connected_watts_strogatz_graph#

connected_watts_strogatz_graph(n, k, p, tries=100, seed=None, *, create_using=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.

create_usingGraph constructor, optional (default=nx.Graph)

Graph type to create. If graph instance, then cleared before populated. Multigraph and directed types are not supported and raise a NetworkXError.

See also

newman_watts_strogatz_graph

watts_strogatz_graph

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.