How to generate a triangle free graph in Networkx (with randomseed)? - networkx

Question

After checking the documentation on triangles of networkx, I've wondered if there is a more efficient way of generating a triangle free graph than to randomly spawn graphs until a triangle free one happens to emerge, (in particular if one would like to use a constant random seed).
Below is code that spawns graphs until they are triangle free, yet with varying random seeds. For a graph of 10 nodes it already takes roughly 20 seconds.
def create_triangle_free_graph(show_graphs):
seed = 42
nr_of_nodes = 10
probability_of_creating_an_edge = 0.85
nr_of_triangles = 1 # Initialise at 1 to initiate while loop.
while nr_of_triangles > 0:
graph = nx.fast_gnp_random_graph(
nr_of_nodes, probability_of_creating_an_edge
)
triangles = nx.triangles(G).values()
nr_of_triangles = sum(triangles) / 3
print(f"nr_of_triangles={nr_of_triangles}")
return graph
Hence, I would like to ask:
Are there faster ways to generate triangle free graphs (using random seeds) in networkx?

Answer 1

A triangle exists in a graph iff two vertices connected by an edge share one or more neighbours. A triangle-free graph can be expanded by adding edges between nodes that share no neighbours. The empty graph is triangle-free, so there is a straightforward algorithm to create triangle-free graphs.
#!/usr/bin/env python
"""
Create a triangle free graph.
"""
import random
import networkx as nx
from itertools import combinations
def triangle_free_graph(total_nodes):
"""Construct a triangle free graph."""
nodes = range(total_nodes)
g = nx.Graph()
g.add_nodes_from(nodes)
edge_candidates = list(combinations(nodes, 2))
random.shuffle(edge_candidates)
for (u, v) in edge_candidates:
if not set(n for n in g.neighbors(u)) & set(n for n in g.neighbors(v)):
g.add_edge(u, v)
return g
g = triangle_free_graph(10)
print(nx.triangles(g))
The number of edges in the resulting graph is highly dependent on the ordering of edge_candidates. To get a graph with the desired edge density, repeat the process until a graph with equal or higher density is found (and then remove superfluous edges), or until your patience runs out.
cutoff = 0.85
max_iterations = 1e+4
iteration = 0
while nx.density(g) < cutoff:
g = triangle_free_graph(10)
iteration += 1
if iteration == max_iterations:
import warnings
warnings.warn("Maximum number of iterations reached!")
break
# TODO: remove edges until the desired density is achieved