centrality_improved_closeness: Improved closeness centrality

In cograph: Analysis and Visualization of Complex Networks

Improved closeness centrality

Description

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.

Usage

centrality_improved_closeness(x, icc_alpha = 0.2, ...)

Arguments

x	Network input accepted by centrality.
icc_alpha	Multiplicity exponent between zero and one, default 0.2.
...	Additional arguments to centrality. With normalized = 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.

Value

Named numeric vector in input node order.

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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11424-021-0111-7")}.

Examples

centrality_improved_closeness(igraph::make_ring(4), icc_alpha = 0.2)

cograph documentation built on Sept. 30, 2026, 5:08 p.m.