networkx.generators.random_graphs.connected_watts_strogatz_graph

connected_watts_strogatz_graph(n, k, p, tries=100, seed=None)[source]

Returns a connected Watts–Strogatz small-world graph.

Attempts to generate a connected graph by repeated generation of Watts–Strogatz small-world graphs. An exception is raised if the maximum number of tries is exceeded.

Parameters
nint

The number of nodes

kint

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

pfloat

The probability of rewiring each edge

triesint

Number of attempts to generate a connected graph.

seedinteger, random_state, or None (default)

Indicator of random number generation state. See Randomness.

Notes

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 \(w\). The entire process is repeated until a connected graph results.

References

1

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