square_clustering

square_clustering(G, nodes=None)

Compute the squares clustering coefficient for nodes.

For each node return the fraction of possible squares that exist at the node [1]

$$C_4(v)=\frac{\sum _{u=1}^{k_v}\sum _{w=u+1}^{k_v}{q}_v(u,w)}{\sum _{u=1}^{k_v}\sum _{w=u+1}^{k_v}[a_v(u,w)+q_v(u,w)]},$$

where $q_v(u,w)$ are the number of common neighbors of $u$ and $w$ other than $v$ (ie squares), and $a_v(u,w)=(k_u-(1+q_v(u,w)+{\theta }_{uv}))+(k_w-(1+q_v(u,w)+{\theta }_{uw}))$, where ${\theta }_{uw}=1$ if $u$ and $w$ are connected and 0 otherwise. [2]

Parameters:

Ggraph

nodescontainer of nodes, optional (default=all nodes in G)

Compute clustering for nodes in this container.

Returns:

c4dictionary

A dictionary keyed by node with the square clustering coefficient value.

Notes

While $C_3(v)$ (triangle clustering) gives the probability that two neighbors of node v are connected with each other, $C_4(v)$ is the probability that two neighbors of node v share a common neighbor different from v. This algorithm can be applied to both bipartite and unipartite networks.

References

[1]

Pedro G. Lind, Marta C. González, and Hans J. Herrmann. 2005 Cycles and clustering in bipartite networks. Physical Review E (72) 056127.

[2]

Zhang, Peng et al. Clustering Coefficient and Community Structure of Bipartite Networks. Physica A: Statistical Mechanics and its Applications 387.27 (2008): 6869–6875. https://arxiv.org/abs/0710.0117v1

Examples

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