This is a short post on how to quickly calculate the Fiedler vector for large graphs with the igraph package.

#used libraries
library(igraph)    # for network data structures and tools
library(microbenchmark)    # for benchmark results

Fiedler Vector

Fiedler Vector with eigen

My goto approach at the start was using the eigen() function to compute the whole spectrum of the Laplacian Matrix.

g <- sample_gnp(n = 100,p = 0.1,directed = FALSE,loops = FALSE)
M <- laplacian_matrix(g,sparse = FALSE)
spec <- eigen(M)
comps <- sum(round(spec$values,8)==0)
fiedler <- spec$vectors[,comps-1]

While this is easy to implement, it comes with the huge drawback of computing many unnecessary eigenvectors. We just need one, but we calculate all 100 in the example. The bigger the graph, the bigger the overheat from computing all eigenvectors.

# 100 nodes
g <- sample_gnp(n = 100,p = 0.1,directed = FALSE,loops = FALSE)
M <- laplacian_matrix(g,sparse = FALSE)
system.time(eigen(M))
##    user  system elapsed 
##   0.003   0.000   0.004
# 1000 nodes
g <- sample_gnp(n = 1000,p = 0.02,directed = FALSE,loops = FALSE)
M <- laplacian_matrix(g,sparse = FALSE)
system.time(eigen(M))
##    user  system elapsed 
##   1.659   0.011   1.672
# 2500 nodes
g <- sample_gnp(n = 2500,p = 0.01,directed = FALSE,loops = FALSE)
M <- laplacian_matrix(g,sparse = FALSE)
system.time(eigen(M))
##    user  system elapsed 
##  21.153   0.119  21.276

It would thus be useful to have a function that computes only a small number of eigenvectors, which should speed up the calculations considerably.

Fiedler Vector with arpack

What I found after some digging is that igraph provides an interface to the ARPACK library for calculating eigenvectors of sparse matrices via the function arpack().

The function below is an implementation to calculate the Fiedler vector for connected graphs.

fiedler_vector <- function(g){
  M <- laplacian_matrix(g, sparse = TRUE)
  f <- function(x,extra = NULL){
    as.vector(M%*%x)
  }
  fvec <- arpack(f,sym = TRUE,options=list(n = vcount(g),nev = 2,ncv = 8, 
                                           which = "SM",maxiter = 2000))
  return(fvec$vectors[,2])
}

The parameters n and maxiter should be self explanatory. nev specifies the number of eigenvectors to return and which if it should be the largest (“LM”) or smallest (“SM”) one’s. Since the Fiedler vector of connected graphs is the second smallest, we need to return the two smallest eigenvalues.

Let’s see how much we gain.

g <- sample_gnp(n = 2500,p = 0.01,directed = FALSE,loops = FALSE)
system.time(fiedler_vector(g))
##    user  system elapsed 
##   0.793   0.016   0.809

The speed up is enormous (20x) and a nice feature of the arpack() function is that its performance mostly depends on the sparsity of the graph.

g <- sample_gnp(n = 10000,p = 0.005,directed = FALSE,loops = FALSE)
system.time(fiedler_vector(g))
##    user  system elapsed 
##   0.593   0.012   0.605