Node Vulnerability

Computes the vulnerability of each node, defined as the relative drop in global efficiency when that node is removed from the network.

Usage

vulnerability(
  x,
  directed = NULL,
  normalized = TRUE,
  weighted = FALSE,
  invert_weights = TRUE,
  alpha = 1,
  digits = NULL,
  ...
)

Arguments

x: Network input: matrix, igraph, network, cograph_network, or tna object.
directed: Logical or NULL. If NULL (default), auto-detect from matrix symmetry. Set TRUE to force directed, FALSE to force undirected.
normalized: Logical. If TRUE (default), return the proportional drop. If FALSE, return the raw efficiency difference.
weighted: Logical. If TRUE, honor edge weights when computing shortest paths (Dijkstra); distance is 1/weight^alpha per the usual qgraph/tna convention when invert_weights = TRUE. If FALSE (default, matches prior behavior), all edges are treated as unit length.
invert_weights: Logical. If TRUE (default) and weights are present, invert weights to distances via 1/weight^alpha so that higher weight = shorter path (matches centrality()'s default). Ignored when weighted = FALSE.
alpha: Weight-to-distance exponent (default 1).
digits: Integer or NULL. Round scores to this many decimal places. Default NULL (no rounding).
...: Currently unused; directed is already an explicit argument above and to_igraph accepts no others.

Value

A data frame of class "cograph_vulnerability" with one row per node and columns:

node: Node labels.
vulnerability: Vulnerability scores, sorted descending.

The original input network ("network") and the normalization mode ("normalized") are stored as attributes. Scores are NA for a network with at most one node, and all zero when the full network already has zero global efficiency.

Details

$$V(i) = \frac{E_{global} - E_{global \setminus i}}{E_{global}}$$

where $E_{global}$ is the global efficiency of the full network and $E_{global \setminus i}$ is the global efficiency after removing node i and all its edges.

Global efficiency is defined as:

$$E_{global} = \frac{1}{n(n-1)} \sum_{i \neq j} \frac{1}{d(i,j)}$$

$E_{global \setminus i}$ is computed on the reduced graph but keeps the original $n(n-1)$ denominator, so vulnerability is bounded below by zero (Latora & Marchiori 2007); re-normalizing by $(n-1)(n-2)$ would let a node removal appear to raise efficiency.

Nodes with high vulnerability are critical to the network's communication efficiency. Removing them causes the greatest drop in global efficiency.

Performance note: This function computes all-pairs shortest paths once for the full graph and once per node removal, giving O(n) calls to the shortest-path algorithm. A warning is issued for networks with more than 500 nodes.

References

Latora, V. & Marchiori, M. (2007). A measure of centrality based on network efficiency. New Journal of Physics, 9(6), 188. doi:10.1088/1367-2630/9/6/188

Examples

# Star network: hub is most vulnerable
star <- matrix(c(0,1,1,1, 1,0,0,0, 1,0,0,0, 1,0,0,0), 4, 4)
rownames(star) <- colnames(star) <- c("hub", "a", "b", "c")
cograph::vulnerability(star)
#> Node Vulnerability (normalized)
#>  node vulnerability
#>   hub     1.0000000
#>     a     0.4444444
#>     b     0.4444444
#>     c     0.4444444

# Complete graph: all nodes equally vulnerable
k4 <- matrix(1, 4, 4); diag(k4) <- 0
rownames(k4) <- colnames(k4) <- c("A", "B", "C", "D")
cograph::vulnerability(k4)
#> Node Vulnerability (normalized)
#>  node vulnerability
#>     A           0.5
#>     B           0.5
#>     C           0.5
#>     D           0.5