Luan et al.'s improved closeness is \(ICC(i)=(n-1)/\sum_{j\ne i}d_{ij}/\sigma_{ij}^{\alpha}\), where d is the hop distance and sigma counts shortest paths. Multiple shortest paths reduce the effective distance to a partner. At alpha zero this is ordinary normalized closeness on a connected graph; on a tree it is independent of alpha because each pair has one shortest path. Scores need not be bounded by one.
Arguments
- x
Network input accepted by
centrality.- icc_alpha
Multiplicity exponent between zero and one, default 0.2.
- ...
Additional arguments to
centrality. Withnormalized = TRUE, positive final scores are divided by their maximum. The published n-1 factor is present in raw scores already.
Details
Uses the simple undirected unweighted skeleton: either direction creates
an edge, parallel edges count once and self-loops are removed. Weights,
mode and path-weight inversion do not affect the result. These
are explicit cograph projections to the published domain.
In a disconnected graph, every node has an unreachable partner and therefore scores zero under the global infinite-distance convention. Singletons score zero by an explicit cograph convention for the otherwise undefined zero-over-zero expression. For within-component scores, supply each component separately. Empty input returns an empty vector.
Breadth-first traversal counts shortest paths in logarithmic form, avoiding overflow when the number of paths exceeds double precision. Extremely small effective-distance terms can underflow to zero, but direct-neighbor terms remain one and keep the denominator positive. Computation costs O(n times (n+m)) with an additional dense adjacency representation. Default alpha 0.2 is a setting studied in the source, not an estimate or a guarantee of optimal spreading predictions.
References
Luan, Y., Bao, Z., & Zhang, H. (2021). Identifying Influential Spreaders in Complex Networks by Considering the Impact of the Number of Shortest Paths. Journal of Systems Science and Complexity, 34, 2168-2181, equation 7. doi:10.1007/s11424-021-0111-7 .
Examples
centrality_improved_closeness(igraph::make_ring(4), icc_alpha = 0.2)
#> 1 2 3 4
#> 0.8019029 0.8019029 0.8019029 0.8019029