The extended hybrid characteristic centrality of Liu and Zheng is the
closed-neighborhood sum of centrality_hcc:
\(EHCC(u)=HCC(u)+\sum_{v\in\phi(u)}HCC(v)\), the focal node counted
once and each neighbor of the open 1-order neighborhood once. It
rewards a node whose neighbors are themselves high in both the extended
degree and the E-shell hierarchy, which a node can be without being high
itself.
Arguments
- x
Network input accepted by
centrality.- ...
Additional arguments to
centrality, includinghcc_delta.
Details
Everything recorded on centrality_hcc carries over
unchanged: the source's \(\arg\max\)/\(\arg\min\) typo in step 3 of
the E-shell procedure, the original-graph reading of \(k^{ex}\) and
\(k^{ex}_{max}\) against the residual-graph peel, the global and
therefore not component-local normalizers, the hcc_delta domain
\([0,1]\), the \(0/0\) of an edgeless graph written as zero, the
simple undirected unweighted skeleton, and the ignored weights, mode,
cutoff and inversion. Because HCC lies in \([0,2]\), EHCC lies in
\([0,2(1+k_{max})]\), and an isolate scores exactly its own HCC.
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. Equation (5) on page 3. doi:10.1038/s41598-023-30308-5 .
See also
centrality_hcc for the summand and
list_centralities for the catalogue.
Examples
# On a regular graph every node scores 2, so EHCC is 2 (1 + k).
centrality_ehcc(igraph::make_ring(6))
#> 1 2 3 4 5 6
#> 6 6 6 6 6 6
# The star's center collects every leaf's score as well as its own.
centrality_ehcc(igraph::make_star(6, mode = "undirected"))
#> 1 2 3 4 5 6
#> 7.5 3.1 3.1 3.1 3.1 3.1