`boxes$s2d = GraphicsGroupBox[{{ Directive[ Opacity[0.7], Hue[0.6, 0.7, 0.5]], FontFamily -> "Arial", LineBox[{ DynamicLocation["VertexID$1", Automatic, Center], DynamicLocation["VertexID$2", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$1", Automatic, Center], DynamicLocation["VertexID$4", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$2", Automatic, Center], DynamicLocation["VertexID$3", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$2", Automatic, Center], DynamicLocation["VertexID$5", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$3", Automatic, Center], DynamicLocation["VertexID$6", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$4", Automatic, Center], DynamicLocation["VertexID$5", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$4", Automatic, Center], DynamicLocation["VertexID$7", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$5", Automatic, Center], DynamicLocation["VertexID$6", Automatic, Center]}], BezierCurveBox[{ DynamicLocation["VertexID$5", Automatic, Center], { 2.5000000000000004`, 2.1378065801163424`}, DynamicLocation["VertexID$8", Automatic, Center]}], BezierCurveBox[{ DynamicLocation["VertexID$5", Automatic, Center], { 2.5000000000000004`, 1.862193419883658}, DynamicLocation["VertexID$8", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$6", Automatic, Center], DynamicLocation["VertexID$9", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$7", Automatic, Center], DynamicLocation["VertexID$8", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$7", Automatic, Center], DynamicLocation["VertexID$10", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$8", Automatic, Center], DynamicLocation["VertexID$9", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$8", Automatic, Center], DynamicLocation["VertexID$11", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$9", Automatic, Center], DynamicLocation["VertexID$12", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$10", Automatic, Center], DynamicLocation["VertexID$11", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$11", Automatic, Center], DynamicLocation["VertexID$12", Automatic, Center]}]}, { Directive[ Hue[0.6, 0.2, 0.8], EdgeForm[ Directive[ GrayLevel[0], Opacity[0.7]]]], FontFamily -> "Arial", TagBox[ DiskBox[{1., 1.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$1"], TagBox[ InsetBox[ BoxData[ FormBox[ PaneBox["2", Alignment -> Center, ImageMargins -> 2], TraditionalForm]], {1., 2.}, BaseStyle -> "Graphics"], "DynamicName", BoxID -> "VertexID$2"], TagBox[ DiskBox[{1., 3.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$3"], TagBox[ DiskBox[{2., 1.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$4"], TagBox[ DiskBox[{2., 2.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$5"], TagBox[ DiskBox[{2., 3.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$6"], TagBox[ InsetBox[ BoxData[ FormBox[ PaneBox["7", Alignment -> Center, ImageMargins -> 2], TraditionalForm]], {3., 1.}, BaseStyle -> "Graphics"], "DynamicName", BoxID -> "VertexID$7"], TagBox[ DiskBox[{3., 2.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$8"], TagBox[ DiskBox[{3., 3.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$9"], TagBox[ DiskBox[{4., 1.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$10"], TagBox[ DiskBox[{4., 2.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$11"], TagBox[ DiskBox[{4., 3.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$12"]}}], $CellContext`flag}, TagBox[ DynamicBox[GraphComputation`NetworkGraphicsBox[ 3, Typeset`graph, Typeset`boxes, $CellContext`flag], {CachedValue :> Typeset`boxes, SingleEvaluation -> True, SynchronousUpdating -> False, TrackedSymbols :> {$CellContext`flag}}, ImageSizeCache->{{-1., 89.24835191419872}, {-35.275820010886434`, 32.}}], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False, UnsavedVariables:>{$CellContext`flag}]], BaseStyle->(FontFamily -> "Arial"), DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FrameTicks->None, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], LabelStyle->{FontFamily -> "Arial"}]\), 2, 7, {3, 4}, 2, VertexSize -> 0.5, VertexLabels -> Placed[Automatic, Center], "Disinherit" -> VertexShapeFunction]$

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Mixed graphs:

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$ResourceFunction["WeakPathGraphs"][\!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {{{5, 8}}, {{1, 2}, {1, 4}, {2, 3}, {2, 5}, {3, 6}, {4, 5}, {4, 7}, {5, 6}, {6, 9}, {7, 8}, {7, 10}, {8, 9}, {8, 11}, {9, 12}, {10, 11}, {11, 12}}}, {EdgeStyle -> { Arrowheads[0.08]}, PerformanceGoal -> "Q", VertexCoordinates -> {{1., 1.}, {1., 2.}, {1., 3.}, {2., 1.}, {2., 2.}, {2., 3.}, {3., 1.}, {3., 2.}, {3., 3.}, {4., 1.}, {4., 2.}, {4., 3.}}, VertexShapeFunction -> {2 -> "Name", 7 -> "Name"}}]], Typeset`boxes, Typeset`boxes$s2d = GraphicsGroupBox[{{ Arrowheads[0.028937842778793414`], Directive[ Opacity[0.7], Hue[0.6, 0.7, 0.5]], FontFamily -> "Arial", Arrowheads[0.08], LineBox[{ DynamicLocation["VertexID$1", Automatic, Center], DynamicLocation["VertexID$2", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$1", Automatic, Center], DynamicLocation["VertexID$4", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$2", Automatic, Center], DynamicLocation["VertexID$3", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$2", Automatic, Center], DynamicLocation["VertexID$5", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$3", Automatic, Center], DynamicLocation["VertexID$6", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$4", Automatic, Center], DynamicLocation["VertexID$5", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$4", Automatic, Center], DynamicLocation["VertexID$7", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$5", Automatic, Center], DynamicLocation["VertexID$6", Automatic, Center]}], ArrowBox[{ DynamicLocation["VertexID$5", Automatic, Center], DynamicLocation["VertexID$8", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$6", Automatic, Center], DynamicLocation["VertexID$9", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$7", Automatic, Center], DynamicLocation["VertexID$8", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$7", Automatic, Center], DynamicLocation["VertexID$10", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$8", Automatic, Center], DynamicLocation["VertexID$9", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$8", Automatic, Center], DynamicLocation["VertexID$11", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$9", Automatic, Center], DynamicLocation["VertexID$12", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$10", Automatic, Center], DynamicLocation["VertexID$11", Automatic, Center]}], LineBox[{ DynamicLocation["VertexID$11", Automatic, Center], DynamicLocation["VertexID$12", Automatic, Center]}]}, { Directive[ Hue[0.6, 0.2, 0.8], EdgeForm[ Directive[ GrayLevel[0], Opacity[0.7]]]], FontFamily -> "Arial", TagBox[ DiskBox[{1., 1.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$1"], TagBox[ InsetBox[ BoxData[ FormBox[ PaneBox["2", Alignment -> Center, ImageMargins -> 2], TraditionalForm]], {1., 2.}, BaseStyle -> "Graphics"], "DynamicName", BoxID -> "VertexID$2"], TagBox[ DiskBox[{1., 3.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$3"], TagBox[ DiskBox[{2., 1.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$4"], TagBox[ DiskBox[{2., 2.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$5"], TagBox[ DiskBox[{2., 3.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$6"], TagBox[ InsetBox[ BoxData[ FormBox[ PaneBox["7", Alignment -> Center, ImageMargins -> 2], TraditionalForm]], {3., 1.}, BaseStyle -> "Graphics"], "DynamicName", BoxID -> "VertexID$7"], TagBox[ DiskBox[{3., 2.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$8"], TagBox[ DiskBox[{3., 3.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$9"], TagBox[ DiskBox[{4., 1.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$10"], TagBox[ DiskBox[{4., 2.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$11"], TagBox[ DiskBox[{4., 3.}, 0.030239520958083826`], "DynamicName", BoxID -> "VertexID$12"]}}], $CellContext`flag}, TagBox[ DynamicBox[GraphComputation`NetworkGraphicsBox[ 3, Typeset`graph, Typeset`boxes, $CellContext`flag], {CachedValue :> Typeset`boxes, SingleEvaluation -> True, SynchronousUpdating -> False, TrackedSymbols :> {$CellContext`flag}}, ImageSizeCache->{{-1., 89.24835191419872}, {-35.275820010886434`, 32.}}], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False, UnsavedVariables:>{$CellContext`flag}]], BaseStyle->(FontFamily -> "Arial"), DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FrameTicks->None, GridLinesStyle->Directive[ GrayLevel[0.5, 0.4]], LabelStyle->{FontFamily -> "Arial"}]\), 2, 7, {1, \[Infinity]}, 1, VertexSize -> 0.5, VertexLabels -> Placed[Automatic, Center]]$

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Weighted graphs:

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$ResourceFunction["WeakPathGraphs"][\!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{1, 2, 3, 4}, {Null, SparseArray[ Automatic, {4, 4}, 0, {1, {{0, 3, 5, 8, 10}, {{2}, {3}, {4}, {1}, {3}, {1}, { 2}, {4}, {1}, {3}}}, Pattern}]}, {EdgeLabels -> {"EdgeWeight"}, VertexShapeFunction -> {"Name"}, EdgeWeight -> {0.5, 1.2, 0.2, 0.9, 0.62}}]], Typeset`boxes, Typeset`boxes$s2d = GraphicsGroupBox[{{ Directive[ Opacity[0.7], Hue[0.6, 0.7, 0.5]], { TagBox[ LineBox[{ DynamicLocation["VertexID$1", Automatic, Center], DynamicLocation["VertexID$2", Automatic, Center]}], "DynamicName", BoxID -> "EdgeLabelID$1"], InsetBox[ FormBox["\"0.5\"", TraditionalForm], Offset[{0, 2}, DynamicLocation["EdgeLabelID$1", Automatic, Scaled[0.5]]], {0, -1}, BaseStyle -> "Graphics"]}, { TagBox[ LineBox[{ DynamicLocation["VertexID$1", Automatic, Center], DynamicLocation["VertexID$3", Automatic, Center]}], "DynamicName", BoxID -> "EdgeLabelID$2"], InsetBox[ FormBox["\"1.2\"", TraditionalForm], Offset[{0, 2}, DynamicLocation["EdgeLabelID$2", Automatic, Scaled[0.5]]], {0, -1}, BaseStyle -> "Graphics"]}, { TagBox[ LineBox[{ DynamicLocation["VertexID$1", Automatic, Center], DynamicLocation["VertexID$4", Automatic, Center]}], "DynamicName", BoxID -> "EdgeLabelID$3"], InsetBox[ FormBox["\"0.2\"", TraditionalForm], Offset[{0, 2}, DynamicLocation["EdgeLabelID$3", Automatic, Scaled[0.5]]], {0, -1}, BaseStyle -> "Graphics"]}, { TagBox[ LineBox[{ DynamicLocation["VertexID$2", Automatic, Center], DynamicLocation["VertexID$3", Automatic, Center]}], "DynamicName", BoxID -> "EdgeLabelID$4"], InsetBox[ FormBox["\"0.9\"", TraditionalForm], Offset[{0, 2}, DynamicLocation["EdgeLabelID$4", Automatic, Scaled[0.5]]], {0, -1}, BaseStyle -> "Graphics"]}, { TagBox[ LineBox[{ DynamicLocation["VertexID$3", Automatic, Center], DynamicLocation["VertexID$4", Automatic, Center]}], "DynamicName", BoxID -> "EdgeLabelID$5"], InsetBox[ FormBox["\"0.62\"", TraditionalForm], Offset[{0, 2}, DynamicLocation["EdgeLabelID$5", Automatic, Scaled[0.5]]], {0, -1}, BaseStyle -> "Graphics"]}}, { Directive[ Hue[0.6, 0.2, 0.8], EdgeForm[ Directive[ GrayLevel[0], Opacity[0.7]]]], TagBox[ InsetBox[ BoxData[ FormBox[ PaneBox["1", Alignment -> Center, ImageMargins -> 2], TraditionalForm]], {0.9329038775049628, 0.868997056961016}, BaseStyle -> "Graphics"], "DynamicName", BoxID -> "VertexID$1"], TagBox[ InsetBox[ BoxData[ FormBox[ PaneBox["2", Alignment -> Center, ImageMargins -> 2], TraditionalForm]], {0., 0.4347904415891483}, BaseStyle -> "Graphics"], "DynamicName", BoxID -> "VertexID$2"], TagBox[ InsetBox[ BoxData[ FormBox[ PaneBox["3", Alignment -> Center, ImageMargins -> 2], TraditionalForm]], {0.9339517502051011, 0.}, BaseStyle -> "Graphics"], "DynamicName", BoxID -> "VertexID$3"], TagBox[ InsetBox[ BoxData[ FormBox[ PaneBox["4", Alignment -> Center, ImageMargins -> 2], TraditionalForm]], {1.8680693714551628`, 0.43527754789663736`}, BaseStyle -> "Graphics"], "DynamicName", BoxID -> "VertexID$4"]}}], $CellContext`flag}, TagBox[ DynamicBox[GraphComputation`NetworkGraphicsBox[ 3, Typeset`graph, Typeset`boxes, $CellContext`flag], {CachedValue :> Typeset`boxes, SingleEvaluation -> True, SynchronousUpdating -> False, TrackedSymbols :> {$CellContext`flag}}, ImageSizeCache->{{12.122578125, 124.00632812500001`}, {-35.932423961161355`, 30.462325}}], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False, UnsavedVariables:>{$CellContext`flag}]], DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FrameTicks->None, ImagePadding->15, ImageSize->{136.12890625, Automatic}]\), 2, 4]$

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Weighted graphs with EdgeWeight disinherited:

In[10]:=

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Small graphs:

In[11]:=

$ResourceFunction["WeakPathGraphs"][Graph[{1 \[DirectedEdge] 2}], 1, 2, VertexLabels -> Placed[Automatic, Center], VertexSize -> 0.6]$

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Options (2)

By default, the weak path graphs created by this function inherit options from the underlying graph. One can use any option available to Graph to override those options or to add other options:

In[12]:=

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In[13]:=

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One can also choose to prevent inheritance altogether:

In[14]:=

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One can also choose to inherit only listed options:

In[15]:=

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And one can combine inheritance of listed options with additional options:

In[16]:=

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Instead of a list of options you want inherited, you can specify options you want disinherited:

In[17]:=

gr = Graph[{"sodium" -> "sbp", "age" -> "sodium", "age" -> "sbp", "sodium" -> "pro", "sbp" -> "pro"}, VertexLabels -> Placed["Name", Center], EdgeShapeFunction -> "CarvedArrow", VertexSize -> 0.5, VertexShapeFunction -> "Diamond", ImageSize -> 200, Background -> Green, AspectRatio -> 1]

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In[18]:=

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Applications (3)

One can show the weak paths in the context of the original graph:

In[19]:=

$With[{g = Graph[{"sodium" -> "sbp", "age" -> "sodium", "age" -> "sbp", "sodium" -> "pro", "sbp" -> "pro"}, VertexLabels -> Placed["Name", Center], VertexSize -> 0.5, ImageSize -> 100]}, Map[HighlightGraph[g, #] &, ResourceFunction["WeakPathGraphs"][g, "sodium", "sbp", {1, \[Infinity]}, All]]]$

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Find links between blogs that exist only when the orientation of the graph is disregarded, but show the orientation of the edges on the path:

In[20]:=

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There are no conventional paths:

In[21]:=

$FindPath[blogs, "talkingpointsmemo.com", "realclearpolitics.com", {1, \[Infinity]}]$

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There are paths if orientation is disregarded:

In[22]:=

ResourceFunction[
"WeakPathGraphs"][blogs, "talkingpointsmemo.com", "realclearpolitics.com", {1, 3}, 1, VertexLabels -> Placed["Name", Center], VertexLabelStyle -> Directive[Red, Italic, 8], GraphLayout -> "SpringEmbedding", ImageSize -> 600]

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Find "backdoor" paths as well as directed paths in a causal network:

In[23]:=

$cancerGraph = \!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{"anxiety", "smoking", "peer pressure", "yellow fingers", "lung cancer", "genetics", "attention disorder", "fatigue", "coughing", "allergy", "car accident", "born on an even day"}, {{{1, 2}, {3, 2}, {2, 4}, {2, 5}, { 6, 5}, {6, 7}, {5, 8}, {5, 9}, {10, 9}, {8, 11}, {7, 11}, { 9, 8}}, Null}, {GraphLayout -> "LayeredDigraphEmbedding", VertexLabels -> {"Name"}}]]}, TagBox[GraphicsGroupBox[{ {Hue[0.6, 0.7, 0.5], Opacity[0.7], Arrowheads[Medium], ArrowBox[{{0.5000100000000001, -0.5000099999999996}, { 1.50001, -1.5000099999999996`}}, 0.043048128342245986`], ArrowBox[{{1.50001, -1.5000099999999996`}, { 1.50001, -2.5000099999999996`}}, 0.043048128342245986`], ArrowBox[{{1.50001, -1.5000099999999996`}, { 2.50001, -2.5000099999999996`}}, 0.043048128342245986`], ArrowBox[{{1.50001, -0.5000099999999996}, { 1.50001, -1.5000099999999996`}}, 0.043048128342245986`], ArrowBox[BezierCurveBox[CompressedData[" 1:eJxTTMoPSmViYGCQAWIQbZmka8fKwOJgAaEPQGmHJ5NCKiQTWA7A5DskRY55 nWI5YAXlK33LVp9pxwqX36QRM4dlLytc3nrjE5VeVza4/O5+1oNGNxB8wyP7 Cr6WsMP58/0ETK5Kc8D5LPq/uC6d4YC7J8VqQ/bdJk64/DHXtDMvLLjg9qkH yuh+eccFl2+PudT7fwk3XP5Fesc7rmgeuLxXsZ2/mCAvnL+q7st6xeMIPnfX KgHdWj44P3tqQqGFMT/cPedLvTxMn/DD5eUZzrV2uQrA7Yueteru68kCcPnZ nLtX291FyN/z/f++SVEQLq+RXr5qRwKCXxKqfPv+LAT/mARz04/zCL7MRrH5 TAxCBwA3bIq1 "]], 0.043048128342245986`], ArrowBox[{{2.50001, -2.5000099999999996`}, { 1.50001, -3.5000099999999996`}}, 0.043048128342245986`], ArrowBox[{{3.50001, -1.5000099999999996`}, { 2.50001, -2.5000099999999996`}}, 0.043048128342245986`], ArrowBox[{{3.50001, -1.5000099999999996`}, { 3.50001, -2.5000099999999996`}}, 0.043048128342245986`], ArrowBox[BezierCurveBox[CompressedData[" 1:eJxTTMoPSmViYGBQBWIQbZmka8fKwONgAaZZDlhA+U8mhVRIJiD4HZIix7xO sRyAqVf6lq0+044VLr9JI2YOy17WA1ZQvvXGJyq9rmxw+d39rAeNbrDB9Rse 2VfwtYQdzp/vJ2ByVZoDzmfR/8V16QwHXH+K1Ybsu02ccP4x17QzLyy44OrV A2V0v7zjgsu3x1zq/b+EG+6eF+kd77iieeDyXsV2/mKCvHD9q+q+rFc8juBz d60S0K3lg/ObxGtZM9ME4PoNZW9szTuI4D9WMk4rkRGEq5+i0SdWVS4Il3fR e3ms/pIg3D1fjF3KW3WF4PJLLeerd3cIwfWH2v+6PuExgs/qGtoxzU4Yzt/m tcFizkxhuP4XVqpLpD4h+AoM51q7XEXg6mNmrbr7erIIXH4W5+7VdndF4O65 5/v/fZOiKFxeI7181Y4EUbj+klDl2/dnIfjHJJibfpxH8GU2is1nYhA7AACW DLgP "]], 0.043048128342245986`], ArrowBox[{{2.50001, -4.50001}, {3.50001, -5.50001}}, 0.043048128342245986`], ArrowBox[{{1.50001, -3.5000099999999996`}, { 2.50001, -4.50001}}, 0.043048128342245986`], ArrowBox[{{0.5000100000000001, -2.5000099999999996`}, { 1.50001, -3.5000099999999996`}}, 0.043048128342245986`]}, {Hue[0.6, 0.2, 0.8], EdgeForm[{GrayLevel[0], Opacity[ 0.7]}], { DiskBox[{0.5000100000000001, -0.5000099999999996}, 0.043048128342245986], InsetBox["\<\"anxiety\"\>", Offset[{0, 2}, {0.5000100000000001, -0.45696187165775365}], ImageScaled[{0.5, 0}], BaseStyle->"Graphics"]}, { DiskBox[{1.50001, -1.5000099999999996}, 0.043048128342245986], InsetBox["\<\"smoking\"\>", Offset[{0, 2}, {1.50001, -1.4569618716577537}], ImageScaled[{0.5, 0}], BaseStyle->"Graphics"]}, { DiskBox[{1.50001, -0.5000099999999996}, 0.043048128342245986], InsetBox["\<\"peer pressure\"\>", Offset[{0, 2}, {1.50001, -0.45696187165775365}], ImageScaled[{0.5, 0}], BaseStyle->"Graphics"]}, { DiskBox[{1.50001, -2.5000099999999996}, 0.043048128342245986], InsetBox["\<\"yellow fingers\"\>", Offset[{0, -2}, {1.50001, -2.5430581283422455}], ImageScaled[{0.5, 1}], BaseStyle->"Graphics"]}, { DiskBox[{2.50001, -2.5000099999999996}, 0.043048128342245986], InsetBox["\<\"lung cancer\"\>", Offset[{0, 2}, {2.50001, -2.4569618716577537}], ImageScaled[{0.5, 0}], BaseStyle->"Graphics"]}, { DiskBox[{3.50001, -1.5000099999999996}, 0.043048128342245986], InsetBox["\<\"genetics\"\>", Offset[{0, 2}, {3.50001, -1.4569618716577537}], ImageScaled[{0.5, 0}], BaseStyle->"Graphics"]}, { DiskBox[{3.50001, -2.5000099999999996}, 0.043048128342245986], InsetBox["\<\"attention disorder\"\>", Offset[{0, 2}, {3.50001, -2.4569618716577537}], ImageScaled[{0.5, 0}], BaseStyle->"Graphics"]}, { DiskBox[{2.50001, -4.50001}, 0.043048128342245986], InsetBox["\<\"fatigue\"\>", Offset[{0, 2}, {2.50001, -4.456961871657754}], ImageScaled[{0.5, 0}], BaseStyle->"Graphics"]}, { DiskBox[{1.50001, -3.5000099999999996}, 0.043048128342245986], InsetBox["\<\"coughing\"\>", Offset[{0, 2}, {1.50001, -3.4569618716577537}], ImageScaled[{0.5, 0}], BaseStyle->"Graphics"]}, { DiskBox[{0.5000100000000001, -2.5000099999999996}, 0.043048128342245986], InsetBox["\<\"allergy\"\>", Offset[{0, 2}, {0.5000100000000001, -2.4569618716577537}], ImageScaled[{0.5, 0}], BaseStyle->"Graphics"]}, { DiskBox[{3.50001, -5.50001}, 0.043048128342245986], InsetBox["\<\"car accident\"\>", Offset[{0, -2}, {3.50001, -5.5430581283422455}], ImageScaled[{0.5, 1}], BaseStyle->"Graphics"]}, { DiskBox[{4.500030000000001, -0.50001}, 0.043048128342245986], InsetBox["\<\"born on an even day\"\>", Offset[{2, 2}, {4.543078128342247, -0.456961871657754}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}}}], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False]], DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FormatType->TraditionalForm, FrameTicks->None, ImageSize->{672.59375, Automatic}]\);$

In[24]:=

ResourceFunction["WeakPathGraphs"][cancerGraph, "lung cancer", "car accident", VertexLabels -> Placed["Name", Center], GraphLayout -> "SpringEmbedding", ImageSize -> 300, VertexSize -> 0.5] // Column

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Show that deleting all incoming edges to attention disorder means that the only paths remaining between lung cancer and car accident are direct causal chains:

In[25]:=

$ResourceFunction["WeakPathGraphs"][ EdgeDelete[ cancerGraph, _ \[DirectedEdge] "attention disorder"], "lung cancer", "car accident", VertexLabels -> Placed["Name", Center], GraphLayout -> "SpringEmbedding", ImageSize -> 300, VertexSize -> 0.5]$

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Characterize all the "triples" on direction tolerant paths between two vertices as "chains," "forks" or "colliders":

In[26]:=

$tripleType[gr_Graph] := Map[#[[1]] -> Switch[Rest[#], {1, 1}, "chain", {0, 2}, "fork", {2, 0}, "collider", _, "other"] &, Thread[List[VertexList[gr], VertexInDegree[gr], VertexOutDegree[gr]]]]$

In[27]:=

AssociationMap[tripleType, ResourceFunction["WeakPathGraphs"][cancerGraph, "smoking", "car accident", VertexLabels -> Placed["Name", Center], GraphLayout -> "SpringEmbedding", ImageSize -> 300, VertexSize -> 0.5]]

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Neat Examples (1)

Determine whether two nodes of a graph are "d-separated" from each other by seeing if there are colliders on all the weak paths between two vertices:

In[28]:=

$cancerGraph = Graph[{"anxiety", "smoking", "peer pressure", "yellow fingers", "lung cancer", "genetics", "attention disorder", "fatigue", "coughing", "allergy", "car accident", "born on an even day"}, {"anxiety" \[DirectedEdge] "smoking", "peer pressure" \[DirectedEdge] "smoking", "smoking" \[DirectedEdge] "yellow fingers", "smoking" \[DirectedEdge] "lung cancer", "genetics" \[DirectedEdge] "lung cancer", "genetics" \[DirectedEdge] "attention disorder", "lung cancer" \[DirectedEdge] "fatigue", "lung cancer" \[DirectedEdge] "coughing", "allergy" \[DirectedEdge] "coughing", "fatigue" \[DirectedEdge] "car accident", "attention disorder" \[DirectedEdge] "car accident", "coughing" \[DirectedEdge] "fatigue"}, VertexLabels -> "Name", GraphLayout -> "LayeredDigraphEmbedding", ImageSize -> 500]$

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In[29]:=

$colliderPresentQ[g_Graph] := MemberQ[Thread[List[VertexInDegree[g], VertexOutDegree[g]]], {2, 0}]$

In[30]:=

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Publisher

Seth J. Chandler

Version History

1.1.1 – 23 June 2022
1.1.0 – 13 July 2021

Related Resources

Author Notes

Here are the details on how a weak path graph p inherit options from their underlying graph g when the "Inherit" option is set to Automatic. If the option does not directly relate to an edge or vertex (such as Background or AspectRatio), it is passed on to the p unchanged. If the option does relate to a edges or vertices then we examine the form of the option setting. If the form is not a List at all, it is passed on unchanged to p. If it is a List of rules, only those rules for which the left hand side is either (a) a vertex (or edge) in p or (b) an expression containing a pattern are passed on to p. If the right hand side is a List whose length is equal to the number of edges in g (if it is an edge-related option) or the number of vertices in g (if it a vertex-related option) then the edges (or vertices) of g are threaded with the list such that a list of rules from edge (vertex) to value is created. That list is then filtered so that only those rules pertaining to edges or vertices in p are passed down to p. If the right hand side is a list with a single item (like {VertexSize->0.6},it is passed down to p without change. If none of these conditions match, the option is passed down to p without change but a warning message is issued indicating that the option setting is not recognized.

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License

Wolfram Function Repository

WeakPathGraphs

Details and Options