nPCA {CASCORE}

Normalized Principle Component Analysis.

Description

Normalized Principle Component Analysis (nPCA), also known as spectral clustering on the graph Laplacian, is a classical spectral clustering method that applies k-means on the first K leading (unit-norm) eigenvectors of the degree-corrected normalized graph laplacian.

Usage

nPCA(Adj, K, tau = NULL, itermax = 100, startn = 10)

Arguments

Adj

A 0/1 adjacency matrix.

K

A positive integer, indicating the number of underlying communities in graph Adj.

tau

An optional regularization parameter for suitable degree normalization. The default value is the average degree of graph Adj.

itermax

k-means parameter, indicating the maximum number of iterations allowed. The default value is 100.

startn

k-means parameter. If centers is a number, how many random sets should be chosen? The default value is 10.

Value

estall

A lavel vector.

References

Chung, F. R., & Graham, F. C. (1997). Spectral graph theory (Vol. 92). American Mathematical Soc..

Examples

# Simulate the Network
n = 10; K = 2;
theta = 0.4 + (0.45-0.05)*(seq(1:n)/n)^2; Theta = diag(theta);
P  = matrix(c(0.8, 0.2, 0.2, 0.8), byrow = TRUE, nrow = K)
set.seed(2022)
l = sample(1:K, n, replace=TRUE); # node labels
Pi = matrix(0, n, K) # label matrix
for (k in 1:K){
  Pi[l == k, k] = 1
}
Omega = Theta %*% Pi %*% P %*% t(Pi) %*% Theta;
Adj = matrix(runif(n*n, 0, 1), nrow = n);
Adj = Omega - Adj;
Adj = 1*(Adj >= 0)
diag(Adj) = 0
Adj[lower.tri(Adj)] = t(Adj)[lower.tri(Adj)]
nPCA(Adj, 2)