# 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 . 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



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