watts_strogatz_graph(n, k, p, seed=None)[source]#

Returns a Watts–Strogatz small-world graph.


The number of nodes


Each node is joined with its k nearest neighbors in a ring topology.


The probability of rewiring each edge

seedinteger, random_state, or None (default)

Indicator of random number generation state. See Randomness.


First create a ring over \(n\) nodes [1]. Then each node in the ring is joined to its \(k\) nearest neighbors (or \(k - 1\) neighbors if \(k\) is odd). Then shortcuts are created by replacing some edges as follows: for each edge \((u, v)\) in the underlying “\(n\)-ring with \(k\) nearest neighbors” with probability \(p\) replace it with a new edge \((u, w)\) with uniformly random choice of existing node \(w\).

In contrast with newman_watts_strogatz_graph(), the random rewiring does not increase the number of edges. The rewired graph is not guaranteed to be connected as in connected_watts_strogatz_graph().



Duncan J. Watts and Steven H. Strogatz, Collective dynamics of small-world networks, Nature, 393, pp. 440–442, 1998.