vulnerability: Node Vulnerability

In cograph: Analysis and Visualization of Complex Networks

Node Vulnerability

Description

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).
...	Additional arguments passed to to_igraph.

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)}

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.

Value

A data frame of class "cograph_vulnerability" with columns:

node

Node labels.

vulnerability

Vulnerability scores, sorted descending.

The original input network and normalization mode are stored as attributes.

References

Latora, V. & Marchiori, M. (2007). A measure of centrality based on network efficiency. New Journal of Physics, 9(6), 188. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1088/1367-2630/9/6/188")}

See Also

network_global_efficiency, robustness, centrality

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)

# 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)

cograph documentation built on May 31, 2026, 5:06 p.m.