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navigable_small_world_graph

navigable_small_world_graph(n, p=1, q=1, r=2, dim=2, seed=None)

Return a navigable small-world graph.

A navigable small-world graph is a directed grid with additional long-range connections that are chosen randomly. From [R291]:

Begin with a set of nodes that are identified with the set of lattice points in an \(n \times n\) square, \({(i,j): i\in {1,2,\ldots,n}, j\in {1,2,\ldots,n}}\) and define the lattice distance between two nodes \((i,j)\) and \((k,l)\) to be the number of “lattice steps” separating them: \(d((i,j),(k,l)) = |k-i|+|l-j|\).

For a universal constant \(p\), the node \(u\) has a directed edge to every other node within lattice distance \(p\) (local contacts) .

For universal constants \(q\ge 0\) and \(r\ge 0\) construct directed edges from \(u\) to \(q\) other nodes (long-range contacts) using independent random trials; the i’th directed edge from \(u\) has endpoint \(v\) with probability proportional to \(d(u,v)^{-r}\).

Parameters :

n : int

The number of nodes.

p : int

The diameter of short range connections. Each node is connected to every other node within lattice distance p.

q : int

The number of long-range connections for each node.

r : float

Exponent for decaying probability of connections. The probability of connecting to a node at lattice distance d is 1/d^r.

dim : int

Dimension of grid

seed : int, optional

Seed for random number generator (default=None).

References

[R291]

(1, 2) J. Kleinberg. The small-world phenomenon: An algorithmic perspective. Proc. 32nd ACM Symposium on Theory of Computing, 2000.