Detect Core-Periphery Structure

Identifies core-periphery structure in a network using either continuous (Borgatti-Everett) or discrete methods. Core nodes are densely interconnected, while periphery nodes connect primarily to the core.

Usage

core_periphery(
  x,
  method = c("continuous", "discrete"),
  directed = NULL,
  iter = 100,
  digits = NULL,
  ...
)

Arguments

x: Network input: matrix, igraph, network, cograph_network, or tna object
method: Character string; either "continuous" (default, Borgatti-Everett model) or "discrete" (binary core/periphery assignment).
directed: Logical or NULL. If NULL (default), auto-detect from matrix symmetry. Set TRUE to force directed, FALSE to force undirected.
iter: Integer; maximum number of iterations for the continuous algorithm. Default 100.
digits: Integer or NULL. Round numeric outputs 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 with class "cograph_core_periphery", one row per node, and columns:

node: Node label.
role: Character: "core" or "periphery".
coreness: Numeric continuous coreness score, rescaled to \([0, 1]\). Reported for both methods.

The attributes "fitness", "core_density", "periphery_density" and "network" (the original input) carry the remaining results.

Details

Continuous method (Borgatti-Everett): Seeks a coreness vector c (rescaled to the 0-1 range) whose ideal rank-1 pattern matrix (the outer product of the vector with itself) correlates as highly as possible with the adjacency matrix. The vector is approximated by initializing from the dominant eigenvector of the adjacency matrix and refining it by power iteration until convergence or iter steps; the achieved correlation is reported as the "fitness" attribute rather than being optimized directly.

Discrete method: Produces a binary core / periphery assignment. Starts from the continuous solution thresholded at the median, then greedily flips the single node assignment that most improves fitness until no flip improves it. The discrete fitness being maximized is density(core) - density(periphery); the "fitness" attribute reported for method = "discrete" is the correlation between the adjacency matrix and the ideal block pattern of that assignment.

References

Borgatti, S.P. & Everett, M.G. (2000). Models of core/periphery structures. Social Networks, 21(4), 375-395. doi:10.1016/S0378-8733(99)00019-2

Examples

# Core-periphery in a simple network
adj <- matrix(c(
  0, 1, 1, 1, 0,
  1, 0, 1, 1, 0,
  1, 1, 0, 1, 1,
  1, 1, 1, 0, 1,
  0, 0, 1, 1, 0
), 5, 5)
rownames(adj) <- colnames(adj) <- LETTERS[1:5]
cp <- cograph::core_periphery(adj)
cp
#> Core-Periphery | Core: 4  Periphery: 1  Fitness: 0.569
#> Core density: 1.000 | Periphery density: 0.000
#> 
#>  node      role  coreness
#>     A      core 0.6504481
#>     B      core 0.6504481
#>     C      core 1.0000000
#>     D      core 1.0000000
#>     E periphery 0.0000000

# Discrete assignment
cp_disc <- cograph::core_periphery(adj, method = "discrete")
cp_disc
#> Core-Periphery | Core: 4  Periphery: 1  Fitness: 0.612
#> Core density: 1.000 | Periphery density: 0.000
#> 
#>  node      role  coreness
#>     A      core 0.6504481
#>     B      core 0.6504481
#>     C      core 1.0000000
#>     D      core 1.0000000
#>     E periphery 0.0000000