Erdős-Rényi Graph

This example demonstrates how to generate Erdős–Rényi graphs using igraph.GraphBase.Erdos_Renyi(). There are two variants of graphs:

• Erdos_Renyi(n, p) will generate a graph from the so-called \(G(n,p)\) model where each edge between any two pair of nodes has an independent probability p of existing.

• Erdos_Renyi(n, m) will pick a graph uniformly at random out of all graphs with n nodes and m edges. This is referred to as the \(G(n,m)\) model.

We generate two graphs of each, so we can confirm that our graph generator is truly random.

import igraph as ig
import matplotlib.pyplot as plt
import random

First, we set a random seed for reproducibility

random.seed(0)

Then, we generate two \(G(n,p)\) Erdős–Rényi graphs with identical parameters:

g1 = ig.Graph.Erdos_Renyi(n=15, p=0.2, directed=False, loops=False)
g2 = ig.Graph.Erdos_Renyi(n=15, p=0.2, directed=False, loops=False)

For comparison, we also generate two \(G(n,m)\) Erdős–Rényi graphs with a fixed number of edges:

g3 = ig.Graph.Erdos_Renyi(n=20, m=35, directed=False, loops=False)
g4 = ig.Graph.Erdos_Renyi(n=20, m=35, directed=False, loops=False)

We can print out summaries of each graph to verify their randomness

ig.summary(g1)
ig.summary(g2)
ig.summary(g3)
ig.summary(g4)

# IGRAPH U--- 15 18 --
# IGRAPH U--- 15 21 --
# IGRAPH U--- 20 35 --
# IGRAPH U--- 20 35 --

IGRAPH U--- 15 23 --
IGRAPH U--- 15 28 --
IGRAPH U--- 20 35 --
IGRAPH U--- 20 35 --

Finally, we can plot the graphs to illustrate their structures and differences:

fig, axs = plt.subplots(2, 2)
# Probability
ig.plot(
    g1,
    target=axs[0, 0],
    layout="circle",
    vertex_color="lightblue"
)
ig.plot(
    g2,
    target=axs[0, 1],
    layout="circle",
    vertex_color="lightblue"
)
axs[0, 0].set_ylabel('Probability')
# N edges
ig.plot(
    g3,
    target=axs[1, 0],
    layout="circle",
    vertex_color="lightblue",
    vertex_size=0.15
)
ig.plot(
    g4,
    target=axs[1, 1],
    layout="circle",
    vertex_color="lightblue",
    vertex_size=0.15
)
axs[1, 0].set_ylabel('N. edges')
plt.show()