# GraphPropertyDistribution

GraphPropertyDistribution[expr,xgdist]

represents the distribution of the property expr where the random variable x follows the graph distribution gdist.

GraphPropertyDistribution[expr,{x1gdist1,x2gdist2,}]

represents the distribution where x1, x2, are independent and follow the graph distributions gdist1, gdist2, .

# Details and Options • GraphPropertyDistribution is a transformation from the space of graphs to some property that is often of much lower dimension.
• GraphPropertyDistribution is typically used to study properties of distributions of graphs such as edge count, vertex count, length of shortest path, number of connected components or any other property that can be computed for graphs.
• xdist can be entered as x dist dist or x \[Distributed]dist.
• GraphPropertyDistribution will simplify to known special distributions whenever possible.
• GraphPropertyDistribution can be used with such functions as NProbability, NExpectation, and RandomVariate.

# Examples

open allclose all

## Basic Examples(3)

Obtain property distributions of graph models:

Simulate the distribution of a graph property:

Compute the mean:

Plot probability density functions:

## Scope(14)

### Basic Uses(5)

Obtain symbolic property distributions of graph models:

Compute the probability of an event for a graph property distribution:

Compute the expectation of an expression for a graph property distribution:

Simulate a property distribution:

Generate a probability histogram:

Create an empirical distribution of graph property data:

Visualize distribution functions:

Compute stochastic properties:

Compute moments and quantiles:

### Distribution Properties(4)

GraphPropertyDistribution works with basic graph properties, such as EdgeCount:

Predicates, such as ConnectedGraphQ:

Graph measures and metrics, such as GraphDiameter:

GraphPropertyDistribution works with any expression, such as maximum eigenvector centrality:

Size of giant component:

### Graph Distributions(3)

GraphPropertyDistribution works with all graph distributions, such as BarabasiAlbertGraphDistribution:

Symbolic graph distributions:

Combine independent distributions:

### Automatic Simplifications(2)

GraphPropertyDistribution will simplify to known distributions whenever possible:

Special transformations of graph property distributions:

## Options(1)

### Assumptions(1)

Compute the edge count of the Price distribution:

Use Assumptions to specify the condition :

## Applications(5)

After 20 children have spent their first week in kindergarten, the probability that two children have made friends is 0.2. Find the probability that the social network is connected:

Directly compute the probability:

This represents a social network of 100 persons in a small village where the average number of relations per person is 20. Find the expected number of relations of the least-connected person:

The number of relations is given by the VertexDegree:

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 the lily leaf density and SpatialGraphDistribution:

Sample a random pond:

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

Use simulation to find the sizes of the largest collections of lily pads for similar ponds:

Find the number of times the frog would have to swim to visit all lily pads:

Simulate to get results for similar lily ponds:

In a medical study of an outbreak of influenza in a group of seven subjects, each subject has reported his or her number of potentially contagious interactions within the group. Model the interactions as a DegreeGraphDistribution:

Simulate to see whether the first two subjects have interacted:

Find the probability that the first two subjects have interacted:

In a piece of brain cortex with 100 neurons, the neurons are connected by synapses if they are at a distance less than 0.2:

Probability that the network is connected:

## Properties & Relations(4)

GraphPropertyDistribution uses local names for the variables in the input:

Use NProbability to compute the probability of an event:

Use NExpectation to compute the expectation of an expression:

Use RandomVariate to simulate a property distribution:

## Possible Issues(1)

Monte Carlo methods are used to numerically approximate statistical properties: