clustering

clustering(G, nodes=None, weight=None)

Compute the clustering coefficient for nodes.

For unweighted graphs, the clustering of a node \(u\) is the fraction of possible triangles through that node that exist,

\[c_u = \frac{2 T(u)}{deg(u)(deg(u)-1)},\]

where \(T(u)\) is the number of triangles through node \(u\) and \(deg(u)\) is the degree of \(u\).

For weighted graphs, there are several ways to define clustering [1]. the one used here is defined as the geometric average of the subgraph edge weights [2],

\[c_u = \frac{1}{deg(u)(deg(u)-1))} \sum_{vw} (\hat{w}_{uv} \hat{w}_{uw} \hat{w}_{vw})^{1/3}.\]

The edge weights \(\hat{w}_{uv}\) are normalized by the maximum weight in the network \(\hat{w}_{uv} = w_{uv}/\max(w)\).

The value of \(c_u\) is assigned to 0 if \(deg(u) < 2\).

Additionally, this weighted definition has been generalized to support negative edge weights [3].

For directed graphs, the clustering is similarly defined as the fraction of all possible directed triangles or geometric average of the subgraph edge weights for unweighted and weighted directed graph respectively [4].

\[c_u = \frac{T(u)}{2(deg^{tot}(u)(deg^{tot}(u)-1) - 2deg^{\leftrightarrow}(u))},\]

where \(T(u)\) is the number of directed triangles through node \(u\), \(deg^{tot}(u)\) is the sum of in degree and out degree of \(u\) and \(deg^{\leftrightarrow}(u)\) is the reciprocal degree of \(u\).

Parameters:

Ggraph

nodesnode, iterable of nodes, or None (default=None)

If a singleton node, return the number of triangles for that node. If an iterable, compute the number of triangles for each of those nodes. If None (the default) compute the number of triangles for all nodes in G.

weightstring or None, optional (default=None)

The edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1.

Returns:

outfloat, or dictionary

Clustering coefficient at specified nodes

Notes

Self loops are ignored.

References

[1]

Generalizations of the clustering coefficient to weighted complex networks by J. Saramäki, M. Kivelä, J.-P. Onnela, K. Kaski, and J. Kertész, Physical Review E, 75 027105 (2007). http://jponnela.com/web_documents/a9.pdf

[2]

Intensity and coherence of motifs in weighted complex networks by J. P. Onnela, J. Saramäki, J. Kertész, and K. Kaski, Physical Review E, 71(6), 065103 (2005).

[3]

Generalization of Clustering Coefficients to Signed Correlation Networks by G. Costantini and M. Perugini, PloS one, 9(2), e88669 (2014).

[4]

Clustering in complex directed networks by G. Fagiolo, Physical Review E, 76(2), 026107 (2007).

Examples

>>> G = nx.complete_graph(5)
>>> print(nx.clustering(G, 0))
1.0
>>> print(nx.clustering(G))
{0: 1.0, 1: 1.0, 2: 1.0, 3: 1.0, 4: 1.0}
----

Additional backends implement this function

cugraphGPU-accelerated backend.

Directed graphs and weight parameter are not yet supported.

graphblas : OpenMP-enabled sparse linear algebra backend.

parallelA networkx backend that uses joblib to run graph algorithms in parallel. Find the nx-parallel’s configuration guide here

The nodes are chunked into node_chunks and then the clustering coefficient for all node_chunks is computed in parallel over n_jobs number of CPU cores.

Additional parameters:

get_chunksstr, function (default = “chunks”)

A function that takes in a list of all the nodes (or nbunch) as input and returns an iterable node_chunks. The default chunking is done by slicing the nodes into n_jobs number of chunks.

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