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PriceGraphDistribution[n,k,a]

represents a de Solla Price graph distribution for n-vertex graphs where a new vertex with k edges is added at each step, using attractiveness parameter a.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
DirectedEdges  
Applications  
Properties & Relations  
Neat Examples  
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PriceGraphDistribution

PriceGraphDistribution[n,k,a]

represents a de Solla Price graph distribution for n-vertex graphs where a new vertex with k edges is added at each step, using attractiveness parameter a.

Details and Options

Examples

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Basic Examples  (2)

Generate a pseudorandom graph:

Wolfram Language code: RandomGraph[PriceGraphDistribution[30, 2, 1]]

In-degree distribution:

Wolfram Language code: RandomGraph[PriceGraphDistribution[10 ^ 5, 2, 2]];
Wolfram Language code: EmpiricalDistribution[VertexInDegree[%]]

Probability density function:

Wolfram Language code: DiscretePlot[PDF[%, k], {k, 10, 30}, PlotRange -> All]

Scope  (4)

Generate simple directed graphs:

Wolfram Language code: RandomGraph[PriceGraphDistribution[7, 3, 1.5]]

Simple undirected graphs:

Wolfram Language code: RandomGraph[PriceGraphDistribution[7, 3, 1.5, DirectedEdges -> False]]

Generate a set of pseudorandom graphs:

Wolfram Language code: RandomGraph[PriceGraphDistribution[10, 3, 1.5], 4]

Compute probabilities and statistical properties:

Wolfram Language code: 𝒟 = GraphPropertyDistribution[Median[VertexInDegree[g]], gPriceGraphDistribution[30, 4, 2]];
Wolfram Language code: NExpectation[x - 4, x𝒟]

Options  (2)

DirectedEdges  (2)

By default, the Price graph is a directed graph:

Wolfram Language code: RandomGraph[PriceGraphDistribution[5, 2, 1.5]]
Wolfram Language code: DirectedGraphQ[%]

With the setting DirectedEdges->False, undirected Price graphs are generated:

Wolfram Language code: RandomGraph[PriceGraphDistribution[5, 2, 1.5, DirectedEdges -> False]]
Wolfram Language code: DirectedGraphQ[%]

Applications  (2)

A citation network can be modeled with PriceGraphDistribution:

Wolfram Language code: g = ExampleData[{"NetworkGraph", "HighEnergyPhysicsTheory"}];
Wolfram Language code: 𝒢 = PriceGraphDistribution[VertexCount[g], Round[EdgeCount[g] / VertexCount[g]], 10]

The model captures the power law nature of the empirical in-degree distribution:

Wolfram Language code: {Histogram[VertexInDegree[g], {"Log", 10}, {"Log", "PDF"}], Histogram[VertexInDegree[RandomGraph[𝒢]], {"Log", 10}, {"Log", "PDF"}]}

Use the undirected Price graph distribution as a model of the Western States Power Grid network:

Wolfram Language code: g = ExampleData[{"NetworkGraph", "PowerGrid"}];
Wolfram Language code: 𝒢 = PriceGraphDistribution[VertexCount[g], Round[EdgeCount[g] / VertexCount[g]], 3.0, DirectedEdges -> False]

The model captures the power law nature of the empirical degree distribution:

Wolfram Language code: f[g_] := Map[{#[[1]], #[[2]] / VertexCount[g]}&, Tally[VertexDegree[g]]]
Wolfram Language code: {ListLogLogPlot[f[g]], ListLogLogPlot[f[RandomGraph[𝒢]]]}

Properties & Relations  (4)

Distribution of the number of vertices:

Wolfram Language code: 𝒟[n_, k_, a_] = GraphPropertyDistribution[VertexCount[g], gPriceGraphDistribution[n, k, a]]

Distribution of the number of edges:

Wolfram Language code: 𝒟[n_, k_, a_] = GraphPropertyDistribution[EdgeCount[g], gPriceGraphDistribution[n, k, a]]

In-degree distribution:

Wolfram Language code: 𝒟[n_, k_, a_] := HistogramDistribution[VertexInDegree[RandomGraph[PriceGraphDistribution[n, k, a]]], {-1 / 2, n, 5}];

The distribution can be approximated by ZipfDistribution:

Wolfram Language code: ℰ = ZipfDistribution[1.1];
Wolfram Language code: DiscretePlot[Evaluate[{PDF[𝒟[10 ^ 4, 3, 1.5], d], PDF[ℰ, d]}], {d, 20, 100}, Joined -> True]

The degree distribution follows a power-law:

Wolfram Language code: PDF[ℰ, d]

Use RandomSample to simulate PriceGraphDistribution:

Wolfram Language code: price[n_, k_, a_] /; n ≤ k + 1 := Graph[Join@@Table[st, {s, n}, {t, 1, s - 1}]]
Wolfram Language code: price[n_, k_, a_] /; n > k + 1 := Module[{g = price[n - 1, k, a]}, Graph[Join[EdgeList[g], Map[n#&, RandomSample[(VertexInDegree[g] + a) -> VertexList[g], k]]]]]
Wolfram Language code: Table[price[6, 2, 1], {4}]

Neat Examples  (1)

Randomly colored vertices:

Wolfram Language code: HighlightGraph[RandomGraph[PriceGraphDistribution[10 ^ 2, 3, 1]], Table[Style[i, RandomChoice[ColorData[45, "ColorList"]]], {i, 10 ^ 2}]]
Wolfram Research (2010), PriceGraphDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/PriceGraphDistribution.html.

Text

Wolfram Research (2010), PriceGraphDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/PriceGraphDistribution.html.

CMS

Wolfram Language. 2010. "PriceGraphDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PriceGraphDistribution.html.

APA

Wolfram Language. (2010). PriceGraphDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PriceGraphDistribution.html

BibTeX

@misc{reference.wolfram_2026_pricegraphdistribution, author="Wolfram Research", title="{PriceGraphDistribution}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PriceGraphDistribution.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_pricegraphdistribution, organization={Wolfram Research}, title={PriceGraphDistribution}, year={2010}, url={https://reference.wolfram.com/language/ref/PriceGraphDistribution.html}, note=[Accessed: 12-June-2026]}