networkx.algorithms.cluster.clustering

clustering(G, nodes=None, weight=None)[source]

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 \(deg(u) < 2\).

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 3.

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

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
  • G (graph)

  • nodes (container of nodes, optional (default=all nodes in G)) – Compute clustering for nodes in this container.

  • weight (string 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

out – Clustering coefficient at specified nodes

Return type

float, or dictionary

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}

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

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