Network Robustness Analysis

Performs a targeted attack or random failure analysis on a network and computes the size of the largest (weakly) connected component after each vertex or edge removal.

In a targeted attack, vertices are ranked by degree or betweenness centrality (edges by edge betweenness) and removed from highest to lowest. In a random failure analysis, vertices or edges are removed in random order.

Usage

robustness(
  x,
  type = c("vertex", "edge"),
  measure = c("betweenness", "degree", "random"),
  strategy = c("sequential", "static"),
  n_iter = 1000,
  mode = "all",
  seed = NULL,
  ...
)

Arguments

x: Network input: matrix, igraph, network, cograph_network, or tna object
type: Character string; either "vertex" or "edge" removals. Default: "vertex"
measure: Character string; rank by "betweenness", "degree", or remove in "random" order. Default: "betweenness". "degree" is not available for edge removals and raises an error.
strategy: Character string. "sequential" (default) recomputes the centrality after each removal. "static" computes the centrality once on the original network and removes elements in that fixed order, as in brainGraph. The argument applies to targeted attacks only.
n_iter: Integer; number of random removal sequences averaged when measure = "random". Default: 1000.
mode: Degree mode for directed networks: "all", "in", or "out". Default "all". Used only when measure = "degree".
seed: Random seed for reproducibility. Default NULL.
...: Passed to to_igraph, whose only other argument is directed; anything else raises an "unused argument" error.

Value

A data frame (class "cograph_robustness") with one row per removal step, from zero removed through all removed (n + 1 rows, where n is the number of vertices or edges), and columns:

removed_pct: Fraction of vertices/edges removed (0 to 1)
comp_size: Size of largest component after removal (averaged over n_iter runs when measure = "random")
comp_pct: Ratio of component size to original maximum
measure: The measure argument: "betweenness", "degree", or "random"
type: A label for the analysis, one of "Targeted vertex attack", "Targeted edge attack", "Random vertex removal" or "Random edge removal"

The last row always has comp_size = 0. The original number of vertices or edges ("n_original") and the original largest-component size ("orig_max") are stored as attributes.

Details

A betweenness attack removes the vertices or edges that lie on the most shortest paths, which bridge different regions of the network. A degree attack removes the most connected vertices first. Random failure removes vertices or edges in random order and averages the component sizes over n_iter sequences. Betweenness is computed with igraph, which treats edge weights as distances.

The sequential strategy updates the ranking after every removal, so the attack follows the vertices that become new bridges or hubs. The static strategy ranks once on the original network, as in Albert et al. (2000).

Scale-free networks are typically robust to random failures and vulnerable to targeted attacks. Random networks degrade more uniformly.

References

Albert, R., Jeong, H., & Barabasi, A.L. (2000). Error and attack tolerance of complex networks. Nature, 406, 378-381. doi:10.1038/35019019

Examples

robustness(regulation_net, measure = "betweenness")
#>    removed_pct comp_size comp_pct     measure                   type
#> 1          0.0        10      1.0 betweenness Targeted vertex attack
#> 2          0.1         9      0.9 betweenness Targeted vertex attack
#> 3          0.2         8      0.8 betweenness Targeted vertex attack
#> 4          0.3         7      0.7 betweenness Targeted vertex attack
#> 5          0.4         6      0.6 betweenness Targeted vertex attack
#> 6          0.5         5      0.5 betweenness Targeted vertex attack
#> 7          0.6         3      0.3 betweenness Targeted vertex attack
#> 8          0.7         1      0.1 betweenness Targeted vertex attack
#> 9          0.8         1      0.1 betweenness Targeted vertex attack
#> 10         0.9         1      0.1 betweenness Targeted vertex attack
#> 11         1.0         0      0.0 betweenness Targeted vertex attack