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# 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:
    n int or iterable

      Number of nodes or iterable of nodes

    radius: float

      Distance threshold value

    dim int, optional

      Dimension of graph

    pos dict, optional

      A dictionary keyed by node with node positions as values.

    p float, 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_dist function, 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 \(\lambda=1\).

    seed integer, random_state, or None (default)

      Indicator of random number generation state. See Randomness.

    pos_name string, 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.

  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

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

      >>> import random
      >>> import math
      >>> n = 100
      >>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)}
      >>> p_dist = lambda dist: math.exp(-dist)
      >>> G = nx.soft_random_geometric_graph(n, 0.2, pos=pos, p_dist=p_dist)

  References

  [1]

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

  [2]

  scipy.stats - https://docs.scipy.org/doc/scipy/reference/tutorial/stats.html

  Examples

  Default Graph:

  G = nx.soft_random_geometric_graph(50, 0.2)

  Custom Graph:

  Create a soft random geometric graph on 100 uniformly distributed nodes 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.