Directed versus undirected graphs (3) 

WolframRicciCurvatureScalar also supports directed graphs:

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In the directed case, the Wolfram–Ricci scalar curvature is computed using volumes of geodesic cones, as opposed to geodesic balls:

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However, WolframRicciCurvatureScalar can be made to treat directed graphs as though they were undirected (and hence revert to using geodesic balls) with the option "UndirectedGraph":

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Fractional dimensional graphs (1) 

WolframRicciCurvatureScalar also supports curvature computations in graphs with non-integer dimensionality, such as this Sierpinski sieve graph:

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CurvatureMethod (3) 

Compute the average Wolfram–Ricci scalar curvature over geodesic balls around vertex 150 with radii between 3 and 9 in a 20-by-20 2-dimensional grid graph with the option value "CurvatureMethod"→Mean (default):

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Return a list of all curvatures over geodesic balls with radii between 3 and 9 instead:

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Compute the maximum and minimum curvatures over geodesic balls with radii between 3 and 9 with the options "CurvatureMethod"→Max and "CurvatureMethod"→Min, respectively:

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TransitivelyReduce (3) 

By default, directed graphs are not transitively-reduced:

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Compute (and visualize) the Wolfram–Ricci curvature scalar, assuming that the graph is not transitively reduced:

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WolframRicciCurvatureScalar can be made to treat unreduced directed graphs as though they were transitively-reduced with the option "TransitivelyReduce":

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UndirectedGraph (3) 

By default, directed graphs are treated as directed (and so curvature computations use the volumes of geodesic cones, as opposed to geodesic balls):

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Compute (and visualize) the Wolfram–Ricci curvature scalar, assuming that the graph is directed:

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WolframRicciCurvatureScalar can be made to treat directed graphs as though they were undirected with the option "UndirectedGraph":

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VertexMethod (2) 

Return an Association showing the geodesic ball volumes around each vertex in a 20-by-20 2-dimensional grid graph (default):

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Compute the average ball volume across all vertices with the option "VertexMethod"→Mean:

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Compute the maximum and minimum ball volumes across all vertices with the options "VertexMethod"→Max and "VertexMethod"→Min, respectively:

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Return an association showing the Wolfram–Ricci scalar curvature at each vertex in a 20-by-20 2-dimensional grid graph (default):

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Compute the average curvature across all vertices with the option "AverageVertices":

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Compute the maximum and minimum curvatures across all vertices with the options "MaxVertices" and "MinVertices", respectively:

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VolumeMethod (3) 

Compute the average volume of geodesic balls around vertex 150 with radii between 3 and 9 in a 20-by-20 2-dimensional grid graph with the option value "VolumeMethod"→Mean (default):

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Return a list of all volumes of geodesic balls with radii between 3 and 9 instead:

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Compute the maximum and minimum volumes of geodesic balls with radii between 3 and 9 with the option values "VolumeMethod"→Max and "VolumeMethod"→Min, respectively:

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