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"SpringElectricalEmbedding" (Graph Layout Method)

  • Vertex layout for GraphLayout.
  • Use spring electrical embedding to lay out vertices of a graph.

Details & Options

  • The spring electrical embedding is a graph-drawing technique to position vertices of a graph so that they minimize mechanical and electrical energy when each vertex has a charge and each edge corresponds to a spring.
  • The spring electrical embedding is typically used to lay out complex large graphs.
  • The layout xi of the vertices vi of the graph is calculated by minimizing the energy function with
  • f_(i)= -C sum_(j!=i)(K^2)/(||x_i-x_j||^2)(x_(j)-x_(i)) +sum_( i j)(||x_i-x_j||)/K(x_(j)-x_(i)).
  • is a constant that regulates the relative strength of the repulsive force and is the relative spring length.
  • Vertices can be embedded in .
  • The following model parameters can be given:
  • "RepulsiveForcePower" repulsive force constant
    "SpringConstant" spring constant
  • The following graph parameters can be given:
  • "EdgeWeighted"whether to use edge weights
    "VertexWeighted"whether to use vertex weights
  • Possible settings to control the energy minimization process include:
  • "EnergyControl"how the energy function is controlled during minimization
    "InferentialDistance"cutoff distance for inference from faraway vertices
    "MaxIterations"maximum number of iterations
    "RandomSeed"seed for the random generator for initial vertex placement
    "StepControl"how step lengths are modified during energy minimization
    "StepLength"initial step length used in moving the vertices
    "Tolerance"tolerance used in terminating the process
  • Possible settings to control the layout include:
  • "Rotation"Automaticlayout rotation

Examples

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Basic Examples  (2)

Lay out a graph:

Wolfram Language code: Graph[{12, 13, 15, 19, 24, 26, 210, 34, 37, 311, 48, 412, 56, 57, 513, 68, 614, 78, 715, 816, 910, 911, 913, 1012, 1014, 1112, 1115, 1216, 1314, 1315, 1416, 1516}, GraphLayout -> "SpringElectricalEmbedding"]

"SpringElectricalEmbedding" works best for most graphs:

Wolfram Language code: KnightTourGraph[10, 10, GraphLayout -> "SpringElectricalEmbedding"]

Options  (11)

"EdgeWeighted"  (1)

With the setting "EdgeWeighted"True, edge weights are used:

Wolfram Language code: edge = RandomInteger[#]# + 1& /@ Range[0, 10]; weight = RandomInteger[{1, 5}, 11];
Wolfram Language code: Graph[edge, EdgeWeight -> weight, GraphLayout -> {"SpringElectricalEmbedding", "EdgeWeighted" -> #}]& /@ {True, False}

"EnergyControl"  (1)

Use the option "EnergyControl"e to specify limitations on the total energy of the system during minimization:

Wolfram Language code: edge = RandomInteger[#]# + 1& /@ Range[0, 20];
Wolfram Language code: Graph[edge, {GraphLayout -> {"SpringElectricalEmbedding", "EnergyControl" -> #, "StepControl" -> "Monotonic"}}]& /@ {Automatic, "Monotonic", "NonMonotonic"}

"InferentialDistance"  (1)

Use "InferentialDistance"d to specify a cutoff distance beyond which the interaction between vertices is assumed to be nonexistent:

Wolfram Language code: edge = RandomInteger[#]# + 1& /@ Range[0, 20];
Wolfram Language code: Graph[edge, GraphLayout -> {"SpringElectricalEmbedding", "InferentialDistance" -> #}]& /@ {Automatic, .1, .5}

"MaxIteration"  (1)

Use "MaxIteration"it to specify a maximum number of iterations to be used in attempting to minimize the energy:

Wolfram Language code: edge = RandomInteger[#]# + 1& /@ Range[0, 20];
Wolfram Language code: Graph[edge, GraphLayout -> {"SpringElectricalEmbedding", "MaxIteration" -> #}]& /@ {Automatic, 1, 2}

"RandomSeed"  (1)

Use "RandomSeed"int to specify a seed for the random number generator that computes the initial vertex placement:

Wolfram Language code: edge = RandomInteger[#]# + 1& /@ Range[0, 20];
Wolfram Language code: Graph[edge, GraphLayout -> {"SpringElectricalEmbedding", "RandomSeed" -> #}]& /@ {1, 5}

"RepulsiveForcePower"  (1)

Use "RepulsiveForcePower"r to control how fast the repulsive force decays over distance:

Wolfram Language code: edge = RandomInteger[#]# + 1& /@ Range[0, 20];
Wolfram Language code: Graph[edge, GraphLayout -> {"SpringElectricalEmbedding", "RepulsiveForcePower" -> #}]& /@ {-1, -2.5}

"SpringConstant"  (1)

Use "SpringConstant"r to control the constant in the attractive force:

Wolfram Language code: edge = RandomInteger[#]# + 1& /@ Range[0, 20];
Wolfram Language code: Graph[edge, GraphLayout -> {"SpringElectricalEmbedding", "SpringConstant" -> #}]& /@ {1, 4.5, 8}

"StepControl"  (1)

Use "StepControl"method to define how step length is modified during energy minimization:

Wolfram Language code: edge = RandomInteger[#]# + 1& /@ Range[0, 20];
Wolfram Language code: Graph[edge, GraphLayout -> {"SpringElectricalEmbedding", "StepControl" -> #}]& /@ {Automatic, "Monotonic", "NonMonotonic", "StrictlyMonotonic"}

"StepLength"  (1)

Use "StepLength"r to specify the initial step length used in moving the vertices around:

Wolfram Language code: edge = RandomInteger[#]# + 1& /@ Range[0, 20];
Wolfram Language code: Graph[edge, GraphLayout -> {"SpringElectricalEmbedding", "StepLength" -> #}]& /@ {1, 4.5}

"Tolerance"  (1)

Use "Tolerance"r to specify the tolerance used in terminating the energy minimization process:

Wolfram Language code: edge = RandomInteger[#]# + 1& /@ Range[0, 20];
Wolfram Language code: Graph[edge, GraphLayout -> {"SpringElectricalEmbedding", "Tolerance" -> #}]& /@ {Automatic, .5}

"Rotation"  (1)

Use "Rotation"r to rotate the layout:

Wolfram Language code: Graph[{13, 15, 17, 23, 26, 27, 28, 34, 38, 46, 58, 78}, GraphLayout -> {"SpringElectricalEmbedding", "Rotation" -> 45 Degree}]