thresholded_random_geometric_graph

thresholded_random_geometric_graph(n, radius, theta, dim=2, pos=None, weight=None, p=2, seed=None)

Returns a thresholded random geometric graph in the unit cube.

The thresholded random geometric graph [1] model places n nodes uniformly at random in the unit cube of dimensions dim. Each node u is assigned a weight \(w_u\). Two nodes u and v are joined by an edge if they are within the maximum connection distance, radius computed by the p-Minkowski distance and the summation of weights \(w_u\) + \(w_v\) is greater than or equal to the threshold parameter theta.

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

theta: float

Threshold value

dimint, optional

Dimension of graph

posdict, optional

A dictionary keyed by node with node positions as values.

weightdict, optional

Node weights as a dictionary of numbers keyed by node.

pfloat, optional (default 2)

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.

seedinteger, random_state, or None (default)

Indicator of random number generation state. See Randomness.

Returns

Graph

A thresholded random geographic 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. Similarly, each node has a nodethre attribute 'weight' that stores the weight of that node as provided or as generated.

Notes

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

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

For example, to use a 2D Gaussian distribution of node positions with mean (0, 0) and standard deviation 2

If weights are not specified they are assigned to nodes by drawing randomly from the exponential distribution with rate parameter \(\lambda=1\). To specify weights from a different distribution, use the weight keyword argument:

::

>>> import random
>>> import math
>>> n = 50
>>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}
>>> w = {i: random.expovariate(5.0) for i in range(n)}
>>> G = nx.thresholded_random_geometric_graph(n, 0.2, 0.1, 2, pos, w)

References

1

http://cole-maclean.github.io/blog/files/thesis.pdf

Examples

Default Graph:

G = nx.thresholded_random_geometric_graph(50, 0.2, 0.1)

Custom Graph:

Create a thresholded random geometric graph on 50 uniformly distributed nodes where nodes are joined by an edge if their sum weights drawn from a exponential distribution with rate = 5 are >= theta = 0.1 and their Euclidean distance is at most 0.2.