Compute weighted clustering coefficients for all variables in a catgraph
Source:R/clustering_coef.R
clustering_coef.RdReturns a data frame of five local clustering coefficients for every node
in a catgraph object. Four are established weighted extensions of
the Watts-Strogatz coefficient; the fifth (redundancy) is a measure
specific to effect-size graphs that quantifies how much of a pairwise
association is explained by indirect paths through other variables.
Arguments
- x
A
catgraphobject.- method
Character vector. Which coefficient(s) to compute. Any subset of
c("watts_strogatz", "barrat", "onnela", "zhang", "redundancy")or"all"(default).- normalize
Logical. If
TRUE(default), each coefficient is normalised to [0, 1] by dividing by its theoretical maximum. For the four classical coefficients the theoretical maximum is already 1. For redundancy the observed maximum is used.
Value
A data.frame with one row per variable. Columns are
variable plus one column per requested method. The data frame
is sorted by the mean clustering coefficient across all requested
methods in descending order.
Details
All five coefficients measure the extent to which the neighbours of a node are also connected to each other — the triangle-closing tendency — but they differ in how they incorporate edge weights.
Watts-Strogatz (unweighted; Watts & Strogatz, 1998):
$$C_i^{WS} = \frac{t_i}{k_i(k_i-1)}$$
where \(t_i\) is the number of triangles through node \(i\) and \(k_i\) is its degree. This is the classical unweighted coefficient included as a baseline.
Barrat et al. (2004):
$$C_i^{B} = \frac{1}{s_i(k_i-1)} \sum_{j,h} \frac{w_{ij}+w_{ih}}{2} a_{ij} a_{ih} a_{jh}$$
where \(s_i = \sum_j w_{ij}\) is the node strength and \(a_{ij}\) is the binary adjacency. Weights the contribution of each triangle by the average weight of the two edges incident to the focal node. Nodes that close triangles through their strongest edges score highly.
Onnela et al. (2005):
$$C_i^{O} = \frac{1}{k_i(k_i-1)} \sum_{j,h} (\hat{w}_{ij} \hat{w}_{ih} \hat{w}_{jh})^{1/3}$$
where \(\hat{w} = w / \max(w)\) are normalised weights. Uses the geometric mean of triangle edge weights, making it sensitive to weak ties: a triangle with one very weak edge scores low.
Zhang & Horvath (2005):
$$C_i^{ZH} = \frac{\sum_{j,h} w_{ij} w_{ih} w_{jh}} {\left(\sum_{j \neq i} w_{ij}\right)^2 - \sum_{j \neq i} w_{ij}^2}$$
The numerator sums products of three edge weights around all triangles; the denominator is the squared strength minus the sum of squared weights. Originally proposed for co-expression networks (WGCNA). Amplifies strong-edge triangles via cubic weighting.
Redundancy (original):
For each edge \((i,j)\) in the graph, the redundancy ratio is defined as the direct weight divided by the maximum indirect path weight:
$$R_{ij} = \frac{w_{ij}}{\max_{k \neq i,j} \min(w_{ik}, w_{kj})}$$
The node-level redundancy coefficient is the mean of \(R_{ij}\) over all edges incident to node \(i\). A value near 1 indicates that the direct association between two variables is no stronger than what their shared neighbours would predict — the edge may be mediated. A value well above 1 indicates a genuine direct association that exceeds indirect explanation. When no indirect path exists (k < 3), the edge is assigned \(R_{ij} = \infty\) and excluded from the node mean.
References
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
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
Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of 'small-world' networks. Nature, 393(6684), 440–442. doi:10.1038/30918
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
Examples
df <- expand_table(Titanic)
cg <- catgraph(df)
cc <- clustering_coef(cg)
cc
#> variable watts_strogatz barrat onnela zhang redundancy
#> 1 Class 1 0.5 0.2861019 0.2530913 1.0000000
#> 2 Sex 1 0.5 0.2795694 0.2482151 0.8088828
#> 3 Age 1 0.5 0.2106029 0.3691475 0.7145339
#> 4 Survived 1 0.5 0.2688767 0.3138760 0.6472822
# Single method
clustering_coef(cg, method = "barrat")
#> variable barrat
#> 1 Sex 0.5
#> 2 Age 0.5
#> 3 Class 0.5
#> 4 Survived 0.5
# Compare all methods
compare_clustering(cg)
#> variable watts_strogatz barrat onnela zhang redundancy mean_cc
#> 1 Class 1 0.5 0.2861019 0.2530913 1.0000000 0.6078387
#> 2 Sex 1 0.5 0.2795694 0.2482151 0.8088828 0.5673334
#> 3 Age 1 0.5 0.2106029 0.3691475 0.7145339 0.5588569
#> 4 Survived 1 0.5 0.2688767 0.3138760 0.6472822 0.5460070