Finding cycle of 3 nodes ( or triangles) in a graph

Question

I am working with complex networks. I want to find group of nodes which forms a cycle of 3 nodes (or triangles) in a given graph. As my graph contains about million edges, u

Answer 1

i just found that nx.edge_disjoint_paths works to count the triangle contains certain edges. faster than nx.enumerate_all_cliques and nx.cycle_basis. It returns the edges disjoint paths between source and target.Edge disjoint paths are paths that do not share any edge.
And result-1 is the number of triangles that contain certain edges or between source node and target node.

edge_triangle_dict = {}
for i in g.edges:
    edge_triangle_dict[i] = len(list(nx.edge_disjoint_paths(g, i[0], i[1]))-1)

Answer 2

This is a more efficient version of Ajay M answer (I would have commented it, but I've not enough reputation).

Indeed the enumerate_all_cliques method of networkx will return all cliques in the graph, irrespectively of their length; hence looping over it may take a lot of time (especially with very dense graphs).

Moreover, once defined for triangles, it's just a matter of parametrization to generalize the method for every clique length so here's a function:

import networkx as nx

def get_cliques_by_length(G, length_clique):
    """ Return the list of all cliques in an undirected graph G with length 
    equal to length_clique. """
    cliques = []
    for c in nx.enumerate_all_cliques(G) :
        if len(c) <= length_clique:
            if len(c) == length_clique:
                cliques.append(c)            
        else:
            return cliques
    # return empty list if nothing is found
    return cliques

To get triangles just use get_cliques_by_length(G, 3).

Caveat: this method works only for undirected graphs. Algorithm for cliques in directed graphs are not provided in networkx

Answer 3

I don't want to sound harsh, but have you tried to Google it? The first link is a pretty quick algorithm to do that: http://www.mail-archive.com/algogeeks@googlegroups.com/msg05642.html

And then there is this article on ACM (which you may have access to): http://portal.acm.org/citation.cfm?id=244866 (and if you don't have access, I am sure if you kindly ask the lady who wrote it, you will get a copy.)

Also, I can imagine a triangle enumeration method based on clique-decomposition, but I don't know if it was described somewhere.

Answer 4

Assuming its an undirected graph, the answer lies in networkx library of python. if you just need to count triangles, use:

import networkx as nx
tri=nx.triangles(g)

But if you need to know the edge list with triangle (triadic) relationship, use

all_cliques= nx.enumerate_all_cliques(g)

This will give you all cliques (k=1,2,3...max degree - 1)

So, to filter just triangles i.e k=3,

triad_cliques=[x for x in all_cliques if len(x)==3 ]

The triad_cliques will give a edge list with only triangles.

Answer 5

Surprised to see no mention of the Networkx triangles function. I know it doesn't necessarily return the groups of nodes that form a triangle, but should be pretty relevant to many who find themselves on this page.

nx.triangles(G) # list of how many triangles each node is part of
sum(nx.triangles(G).values())/3 # total number of triangles

An alternative way to return clumps of nodes would be something like...

for u,v,d in G.edges(data=True):
    u_array = adj_m.getrow(u).nonzero()[1] # get lists of all adjacent nodes
    v_array = adj_m.getrow(v).nonzero()[1]
    # find the intersection of the two sets - these are the third node of the triangle
    np.intersect1d(v_array,u_array)

Answer 6

If you don't care about multiple copies of the same triangle in different order then a list of 3-tuples works:

from itertools import combinations as combos
[(n,nbr,nbr2) for n in G for nbr, nbr2 in combos(G[n],2) if nbr in G[nbr2]]

The logic here is to check each pair of neighbors of every node to see if they are connected. G[n] is a fast way to iterate over or look up neighbors.

If you want to get rid of reorderings, turn each triple into a frozenset and make a set of the frozensets:

set(frozenset([n,nbr,nbr2]) for n in G for nbr, nbr2 in combos(G[n]) if nbr in G[nbr2])

If you don't like frozenset and want a list of sets then:

triple_iter = ((n, nbr, nbr2) for n in G for nbr, nbr2 in combos(G[n],2) if nbr in G[nbr2])
triangles = set(frozenset(tri) for tri in triple_iter)
nice_triangles = [set(tri) for tri in triangles]