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 LowestCommonAncestors

Source Notebook

Find the lowest common ancestors for pairs of vertices in a graph

Contributed by: Nikolay Murzin

ResourceFunction["LowestCommonAncestors"][g,v,w]

returns a list of the lowest common ancestors of vertices v and w in a graph g.

ResourceFunction["LowestCommonAncestors"][g]

returns an Association of lowest common ancestors for each distinct pair of vertices in a graph g.

ResourceFunction["LowestCommonAncestors"][g,{{v1,w1},{v2,w2},…}]

returns the common ancestors for each vertex pair {vi,wi}.

ResourceFunction["LowestCommonAncestors"][g,v]

returns lowest common ancestors for a vertex v with all other vertices in a graph g.

Details

Lowest common ancestor is also known as "least", "most recent" or "last" common ancestor.
The lowest common ancestor for a pair of vertices in a graph is their shared ancestor located farthest from the graph root. The graph root is defined as a graph sink of a subgraph induced by the set of common ancestors shared by this pair.
ResourceFunction["LowestCommonAncestors"] works for arbitrary graphs.
LowestCommonAncestors returns a list of vertices for a pair, as there could be zero or more than one lowest common ancestor.
ResourceFunction["LowestCommonAncestors"][g,{v,w}] is equivalent to ResourceFunction["LowestCommonAncestors"][g,v,w].

Examples

Basic Examples (4) 

Find a list of lowest common ancestors for a pair of vertices in a graph:

In[1]:=
ResourceFunction["LowestCommonAncestors"][\!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{0, 1, 4, 2, 3}, {{{1, 2}, {1, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}}, Null}, {VertexLabels -> {Automatic}}]]}, TagBox[GraphicsGroupBox[{ {Hue[0.6, 0.7, 0.5], Opacity[0.7], Arrowheads[Medium], ArrowBox[{{0., 2.}, {0., 1.}}, 0.02261146496815286], ArrowBox[{{0., 2.}, {1., 1.}}, 0.02261146496815286], ArrowBox[{{0., 1.}, {0., 0.}}, 0.02261146496815286], ArrowBox[{{0., 1.}, {1., 0.}}, 0.02261146496815286], ArrowBox[{{1., 1.}, {0., 0.}}, 0.02261146496815286], ArrowBox[{{1., 1.}, {1., 0.}}, 0.02261146496815286]}, {Hue[0.6, 0.2, 0.8], EdgeForm[{GrayLevel[0], Opacity[ 0.7]}], {DiskBox[{0., 2.}, 0.02261146496815286], InsetBox["0", Offset[{2, 2}, {0.02261146496815286, 2.022611464968153}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{0., 1.}, 0.02261146496815286], InsetBox["1", Offset[{2, 2}, {0.02261146496815286, 1.0226114649681528}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{1., 1.}, 0.02261146496815286], InsetBox["4", Offset[{2, 2}, {1.0226114649681528, 1.0226114649681528}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{0., 0.}, 0.02261146496815286], InsetBox["2", Offset[{2, 2}, {0.02261146496815286, 0.02261146496815286}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{1., 0.}, 0.02261146496815286], InsetBox["3", Offset[{2, 2}, {1.0226114649681528, 0.02261146496815286}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}}}], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False]], DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FormatType->TraditionalForm, FrameTicks->None, ImageSize->{110.79228637069873`, Automatic}]\), 2, 3]
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Find all lowest common ancestors for a list of pairs of vertices:

In[2]:=
ResourceFunction["LowestCommonAncestors"][\!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{0, 1, 4, 2, 3}, {{{1, 2}, {1, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}}, Null}, {VertexLabels -> {Automatic}}]]}, TagBox[GraphicsGroupBox[{ {Hue[0.6, 0.7, 0.5], Opacity[0.7], Arrowheads[Medium], ArrowBox[{{0., 2.}, {0., 1.}}, 0.02261146496815286], ArrowBox[{{0., 2.}, {1., 1.}}, 0.02261146496815286], ArrowBox[{{0., 1.}, {0., 0.}}, 0.02261146496815286], ArrowBox[{{0., 1.}, {1., 0.}}, 0.02261146496815286], ArrowBox[{{1., 1.}, {0., 0.}}, 0.02261146496815286], ArrowBox[{{1., 1.}, {1., 0.}}, 0.02261146496815286]}, {Hue[0.6, 0.2, 0.8], EdgeForm[{GrayLevel[0], Opacity[ 0.7]}], {DiskBox[{0., 2.}, 0.02261146496815286], InsetBox["0", Offset[{2, 2}, {0.02261146496815286, 2.022611464968153}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{0., 1.}, 0.02261146496815286], InsetBox["1", Offset[{2, 2}, {0.02261146496815286, 1.0226114649681528}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{1., 1.}, 0.02261146496815286], InsetBox["4", Offset[{2, 2}, {1.0226114649681528, 1.0226114649681528}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{0., 0.}, 0.02261146496815286], InsetBox["2", Offset[{2, 2}, {0.02261146496815286, 0.02261146496815286}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{1., 0.}, 0.02261146496815286], InsetBox["3", Offset[{2, 2}, {1.0226114649681528, 0.02261146496815286}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}}}], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False]], DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FormatType->TraditionalForm, FrameTicks->None, ImageSize->{110.79228637069873`, Automatic}]\), {{2, 3}, {1, 4}}]
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Find all lowest common ancestors for all pairs that contain a given vertex:

In[3]:=
ResourceFunction["LowestCommonAncestors"][\!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{0, 1, 4, 2, 3}, {{{1, 2}, {1, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}}, Null}, {VertexLabels -> {Automatic}}]]}, TagBox[GraphicsGroupBox[{ {Hue[0.6, 0.7, 0.5], Opacity[0.7], Arrowheads[Medium], ArrowBox[{{0., 2.}, {0., 1.}}, 0.02261146496815286], ArrowBox[{{0., 2.}, {1., 1.}}, 0.02261146496815286], ArrowBox[{{0., 1.}, {0., 0.}}, 0.02261146496815286], ArrowBox[{{0., 1.}, {1., 0.}}, 0.02261146496815286], ArrowBox[{{1., 1.}, {0., 0.}}, 0.02261146496815286], ArrowBox[{{1., 1.}, {1., 0.}}, 0.02261146496815286]}, {Hue[0.6, 0.2, 0.8], EdgeForm[{GrayLevel[0], Opacity[ 0.7]}], {DiskBox[{0., 2.}, 0.02261146496815286], InsetBox["0", Offset[{2, 2}, {0.02261146496815286, 2.022611464968153}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{0., 1.}, 0.02261146496815286], InsetBox["1", Offset[{2, 2}, {0.02261146496815286, 1.0226114649681528}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{1., 1.}, 0.02261146496815286], InsetBox["4", Offset[{2, 2}, {1.0226114649681528, 1.0226114649681528}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{0., 0.}, 0.02261146496815286], InsetBox["2", Offset[{2, 2}, {0.02261146496815286, 0.02261146496815286}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{1., 0.}, 0.02261146496815286], InsetBox["3", Offset[{2, 2}, {1.0226114649681528, 0.02261146496815286}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}}}], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False]], DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FormatType->TraditionalForm, FrameTicks->None, ImageSize->{110.79228637069873`, Automatic}]\), 3]
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Find all lowest common ancestors for every pair of vertices:

In[4]:=
ResourceFunction["LowestCommonAncestors"][\!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{0, 1, 4, 2, 3}, {{{1, 2}, {1, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}}, Null}, {VertexLabels -> {Automatic}}]]}, TagBox[GraphicsGroupBox[{ {Hue[0.6, 0.7, 0.5], Opacity[0.7], Arrowheads[Medium], ArrowBox[{{0., 2.}, {0., 1.}}, 0.02261146496815286], ArrowBox[{{0., 2.}, {1., 1.}}, 0.02261146496815286], ArrowBox[{{0., 1.}, {0., 0.}}, 0.02261146496815286], ArrowBox[{{0., 1.}, {1., 0.}}, 0.02261146496815286], ArrowBox[{{1., 1.}, {0., 0.}}, 0.02261146496815286], ArrowBox[{{1., 1.}, {1., 0.}}, 0.02261146496815286]}, {Hue[0.6, 0.2, 0.8], EdgeForm[{GrayLevel[0], Opacity[ 0.7]}], {DiskBox[{0., 2.}, 0.02261146496815286], InsetBox["0", Offset[{2, 2}, {0.02261146496815286, 2.022611464968153}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{0., 1.}, 0.02261146496815286], InsetBox["1", Offset[{2, 2}, {0.02261146496815286, 1.0226114649681528}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{1., 1.}, 0.02261146496815286], InsetBox["4", Offset[{2, 2}, {1.0226114649681528, 1.0226114649681528}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{0., 0.}, 0.02261146496815286], InsetBox["2", Offset[{2, 2}, {0.02261146496815286, 0.02261146496815286}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{1., 0.}, 0.02261146496815286], InsetBox["3", Offset[{2, 2}, {1.0226114649681528, 0.02261146496815286}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}}}], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False]], DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FormatType->TraditionalForm, FrameTicks->None, ImageSize->{110.79228637069873`, Automatic}]\)]
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In[5]:=
ResourceFunction["LowestCommonAncestors"][\!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, {{{2, 1}, {3, 2}, {7, 4}, {7, 5}, {7, 6}, {8, 3}, {8, 7}, {10, 8}, {10, 9}}, Null}, {GraphLayout -> {"LayeredEmbedding", "RootVertex" -> 10, "Orientation" -> Top}, VertexLabels -> {Automatic}}]]}, TagBox[GraphicsGroupBox[{ {Hue[0.6, 0.7, 0.5], Opacity[0.7], Arrowheads[Medium], ArrowBox[{{{0., 0.8164965809277263}, {0., 0.}}, {{0., 1.6329931618554523`}, {0., 0.8164965809277263}}, {{ 1.6329931618554523`, 1.6329931618554523`}, { 0.8164965809277261, 0.8164965809277263}}, {{ 1.6329931618554523`, 1.6329931618554523`}, { 1.6329931618554523`, 0.8164965809277263}}, {{ 1.6329931618554523`, 1.6329931618554523`}, { 2.4494897427831783`, 0.8164965809277263}}, {{ 0.8164965809277261, 2.4494897427831783`}, {0., 1.6329931618554523`}}, {{0.8164965809277261, 2.4494897427831783`}, {1.6329931618554523`, 1.6329931618554523`}}, {{1.2247448713915892`, 3.2659863237109046`}, {0.8164965809277261, 2.4494897427831783`}}, {{1.2247448713915892`, 3.2659863237109046`}, {1.6329931618554523`, 2.4494897427831783`}}}, 0.03211708050212858]}, {Hue[0.6, 0.2, 0.8], EdgeForm[{GrayLevel[0], Opacity[ 0.7]}], {DiskBox[{0., 0.}, 0.03211708050212858], InsetBox["1", Offset[{0, -2}, {0., -0.03211708050212858}], ImageScaled[{0.5, 1}], BaseStyle->"Graphics"]}, { DiskBox[{0., 0.8164965809277263}, 0.03211708050212858], InsetBox["2", {-0.04496391270298001, 0.8164965809277263}, ImageScaled[{1.25, 0.49999999999999983}], BaseStyle->"Graphics"]}, { DiskBox[{0., 1.6329931618554523}, 0.03211708050212858], InsetBox["3", {-0.04154123864790748, 1.650200106301193}, ImageScaled[{1.192909649383465, 0.21298742572618223}], BaseStyle->"Graphics"]}, { DiskBox[{0.8164965809277261, 0.8164965809277263}, 0.03211708050212858], InsetBox["4", Offset[{0, -2}, {0.8164965809277261, 0.7843795004255977}], ImageScaled[{0.5, 1}], BaseStyle->"Graphics"]}, { DiskBox[{1.6329931618554523, 0.8164965809277263}, 0.03211708050212858], InsetBox["5", Offset[{0, -2}, {1.6329931618554523, 0.7843795004255977}], ImageScaled[{0.5, 1}], BaseStyle->"Graphics"]}, { DiskBox[{2.4494897427831783, 0.8164965809277263}, 0.03211708050212858], InsetBox["6", Offset[{0, -2}, {2.4494897427831783, 0.7843795004255977}], ImageScaled[{0.5, 1}], BaseStyle->"Graphics"]}, { DiskBox[{1.6329931618554523, 1.6329931618554523}, 0.03211708050212858], InsetBox["7", {1.6647874494364094, 1.6647874494364094}, ImageScaled[{-0.030330085889910485, -0.03033008588991093}], BaseStyle->"Graphics"]}, { DiskBox[{0.8164965809277261, 2.4494897427831783}, 0.03211708050212858], InsetBox["8", {0.7800199581325175, 2.4757806049793345}, ImageScaled[{1.1084316388816706, 0.061467286502176144}], BaseStyle->"Graphics"]}, { DiskBox[{1.6329931618554523, 2.4494897427831783}, 0.03211708050212858], InsetBox["9", Offset[{0, -2}, {1.6329931618554523, 2.41737266228105}], ImageScaled[{0.5, 1}], BaseStyle->"Graphics"]}, { DiskBox[{1.2247448713915892, 3.2659863237109046}, 0.03211708050212858], InsetBox["10", Offset[{0, 2}, {1.2247448713915892, 3.298103404213033}], ImageScaled[{0.5, 0}], BaseStyle->"Graphics"]}}}], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False]], DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FormatType->TraditionalForm, FrameTicks->None, ImageMargins->0., ImageSize->{285.0077714461818, Automatic}]\)]
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Applications (1) 

Measure an average time and spatial distance between pairs of events in a causal graph using longest distance to their lowest common ancestor:

In[6]:=
With[{causalGraph = \!\(\* GraphicsBox[ NamespaceBox["NetworkGraphics", DynamicModuleBox[{Typeset`graph = HoldComplete[ Graph[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, { SparseArray[ Automatic, {10, 10}, 0, {1, {{0, 0, 0, 0, 3, 4, 8, 9, 11, 15, 15}, {{5}, {6}, { 7}, {10}, {1}, {2}, {3}, {10}, {1}, {5}, {6}, {2}, { 3}, {7}, {10}}}, Pattern}], Null}, {EdgeStyle -> { Hue[0, 1, 0.56]}, VertexLabels -> {Automatic}, VertexStyle -> { Directive[ Hue[0.11, 1, 0.97], EdgeForm[{ Hue[0.11, 1, 0.97], Opacity[1]}]]}}]]}, TagBox[GraphicsGroupBox[{ {Hue[0, 1, 0.56], Opacity[0.7], Arrowheads[Medium], ArrowBox[{{-2., 3.}, {-1., 2.}}, 0.03700564971751412], ArrowBox[{{-2., 3.}, {-2., 2.}}, 0.03700564971751412], ArrowBox[BezierCurveBox[CompressedData[" 1:eJxTTMoPSmViYGCQAWIQDQUHIBSHw0W3D3fDTjMcmH6HfWLILHaHZpbQbS4H GQ/sr726JFae3UHCUq/z5EOmA6WPM6yl+9kcep5Uhz3VYTmwm/dQSO9LVodH bHZyMxaxHuj/8+DDZj1WB4kFJQ8eOrEf+LL+KH9/HIuDwQb5uYd4OQ+8UC/Y KFfJ7KBlbh9ixcx9ICflwdWkOiYHFtPLrJ7KvAfqUsVqo/MYHU4k+c0R2Mt7 YFmFw7wocUaHhcxrth9w4TsQvOxEqPZWBodJ4m/CI3fxHeh4t27xWXcGh2nz +GpvyfIfaF30jL953n/7TTOE1V3y+Q9wzNVm5Zv+z/4J6+/Qaev4D8w6o6Nu U/nXXufZEc6Lt/kPmBu+KuV1/WPfY1no9OUb/4FHp5I/Nv76ZQ8JF4EDs2aC wE84//yNo6/nm/+A8z8fVkp98PKbPSOUzx69gTV6w1e4PPu8vLP83V/g/K89 sXv+VX6G8y+p1wLRJzgfrHzKRzg/ItGvROPABzgfQn+wBwCPCq1F "]], 0.03700564971751412], ArrowBox[{{-1., 2.}, {-1., 1.}}, 0.03700564971751412], ArrowBox[BezierCurveBox[CompressedData[" 1:eJxTTMoPSmViYGCQAWIQDQUHoLRDbJG8S1Dmv/16O/mVVpr8t//xLXRyWe2f /bW/V3nutvtnf7l38dscm1/7Xx27WGYm99f+hRlHsOmy7/tLpVtWSl3+bW/3 qfnw+V1f9sv9PPgwPfaX/fn9YvY2VR/3v4jvlpfb+8Pezr6x++uJ1/sv+D9I sv30zT5amXm92OfH+6+fOLD66J8v9g/D0y5uf3pt/59TOr+23flkf+Jlj2y1 +KX9Xh8bbLbqfbL/fdS+9cvm0/uX+0xcHZb10Z6p3a4xK+rIfoG7se6z+z/Y e+rqxG1w2bOf2/WsPJvte6j+TfuFrMK5ZPTeQtUv2W80M9Kv4tMrqPyk/XlJ V15Ldb2wPwPmF+0/MvX4I+avT+1hAWQMBo/h/Hkbu6/b5z+A88/xd09pXH4b zhfPOSiWc/AanC9ZscvUXucSnL8i/c2jr+dPw/lT7F5Otnt5GM5/8PLanw0e O+H8f1pnOvlYV8D5MAAAIRW7qg== "]], 0.03700564971751412], ArrowBox[{{-2., 2.}, {-3., 1.}}, 0.03700564971751412], ArrowBox[{{-2., 2.}, {-2., 1.}}, 0.03700564971751412], ArrowBox[{{-2., 2.}, {-1., 1.}}, 0.03700564971751412], ArrowBox[{{-4., 1.}, {0., 0.}}, 0.03700564971751412], ArrowBox[{{-1., 3.}, {-1., 2.}}, 0.03700564971751412], ArrowBox[{{-1., 3.}, {-2., 2.}}, 0.03700564971751412], ArrowBox[{{-3., 2.}, {-3., 1.}}, 0.03700564971751412], ArrowBox[{{-3., 2.}, {-2., 1.}}, 0.03700564971751412], ArrowBox[{{-3., 2.}, {-4., 1.}}, 0.03700564971751412], ArrowBox[{{-3., 2.}, {-1., 1.}}, 0.03700564971751412]}, {Hue[0.11, 1, 0.97], EdgeForm[{Hue[0.11, 1, 0.97], Opacity[ 1]}], {DiskBox[{0., 0.}, 0.03700564971751412], InsetBox["1", Offset[{2, 2}, {0.03700564971751412, 0.03700564971751412}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{-3., 1.}, 0.03700564971751412], InsetBox["2", Offset[{2, 2}, {-2.962994350282486, 1.0370056497175142}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{-2., 1.}, 0.03700564971751412], InsetBox["3", Offset[{2, 2}, {-1.9629943502824858, 1.0370056497175142}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{-2., 3.}, 0.03700564971751412], InsetBox["4", Offset[{2, 2}, {-1.9629943502824858, 3.037005649717514}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{-1., 2.}, 0.03700564971751412], InsetBox["5", Offset[{2, 2}, {-0.9629943502824859, 2.037005649717514}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{-2., 2.}, 0.03700564971751412], InsetBox["6", Offset[{2, 2}, {-1.9629943502824858, 2.037005649717514}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{-4., 1.}, 0.03700564971751412], InsetBox["7", Offset[{2, 2}, {-3.962994350282486, 1.0370056497175142}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{-1., 3.}, 0.03700564971751412], InsetBox["8", Offset[{2, 2}, {-0.9629943502824859, 3.037005649717514}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{-3., 2.}, 0.03700564971751412], InsetBox["9", Offset[{2, 2}, {-2.962994350282486, 2.037005649717514}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}, {DiskBox[{-1., 1.}, 0.03700564971751412], InsetBox["10", Offset[{2, 2}, {-0.9629943502824859, 1.0370056497175142}], ImageScaled[{0, 0}], BaseStyle->"Graphics"]}}}], MouseAppearanceTag["NetworkGraphics"]], AllowKernelInitialization->False]], DefaultBaseStyle->{"NetworkGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.8, 1., 0.6]}, FormatType->TraditionalForm, FrameTicks->None, ImageSize->{226.421875, Automatic}]\)}, Association@KeyValueMap[ #1 -> Mean@({Subtract[##], Plus[##]} & @@@ Table[Length@ ResourceFunction["FindLongestPath"][causalGraph, from, to] - 1, {from, #2}, {to, #1}]) &, DeleteCases[ ResourceFunction["LowestCommonAncestors"][causalGraph], {}] ] ]
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Version History

  • 1.0.0 – 01 November 2022

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