# Fast Fiedler Vector Computation

**schochastics**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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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 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
## 0.790 0.016 0.812
```

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
## 0.600 0.004 0.603
```

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**schochastics**.

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