Generate and analyze small-world networks according to the revised Watts-Strogatz model where the randomization at β = 1 is truly equal to the Erdős-Rényi network model.

In the Watts-Strogatz model each node rewires its k/2 rightmost edges with probality β. This means each node has halways minimum degree k/2. Also, at β = 1, each edge has been rewired. Hence the probability of it existing is <k/(N-1), contrary to the ER model.

In the adjusted model, each pair of nodes is connected with a certain connection probability. If the lattice distance between the potentially connected nodes is d(i,j) <= k/2 then they are connected with short-range probability p_S = k / (k + β (N-1-k)), otherwise they're connected with long-range probability p_L = β * p_S.

pip install smallworld

Beware: smallworld only works with Python 3!

In the following example you can see how to generate and draw according to the model described above.

from smallworld.draw import draw_network
from smallworld import get_smallworld_graph

import matplotlib.pyplot as pl

# define network parameters
N = 21
k_over_2 = 2
betas = [0, 0.025, 1.0]
labels = [ r'$\beta=0$', r'$\beta=0.025$', r'$\beta=1$']

focal_node = 0

fig, ax = pl.subplots(1,3,figsize=(9,3))


# scan beta values
for ib, beta in enumerate(betas):

    # generate small-world graphs and draw
    G = get_smallworld_graph(N, k_over_2, beta)
    draw_network(G,k_over_2,focal_node=focal_node,ax=ax[ib])

    ax[ib].set_title(labels[ib],fontsize=11)

# show
pl.subplots_adjust(wspace=0.3)
pl.show()