pymnet.lcc_brodka

pymnet.lcc_brodka(net, node, anet=None, threshold=1, undefReturn=0.0)

The “cross-layer clustering coefficient” defined by Brodka et al.

The clustering coefficient for node \(u\) is given by the formula (see Ref. [2]):

\[c_{Br,u,t} = \frac{\sum_{\alpha \in L} \sum_{v \in N(u,t)} \sum_{h \in N(u,t)}( \mathcal{W}_{hv\alpha} + \mathcal{W}_{vh\alpha})}{2 |N(u,t)| |L|},\]

where \(N(u,t) = \{ v : | \{\alpha : \mathcal{A}_{uv\alpha}=1 \wedge \mathcal{A}_{vu\alpha}=1 \}| \leq t \} `, :math:`L\) is the set of layers, \(\mathcal{W}\) is the rank-3 weighted adjacency tensor of the multiplex network, and \(\mathcal{A}\) is the rank-3 unweighted adjacency tensor of the multiplex network.

One can get the “multi-layered clustering coefficient in extendend neighborhood” by setting the threshold to one \(t=1\), and the “multi-layered clustering coefficient in reduced neighborhood” by setting the threshold to the total number of layers \(t=|L|\).

Parameters:

netMultiplexNetwork with aspects=1

The input network.

nodeany object

The focal node. Given as the node index in the network.

anetMultilayerNetwork with aspects=0

The aggregated network. If given, it is used to speed up the calculation. NOTE: for undirected networks the “normal” aggregation strategy (produced for example by the aggregate function in this library) is suitable, but for directed networks the network needs to be aggregated such that thresholding it will produce the neighborhood sets \(N(u,t)\).

thresholdint, string

Threshold for number of layers, see Ref. [2]. If ‘all’, then the threshold is the total number of layers.

Returns:

float or object

The clustering coefficient value.

See also

lcc_aw

Local multiplex clustering coefficient defined by Cozzo et al.

cc_barrett

Local multiplex clustering coefficient defined by Barrett et al.

Notes

This clustering coefficient doesn’t return to the typical unweighted clustering coefficient when the edge weights are binary and there is only a single layer. Further, it is not normalized in a way that it’s values would be between 0 an 1. For example, consider a full multiplex network with n nodes and arbitrary number of layers. In this case this clustering cofficient will take value n-2 for all the nodes.

Multiplying all the weights by a constant c will cause the clustering coefficient values to be multiplied by c.

References

[1] “A Method for Group Extraction in Complex Social Networks”, Piotr Brodka, Katarzyna Musial, Przemyslaw Kazienko,In M. D. Lytras, P. Ordonez De Pablos, A. Ziderman, A. Roulstone, H. Maurer, and J. B. Imber, editors, Knowledge Management, Information Systems, E-Learning, and Sustainability Research, volume 111 of Communications in Computer and Information Science, pages 238-247. Springer Berlin Heidelberg, 2010.

[2] P. Brodka, P. Kazienko, K. Musial, and K. Skibicki. Analysis of Neighbourhoods in Multi-layered Dynamic Social Networks. International Journal of Computational Intelligence Systems, 5(3):582-596, 2012.