Use this Skill for graph construction, centrality measures (betweenness, PageRank), community detection (Louvain), and random graph models via networkx.
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Use this Skill for graph construction, centrality measures (betweenness, PageRank), community detection (Louvain), and random graph models via networkx.
Network Analysis — Graph Theory & Community Detection
One-line summary: Construct and analyze complex networks with networkx and igraph: centrality, shortest paths, community detection (Louvain/Leiden), and random graph models.
When to Use This Skill
When analyzing social, biological, or infrastructure networks
When computing centrality measures to identify important nodes
When detecting communities or clusters in graphs
When fitting random graph models (Erdős-Rényi, Barabási-Albert, Watts-Strogatz)
When studying network robustness, percolation, or spreading processes
When visualizing graph structure for publication figures
import networkx as nx
import igraph as ig
import leidenalg
G = nx.karate_club_graph()
print(f"networkx {nx.__version__}: Karate club n={G.number_of_nodes()} e={G.number_of_edges()}")
print(f"igraph {ig.__version__}")
print(f"leidenalg {leidenalg.__version__}")
Core Workflow
Step 1: Graph Construction and Basic Analysis
import networkx as nx
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
# --- Option 1: Build from edge list ---
edges = [
(0, 1), (0, 2), (1, 2), (1, 3),
(3, 4), (3, 5), (4, 5), (5, 6),
]
G = nx.Graph(edges)
# --- Option 2: From adjacency matrix ---
A = np.array([[0,1,1,0], [1,0,1,1], [1,1,0,0], [0,1,0,0]])
G2 = nx.from_numpy_array(A)
# --- Option 3: From pandas edge list ---
edge_df = pd.DataFrame({"source": [0,0,1,3,3], "target": [1,2,2,4,5], "weight": [1,2,3,1,2]})
G3 = nx.from_pandas_edgelist(edge_df, edge_attr="weight")
# Basic statisticsprint(f"Nodes: {G.number_of_nodes()}, Edges: {G.number_of_edges()}")
print(f"Density: {nx.density(G):.4f}")
print(f"Is connected: {nx.is_connected(G)}")
if nx.is_connected(G):
print(f"Diameter: {nx.diameter(G)}")
print(f"Avg shortest path: {nx.average_shortest_path_length(G):.4f}")
print(f"Avg clustering coefficient: {nx.average_clustering(G):.4f}")
# Degree distribution
degrees = dict(G.degree())
degree_vals = list(degrees.values())
print(f"\nDegree stats: mean={np.mean(degree_vals):.2f}, max={max(degree_vals)}")
Step 2: Centrality Analysis
import networkx as nx
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
# Use a real-world-like network
G = nx.barabasi_albert_graph(100, 3, seed=42)
# Compute all centrality measures
centralities = {
"degree": nx.degree_centrality(G),
"betweenness": nx.betweenness_centrality(G, normalized=True),
"closeness": nx.closeness_centrality(G),
"pagerank": nx.pagerank(G, alpha=0.85),
"eigenvector": nx.eigenvector_centrality(G, max_iter=1000),
}
# Assemble into DataFrame
cent_df = pd.DataFrame(centralities)
cent_df.index.name = "node"
cent_df.reset_index(inplace=True)
print("Top 10 nodes by PageRank:")
print(cent_df.nlargest(10, "pagerank")[["node", "degree", "betweenness", "pagerank"]].to_string(index=False))
# Correlation between centralitiesprint("\nCentrality correlations:")
print(cent_df[["degree", "betweenness", "closeness", "pagerank"]].corr().round(3))
# Visualize network with PageRank as node size
fig, ax = plt.subplots(figsize=(10, 8))
pos = nx.spring_layout(G, seed=42)
node_size = [cent_df.loc[cent_df["node"]==n, "pagerank"].values[0] * 5000 + 50for n in G.nodes()]
nx.draw_networkx(G, pos, node_size=node_size, node_color="steelblue",
edge_color="gray", alpha=0.7, with_labels=False, ax=ax)
ax.set_title("Network with PageRank node sizing")
ax.axis("off")
plt.tight_layout()
plt.savefig("network_centrality.png", dpi=150)
plt.show()
Step 3: Community Detection with Leiden Algorithm
import igraph as ig
import leidenalg
import networkx as nx
import matplotlib.pyplot as plt
import numpy as np
# Convert networkx → igraph
G_nx = nx.karate_club_graph()
G_ig = ig.Graph.from_networkx(G_nx)
# Leiden community detection (resolution parameter controls granularity)
partition = leidenalg.find_partition(
G_ig,
leidenalg.ModularityVertexPartition,
seed=42,
)
communities = list(partition)
n_communities = len(communities)
modularity = G_ig.modularity(partition.membership)
print(f"Communities found: {n_communities}")
print(f"Modularity Q = {modularity:.4f}")
print(f"Community sizes: {sorted([len(c) for c in communities], reverse=True)}")
# Map back to networkx for visualization
node_colors = {}
for comm_id, members inenumerate(communities):
for node in members:
node_colors[node] = comm_id
# Visualize
pos = nx.spring_layout(G_nx, seed=42)
colors = [node_colors[n] for n in G_nx.nodes()]
fig, ax = plt.subplots(figsize=(8, 6))
nx.draw_networkx(G_nx, pos, node_color=colors, cmap=plt.cm.tab20,
node_size=200, edge_color="gray", with_labels=True,
font_size=8, ax=ax)
ax.set_title(f"Karate Club: {n_communities} communities (Q={modularity:.3f})")
ax.axis("off")
plt.tight_layout()
plt.savefig("community_detection.png", dpi=150)
plt.show()
# Compare with ground truth (club membership)
ground_truth = [G_nx.nodes[n]["club"] for n in G_nx.nodes()]
from sklearn.metrics import adjusted_rand_score
pred_labels = [node_colors[n] for n in G_nx.nodes()]
ari = adjusted_rand_score(ground_truth, pred_labels)
print(f"\nAdjusted Rand Index vs. ground truth: {ari:.4f}")
Advanced Usage
Random Graph Model Fitting
import networkx as nx
import numpy as np
import matplotlib.pyplot as plt
deffit_random_graph_model(G):
"""Fit Erdős-Rényi, BA, and WS models and compare degree distributions."""
n = G.number_of_nodes()
m = G.number_of_edges()
p = 2 * m / (n * (n - 1)) # ER probability
k_avg = 2 * m / n
k = max(1, int(round(k_avg / 2))) # BA attachment parameter
G_er = nx.erdos_renyi_graph(n, p, seed=42)
G_ba = nx.barabasi_albert_graph(n, k, seed=42)
G_ws = nx.watts_strogatz_graph(n, min(n-1, 4*k), 0.1, seed=42)
fig, axes = plt.subplots(1, 3, figsize=(15, 4))
for ax, G_model, name inzip(axes, [G_er, G_ba, G_ws],
["Erdős-Rényi", "Barabási-Albert", "Watts-Strogatz"]):
orig_deg = [d for _, d in G.degree()]
model_deg = [d for _, d in G_model.degree()]
ax.hist(orig_deg, bins=20, alpha=0.6, density=True, label="Original")
ax.hist(model_deg, bins=20, alpha=0.6, density=True, label=name)
ax.set_xlabel("Degree"); ax.set_ylabel("Probability"); ax.legend()
c = nx.average_clustering(G_model)
l = nx.average_shortest_path_length(G_model) if nx.is_connected(G_model) elsefloat('inf')
ax.set_title(f"{name}\nC={c:.3f}, L={l:.2f}")
plt.tight_layout()
plt.savefig("random_graph_comparison.png", dpi=150)
plt.show()
G = nx.karate_club_graph()
fit_random_graph_model(G)
Network Robustness Analysis
import networkx as nx
import numpy as np
import matplotlib.pyplot as plt
defrobustness_analysis(G, n_steps=50):
"""Simulate targeted (degree-based) vs. random node removal."""
G_random = G.copy()
G_targeted = G.copy()
sizes_random, sizes_targeted = [1.0], [1.0]
n_total = G.number_of_nodes()
nodes_random = list(G_random.nodes())
np.random.shuffle(nodes_random)
for _ inrange(n_steps):
# Random removalif G_random.number_of_nodes() > 0:
G_random.remove_node(nodes_random.pop())
if G_random.number_of_nodes() > 0and nx.is_connected(G_random):
sizes_random.append(G_random.number_of_nodes() / n_total)
else:
comps = list(nx.connected_components(G_random))
sizes_random.append(max(len(c) for c in comps) / n_total if comps else0)
# Targeted removal (highest degree node)if G_targeted.number_of_nodes() > 0:
max_deg_node = max(G_targeted.degree(), key=lambda x: x[1])[0]
G_targeted.remove_node(max_deg_node)
comps = list(nx.connected_components(G_targeted))
sizes_targeted.append(max(len(c) for c in comps) / n_total if comps else0)
fig, ax = plt.subplots(figsize=(8, 5))
ax.plot(np.linspace(0, 1, len(sizes_random)), sizes_random, 'b-o', ms=3, label="Random removal")
ax.plot(np.linspace(0, 1, len(sizes_targeted)), sizes_targeted, 'r-s', ms=3, label="Targeted removal")
ax.set_xlabel("Fraction of nodes removed")
ax.set_ylabel("Largest component fraction")
ax.legend(); ax.set_title("Network Robustness Analysis")
plt.tight_layout()
plt.savefig("robustness_analysis.png", dpi=150)
plt.show()
G = nx.barabasi_albert_graph(200, 3, seed=42)
robustness_analysis(G)
Troubleshooting
Error: networkx.exception.NetworkXError: Graph is not connected
Cause: Some metrics (diameter, avg path length) require connected graphs.
Fix:
ifnot nx.is_connected(G):
# Work with largest connected component
Gcc = G.subgraph(max(nx.connected_components(G), key=len)).copy()
print(f"Using LCC: {Gcc.number_of_nodes()} / {G.number_of_nodes()} nodes")
Issue: Leiden algorithm returns single large community
Cause: Resolution parameter too low.
Fix:
# Increase resolution for more, smaller communities
partition = leidenalg.find_partition(
G_ig, leidenalg.CPMVertexPartition,
resolution_parameter=0.05# increase from 0 to get more communities
)
