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# 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: n (int) – The number of nodes k (int) – Each node is joined with its k nearest neighbors in a ring topology. p (float) – The probability of rewiring each edge tries (int) – Number of attempts to generate a connected graph. seed (integer, 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.