Modeling Connectivity in Clustered Spatial Graphs

Simulate a clustered lily-pond network and estimate the size of the largest connected component

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A frog in a lily pond is able to jump 1.5 feet to get from one of the 25 lily pads to another. Model the frog's jumping network from a lily leaf density function and

SpatialGraphDistribution

Create a distribution that specifies where lily pads are more likely to be located in the pond:

In[1]:=

lilyDensity=MixtureDistribution[{1,1,1},{BinormalDistribution[{0,0},{1,1},0],BinormalDistribution[{-1,4},{1,1},-1/2],BinormalDistribution[{4,4},{1,1},1/3]}];

Visualize the density distribution:

In[2]:=

ContourPlot[PDF[lilyDensity,{x,y}],{x,-3.5,7},{y,-2.5,7}]

Out[2]=

Define a spatial graph distribution (a random graph model where vertices are placed in space according to a density). Edges occur only between vertices closer to one another than the 1.5 foot jumping distance:

In[3]:=

lilyPond=SpatialGraphDistribution[25,1.5,lilyDensity];

Sample a random pond:

In[4]:=

SeedRandom[1234];g=RandomGraphlilyPond,VertexShape

,VertexSize{"Scaled",0.05},EdgeStyleWhite,BackgroundRGBColor[0.243,0.482,0.725]

Out[4]=

Show this random lily pad graph on top of the underlying lily pad density function that generated it:

In[5]:=

Show[ContourPlot[PDF[lilyDensity,{x,y}],{x,-3.5,7},{y,-2.5,7},AspectRatioAutomatic],g]

Out[5]=

Find the largest collection of lily pads the frog can jump between:

In[6]:=

Length[First@ConnectedComponents[g]]

Out[6]=

Define a distribution to find the sizes of the largest collections of lily pads for similar ponds:

In[7]:=

largestIsland=GraphPropertyDistribution[Length[First[ConnectedComponents[g]]],glilyPond];

Draw 10 random samples from the distribution of largest connected component sizes, showing how many lily pads typically belong to the biggest group in different simulated ponds:

In[8]:=

RandomVariate[largestIsland,10]

Out[8]=

{10,13,18,10,17,11,13,11,17,10}

Plot a histogram of the distribution of this statistic over 10,000 simulated ponds:

In[9]:=

Histogram[RandomVariate[largestIsland,10000]]

Out[9]=

Related Symbols

GraphPropertyDistribution
SpatialGraphDistribution
RandomVariate
ConnectedComponents
MixtureDistribution
BinormalDistribution

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Contributed by: Wolfram Staff