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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.

[…] we begin with a set of nodes […] that are identified with the set of lattice points in an \(n imes n\) square, \(\{(i, j): i \in \{1, 2, \ldots, n\}, j \in \{1, 2, \ldots, n\}\}\), and we 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 \(p \geq 1\), the node \(u\) has a directed edge to every other node within lattice distance \(p\) — these are its local contacts. For universal constants \(q \ge 0\) and \(r \ge 0\) we also construct directed edges from \(u\) to \(q\) other nodes (the long-range contacts) using independent random trials; the \(i`th directed edge from `u\) has endpoint \(v\) with probability proportional to \([d(u,v)]^{-r}\).

—[1]

Parameters:

• n (int) – The number of nodes.
• p (int) – The diameter of short range connections. Each node is joined with every other node within this lattice distance.
• 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

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