Local Efficiency — network_local_efficiency • cograph

Computes the average local efficiency across all nodes, delegating to igraph::average_local_efficiency(). igraph removes the node and measures the distances between its neighbors through the rest of the network, so the value can exceed the one Latora & Marchiori (2001) define, which restricts those distances to the subgraph induced on the neighbors. centrality(x, measures = "local_efficiency") reports the induced-subgraph form, matching networkx, brainGraph and the Brain Connectivity Toolbox. Both measure fault tolerance and local integration; the two agree whenever the neighbors have no detour available.

Usage

network_local_efficiency(
  x,
  weights = NULL,
  invert_weights = NULL,
  alpha = 1,
  ...
)

Arguments

x: Network input: matrix, igraph, network, cograph_network, or tna object
weights: Edge weights (NULL for unweighted). Set to NA to ignore existing weights.
invert_weights: Logical or NULL. Invert weights so higher weights = shorter paths? Default NULL which auto-detects: TRUE for tna objects, FALSE otherwise (matching igraph/sna). Set TRUE for strength/frequency weights (qgraph style).
alpha: Numeric. Exponent for weight inversion. Default 1.
...: Passed to to_igraph, whose only other argument is directed; anything else raises an "unused argument" error.

Value

Numeric average local efficiency. For unweighted simple graphs this is in \([0, 1]\); weighted graphs can exceed 1 when edge distances are below 1.

Examples

# Complete graph: removing any node leaves complete subgraph, so local efficiency = 1
k5 <- matrix(1, 5, 5); diag(k5) <- 0
network_local_efficiency(k5)  # 1
#> [1] 1

# Star: neighbors not connected to each other
star <- matrix(c(0,1,1,1,1, 1,0,0,0,0, 1,0,0,0,0, 1,0,0,0,0, 1,0,0,0,0), 5, 5)
network_local_efficiency(star)  # 0
#> [1] 0

# Per-node values under the Latora definition
centrality(star, measures = "local_efficiency")
#>   node local_efficiency_all
#> 1    1                    0
#> 2    2                    0
#> 3    3                    0
#> 4    4                    0
#> 5    5                    0