Hybrid characteristic centrality (Liu and Zheng 2023) adds the extended degree \(k^{ex}(u) = \delta k(u) + (1 - \delta) \sum_{v \in \phi(u)} k(v)\) to the E-shell position index \(pos(u)\), each divided by its maximum. The E-shell decomposition removes, round by round, the remaining nodes of minimum extended degree, recomputed on the residual graph, and \(pos(u)\) is the round in which \(u\) leaves. $$HCC(u) = \frac{k^{ex}(u)}{k^{ex}_{max}} + \frac{pos(u)}{pos_{max}}$$
Arguments
- x
Network input accepted by
centrality.- ...
Further arguments to
centrality. The measure useshcc_delta(weight \(\delta\) of the node's own degree in the extended degree, default 0.5).
Details
The measure is computed on the simple undirected skeleton of the network,
so direction, weights, loops and parallel edges are ignored. Scores lie in
\([0, 2]\). Equation (4) uses the extended degrees of the original
graph, as the source's worked example requires. The normalizers are
global, so adding a disconnected component can change every score. An
isolate leaves in the first round, and every node of an edgeless graph
scores one. A hcc_delta outside \([0, 1]\) raises a
cograph_bad_parameter error. Step 3 of the source's E-shell
procedure prints \(\arg\max\) where its text and Table 2
require \(\arg\min\), and the minimum is implemented. The
Centrality Zoo describes the E-shell decomposition as a k-shell variant,
which gives different positions on the source's Figure 1.
References
Liu, J. and Zheng, J. (2023). Identifying important nodes in complex networks based on extended degree and E-shell hierarchy decomposition. Scientific Reports, 13, 3197. doi:10.1038/s41598-023-30308-5 .