soft_random_geometric_graph

soft_random_geometric_graph(n, radius, dim=2, pos=None, p=2, p_dist=None, seed=None, *, pos_name='pos')

Returns a soft random geometric graph in the unit cube.

The soft random geometric graph [1] model places n nodes uniformly at random in the unit cube in dimension dim. Two nodes of distance, dist, computed by the p-Minkowski distance metric are joined by an edge with probability p_dist if the computed distance metric value of the nodes is at most radius, otherwise they are not joined.

Edges within radius of each other are determined using a KDTree when SciPy is available. This reduces the time complexity from \(O(n^2)\) to \(O(n)\).

Parameters:

nint or iterable

Number of nodes or iterable of nodes

radius: float

Distance threshold value

dimint, optional

Dimension of graph

posdict, optional

A dictionary keyed by node with node positions as values.

pfloat, optional

Which Minkowski distance metric to use. p has to meet the condition 1 <= p <= infinity.

If this argument is not specified, the \(L^2\) metric (the Euclidean distance metric), p = 2 is used.

This should not be confused with the p of an Erdős-Rényi random graph, which represents probability.

p_distfunction, optional

A probability density function computing the probability of connecting two nodes that are of distance, dist, computed by the Minkowski distance metric. The probability density function, p_dist, must be any function that takes the metric value as input and outputs a single probability value between 0-1. The scipy.stats package has many probability distribution functions implemented and tools for custom probability distribution definitions [2], and passing the .pdf method of scipy.stats distributions can be used here. If the probability function, p_dist, is not supplied, the default function is an exponential distribution with rate parameter \(\lambda=1\).

seedinteger, random_state, or None (default)

Indicator of random number generation state. See Randomness.

pos_namestring, default=”pos”

The name of the node attribute which represents the position in 2D coordinates of the node in the returned graph.

Returns:

Graph

A soft random geometric graph, undirected and without self-loops. Each node has a node attribute 'pos' that stores the position of that node in Euclidean space as provided by the pos keyword argument or, if pos was not provided, as generated by this function.

Notes

This uses a k-d tree to build the graph.

References

[1]

Penrose, Mathew D. “Connectivity of soft random geometric graphs.” The Annals of Applied Probability 26.2 (2016): 986-1028.

Examples

Default Graph:

>>> G = nx.soft_random_geometric_graph(50, 0.2)

Custom Graph:

The pos keyword argument can be used to specify node positions so you can create an arbitrary distribution and domain for positions.

The scipy.stats package can be used to define the probability distribution with the .pdf method used as p_dist.

For example, create a soft random geometric graph on 100 nodes using a 2D Gaussian distribution of node positions with mean (0, 0) and standard deviation 2, where nodes are joined by an edge with probability computed from an exponential distribution with rate parameter \(\lambda=1\) if their Euclidean distance is at most 0.2.

>>> import random
>>> from scipy.stats import expon
>>> n = 100
>>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}
>>> p_dist = lambda x: expon.pdf(x, scale=1)
>>> G = nx.soft_random_geometric_graph(n, 0.2, pos=pos, p_dist=p_dist)