Local transitivity, the clustering coefficient of Watts and Strogatz (1998), is the share of pairs of a node's neighbors that are themselves linked: $$C_i = \frac{2 T_i}{k_i (k_i - 1)},$$ where \(T_i\) is the number of triangles through node \(i\) and \(k_i\) its degree.
Arguments
- x
Network input accepted by
centrality.- transitivity_type
One of
"local"(default),"global","undirected","localundirected","barrat","weighted"or"onnela".- isolates
Value for nodes with fewer than two ties:
"nan"(default) or"zero".- ...
Further arguments to
centrality. The measure usestna_network(defaultNULL, which detects tna input).
Details
Triangles and degrees are counted on the undirected skeleton, so a
reciprocated tie counts once, and edge weights are ignored. The values
equal igraph::transitivity(type = "local") on directed and
undirected networks. "localundirected" gives the same values as
"local", as in igraph. "global" and "undirected"
return the network-level ratio of closed to connected triples for every
node. "barrat" and "weighted" compute the weighted
coefficient of Barrat et al. (2004) and raise a
cograph_directed_unsupported error on directed input.
"onnela" computes the weighted clustering coefficient of Zhang and
Horvath (2005),
\((M^3)_{ii} / (s_i^2 - \sum_j M_{ij}^2)\)
on \(M = W + W^{T}\) with strengths \(s_i\). This is the
value tna::centralities() reports as Clustering, and it is the
default for tna input. The option keeps the name "onnela" used by
tna, but the formula multiplies the raw weights of a triangle and differs
from the geometric-mean form of Onnela et al. (2005). Under the local and
Barrat types a node with fewer than two neighbors is NaN, or 0
with isolates = "zero".
References
Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of 'small-world' networks. Nature, 393(6684), 440-442. doi:10.1038/30918 .
Barrat, A., Barthelemy, M., Pastor-Satorras, R., & Vespignani, A. (2004). The architecture of complex weighted networks. Proceedings of the National Academy of Sciences, 101(11), 3747-3752. doi:10.1073/pnas.0400087101 .
Zhang, B., & Horvath, S. (2005). A general framework for weighted gene co-expression network analysis. Statistical Applications in Genetics and Molecular Biology, 4(1), Article 17. doi:10.2202/1544-6115.1128 .
Onnela, J.-P., Saramaki, J., Kertesz, J., & Kaski, K. (2005). Intensity and coherence of motifs in weighted complex networks. Physical Review E, 71(6), 065103. doi:10.1103/PhysRevE.71.065103 .