Giant Component¶

This example illustrates the sudden appearance of a giant connected component in a binomial random graph.

This example needs Graphviz and PyGraphviz.

p = 0.003, p = 0.006, p = 0.008, p = 0.015
import math

import matplotlib.pyplot as plt
import networkx as nx

n = 150  # 150 nodes
# p value at which giant component (of size log(n) nodes) is expected
p_giant = 1.0 / (n - 1)
# p value at which graph is expected to become completely connected
p_conn = math.log(n) / n

# the following range of p values should be close to the threshold
pvals = [0.003, 0.006, 0.008, 0.015]

fig, axes = plt.subplots(2, 2)
for p, ax, seed in zip(pvals, axes.ravel(), range(len(pvals))):
    #### generate graph ####
    G = nx.binomial_graph(n, p, seed=seed)
    # identify connected/disconnected nodes
    connected = [n for n, d in G.degree() if d > 0]
    disconnected = list(set(G.nodes()) - set(connected))
    # identify largest connected component
    Gcc = sorted(nx.connected_components(G), key=len, reverse=True)
    G0 = G.subgraph(Gcc[0])
    #### draw graph ####
    pos = nx.nx_agraph.graphviz_layout(G)
    ax.set_title(f"p = {p:.3f}")
    # draw largest connected component
    options = {"ax": ax, "edge_color": "tab:red"}
    nx.draw_networkx_edges(G0, pos, width=6.0, **options)
    # draw other connected components
    for Gi in Gcc[1:]:
        if len(Gi) > 1:
            nx.draw_networkx_edges(
                G.subgraph(Gi),
                pos,
                alpha=0.3,
                width=5.0,
                **options,
            )
    # draw connected/disconnected nodes
    options = {"ax": ax, "node_size": 30, "edgecolors": "white"}
    nx.draw(G, pos, nodelist=connected, **options)
    nx.draw(G, pos, nodelist=disconnected, alpha=0.25, **options)
fig.tight_layout()
plt.show()

Total running time of the script: ( 0 minutes 0.822 seconds)

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