NetworkX Graph Generators

API patterns assume import networkx as nx. A generated model is a hypothesis about a graph, not evidence that observed data follow that model. Preserve direction, loops, multiplicity, size, degree constraints and seed/version provenance when constructing null ensembles. See review.md.

Classic Graphs

Complete Graphs

# Complete graph (all nodes connected to all others)
G = nx.complete_graph(n=10)

# Complete bipartite graph
G = nx.complete_bipartite_graph(n1=5, n2=7)

# Complete multipartite graph
G = nx.complete_multipartite_graph(3, 4, 5)  # Three partitions

Cycle and Path Graphs

# Cycle graph (nodes arranged in circle)
G = nx.cycle_graph(n=20)

# Path graph (linear chain)
G = nx.path_graph(n=15)

# Circular ladder graph
G = nx.circular_ladder_graph(n=10)

Regular Graphs

# Empty graph (no edges)
G = nx.empty_graph(n=10)

# Null graph (no nodes)
G = nx.null_graph()

# Star graph (one central node connected to all others)
G = nx.star_graph(n=19)  # Creates 20-node star

# Wheel graph (cycle with central hub)
G = nx.wheel_graph(n=10)

Special Named Graphs

# Bull graph
G = nx.bull_graph()

# Chvatal graph
G = nx.chvatal_graph()

# Cubical graph
G = nx.cubical_graph()

# Diamond graph
G = nx.diamond_graph()

# Dodecahedral graph
G = nx.dodecahedral_graph()

# Heawood graph
G = nx.heawood_graph()

# House graph
G = nx.house_graph()

# Petersen graph
G = nx.petersen_graph()

# Karate club graph (classic social network)
G = nx.karate_club_graph()

Random Graphs

Erdős-Rényi Graphs

# G(n, p) model: n nodes, edge probability p
G = nx.erdos_renyi_graph(n=100, p=0.1, seed=42)

# G(n, m) model: n nodes, exactly m edges
G = nx.gnm_random_graph(n=100, m=500, seed=42)

# Fast version (for large sparse graphs)
G = nx.fast_gnp_random_graph(n=10000, p=0.0001, seed=42)

Watts-Strogatz Small-World

# Small-world network with rewiring
# n nodes, k nearest neighbors, rewiring probability p
G = nx.watts_strogatz_graph(n=100, k=6, p=0.1, seed=42)

# Returns a connected sample or raises after tries attempts
G = nx.connected_watts_strogatz_graph(n=100, k=6, p=0.1, tries=100, seed=42)

Barabási-Albert Preferential Attachment

# Preferential-attachment model; finite samples do not establish a power law
# n nodes, m edges to attach from new node
G = nx.barabasi_albert_graph(n=100, m=3, seed=42)

# Extended version with parameters
G = nx.extended_barabasi_albert_graph(n=100, m=3, p=0.5, q=0.2, seed=42)

Power Law Degree Sequence

# Power law cluster graph
G = nx.powerlaw_cluster_graph(n=100, m=3, p=0.1, seed=42)

# Random power law tree
G = nx.random_powerlaw_tree(n=100, gamma=3, seed=42, tries=1000)

Configuration Model

# Graph with specified degree sequence
degree_sequence = [3, 3, 2, 2, 2, 2]  # Even sum required
G = nx.configuration_model(degree_sequence, seed=42)
assert [G.degree(i) for i in range(len(degree_sequence))] == degree_sequence

# Optional simplification changes the degree sequence and null model
H = nx.Graph(G)
H.remove_edges_from(list(nx.selfloop_edges(H)))

The configuration model is a stub-matching multigraph with possible self-loops. Removing loops/parallel edges does not preserve the specified degrees. havel_hakimi_graph gives one simple realization of a graphical sequence, not a uniform random sample. Degree-preserving swaps need adequate mixing and convergence checks; neither a single sample nor a fixed seed establishes null-model validity.

Random Geometric Graphs

# Nodes in unit square, edges if distance < radius
G = nx.random_geometric_graph(n=100, radius=0.2, seed=42)

# With positions
pos = nx.get_node_attributes(G, 'pos')

Random Regular Graphs

# Every node has exactly d neighbors
G = nx.random_regular_graph(d=3, n=100, seed=42)

Stochastic Block Model

# Community structure model
sizes = [50, 50, 50]  # Three communities
probs = [[0.25, 0.05, 0.02],  # Within and between community probabilities
         [0.05, 0.35, 0.07],
         [0.02, 0.07, 0.40]]
G = nx.stochastic_block_model(sizes, probs, seed=42)

Lattice and Grid Graphs

Grid Graphs

# 2D grid
G = nx.grid_2d_graph(m=5, n=7)  # 5x7 grid

# 3D grid
G = nx.grid_graph(dim=[5, 7, 3])  # 5x7x3 grid

# Hexagonal lattice
G = nx.hexagonal_lattice_graph(m=5, n=7)

# Triangular lattice
G = nx.triangular_lattice_graph(m=5, n=7)

Hypercube

# n-dimensional hypercube
G = nx.hypercube_graph(n=4)

Tree Graphs

Random Trees

# Random labeled tree with n nodes, sampled uniformly over labeled trees
# (nx.random_tree was removed in NetworkX 3.4)
G = nx.random_labeled_tree(100, seed=42)

# Sample uniformly over isomorphism classes instead
G = nx.random_unlabeled_tree(100, seed=42)

# Rooted variant is still an undirected Graph, with G.graph["root"]
G = nx.random_labeled_rooted_tree(100, seed=42)

# Prefix tree (tries)
G = nx.prefix_tree([[0, 1, 2], [0, 1, 3], [0, 4]])

Balanced Trees

# Balanced r-ary tree of height h
G = nx.balanced_tree(r=2, h=5)  # Binary tree, height 5

# Full r-ary tree with n nodes
G = nx.full_rary_tree(r=3, n=100)  # Ternary tree

Barbell and Lollipop Graphs

# Two complete graphs connected by path
G = nx.barbell_graph(m1=5, m2=3)  # Two K_5 graphs with 3-node path

# Complete graph connected to path
G = nx.lollipop_graph(m=7, n=5)  # K_7 with 5-node path

Social Network Models

Karate Club

# Zachary's karate club (classic social network)
G = nx.karate_club_graph()

Davis Southern Women

# Bipartite social network
G = nx.davis_southern_women_graph()

Florentine Families

# Historical marriage network (this generator does not add business ties)
G = nx.florentine_families_graph()

Les Misérables

# Character co-occurrence network
G = nx.les_miserables_graph()

Directed Graph Generators

Random Directed Graphs

# Directed Erdős-Rényi
G = nx.gnp_random_graph(n=100, p=0.1, directed=True, seed=42)

# Directed MultiDiGraph; parallel edges and self-loops can occur
G = nx.scale_free_graph(n=100, seed=42)

DAG (Directed Acyclic Graph)

# Ordered random DAG; this is not uniform over all DAGs
R = nx.gnp_random_graph(n=20, p=0.2, directed=True, seed=42)
G = nx.DiGraph()
G.add_nodes_from(R)  # Preserve isolates
G.add_edges_from((u, v) for u, v in R.edges() if u < v)
assert len(G) == 20 and nx.is_directed_acyclic_graph(G)

Tournament Graphs

# Random tournament (complete directed graph); lives in the tournament module
G = nx.tournament.random_tournament(n=10, seed=42)

Duplication-Divergence Models

Duplication Divergence Graph

# Biological network model (protein interaction networks)
G = nx.duplication_divergence_graph(n=100, p=0.5, seed=42)

Degree Sequence Generators

Valid Degree Sequences

# Check if degree sequence is valid (graphical)
sequence = [3, 3, 2, 2, 2, 2]
is_valid = nx.is_graphical(sequence)

# For directed graphs
in_sequence = [2, 2, 2, 1, 1]
out_sequence = [2, 2, 1, 2, 1]
is_valid = nx.is_digraphical(in_sequence, out_sequence)

Creating from Degree Sequence

# Havel-Hakimi algorithm
G = nx.havel_hakimi_graph(degree_sequence)

# Configuration model (allows multi-edges/self-loops)
G = nx.configuration_model(degree_sequence, seed=42)

# Directed configuration model: sums must match
G = nx.directed_configuration_model(in_sequence, out_sequence, seed=42)

Bipartite Graphs

Random Bipartite

# Random bipartite with two node sets
G = nx.bipartite.random_graph(n=50, m=30, p=0.1, seed=42)

# Configuration model for bipartite
G = nx.bipartite.configuration_model(aseq=[3, 3, 2], bseq=[2, 2, 2, 2], seed=42)

Bipartite Generators

# Complete bipartite
G = nx.complete_bipartite_graph(n1=5, n2=7)

# Gnmk random bipartite (n, m nodes, k edges)
G = nx.bipartite.gnmk_random_graph(n=10, m=8, k=20, seed=42)

Operators on Graphs

Graph Operations

# Union requires disjoint node labels (or supply rename prefixes)
G = nx.union(G1, G2)

# Disjoint union
G = nx.disjoint_union(G1, G2)

# Compose (overlay)
G = nx.compose(G1, G2)

# Complement
G = nx.complement(G1)

# Cartesian product
G = nx.cartesian_product(G1, G2)

# Tensor (Kronecker) product
G = nx.tensor_product(G1, G2)

# Strong product
G = nx.strong_product(G1, G2)

Customization and Seeding

Setting Random Seed

Set a seed and record NetworkX/Python versions and generation parameters. A seed does not promise identical graphs across releases/backends:

G = nx.erdos_renyi_graph(n=100, p=0.1, seed=42)

Converting Graph Types

# Convert to specific type
G_directed = G.to_directed()
G_undirected = G.to_undirected()
G_multi = nx.MultiGraph(G)

Performance Considerations

Fast Generators

For large graphs, use optimized generators:

# Fast ER graph (sparse)
G = nx.fast_gnp_random_graph(n=10000, p=0.0001, seed=42)

Memory Efficiency

Some generators create graphs incrementally to save memory. For very large graphs, consider:

• Using sparse representations
• Generating subgraphs as needed
• Working with adjacency lists or edge lists instead of full graphs

Validation and Properties

Checking Generated Graphs

# Verify properties
print(f"Nodes: {G.number_of_nodes()}")
print(f"Edges: {G.number_of_edges()}")
print(f"Density: {nx.density(G)}")
if len(G):
    connected = nx.is_weakly_connected(G) if G.is_directed() else nx.is_connected(G)
    print(f"Connected (weak if directed): {connected}")
else:
    print("Connectivity undefined for the null graph")

# Degree distribution
degree_sequence = sorted([d for n, d in G.degree()], reverse=True)