## Random graphs with fixed numbers of neighbours In connection with Le Monde puzzle #46, I eventually managed to write an R program that generates graphs with a given number n of nodes and a given number k of edges leaving each of those nodes. (My early attempt was simply too myopic to achieve any level of success when n was larger than 10!) Here is the core of the R code:

```A=42 #number of nodes
L=13 #number of edges
ApL=A+L

if ((A*L)%%2==1){
print("impossible graph")
}else{
con=matrix(0,A,A)
diag(con)=A      #eliminate self-connection
suma=apply(con,1,sum)-A

while (min(suma)<L){

if (sum(suma<L)==1){ #bad news: no correspondence!
#go back:
con=aclrtr(con,L)
diag(con)=A
suma=apply(con,1,sum)-A

}else{

j=sample((1:A)[suma<L],1)
slots=(1:A)[con[j,]==0]  #remaining connections
if (length(slots)==1){
vali=slots
if (sum(con[vali,]>ApL-1)) vali=NULL
}else{
vali=slots[apply(con[slots,],1,sum)<ApL]
}

if (length(vali)==0){

con=aclrtr(con,L)
diag(con)=A
suma=apply(con,1,sum)-A

}else{

if (length(vali)==1){
k=vali
}else{
k=sample(slots[apply(con[slots,],1,sum)<ApL],1)
}
con[k,j]=con[j,k]=1
suma=apply(con,1,sum)-A
}}}}
```

and it uses a sort of annealed backward step to avoid simulating a complete new collection of neighbours when reaching culs-de-sac….

```aclrtr=function(con,L){
#removes a random number of links among the nodes with L links

A=dim(con)
ApL=A+L
while (max(apply(con,1,sum))==ApL){

don=sample(1:(L-1),1)
if (sum(apply(con,1,sum)==ApL)==1){
i=(1:A)[apply(con,1,sum)==ApL]
}else{
i=sample((1:A)[apply(con,1,sum)==ApL],1)
}
off=sample((1:A)[con[i,]==1],don)
con[i,off]=0
con[off,i]=0
}
con
}
```

There is nothing fancy or optimised about this code so I figure there are much better versions to be found elsewhere…

Ps-As noticed previously, sample does not work on a set of length one, which is a bug in my opinion…. Instead, sample(4.5,1) returns a random permutation of (1,2,3,4).

> sample(4.5)
 4 3 1 2

Pps-Following a suggestion by Pierre, I used the R command hcl for a graduation of the colours on the nodes, but besides mimicking the examples in the help documentation, I have trouble producing colours I want (like yellows…) Maybe I should read the whole vignette by the (well-recognised) authors as the abilities to play with colour gradients sound awesome!

### 6 Responses to “Random graphs with fixed numbers of neighbours”

1. Le Monde puzzle  « Xi'an's Og Says:

[…] Le Monde puzzle  This week(end), the Le Monde puzzle can be (re)written as follows (even though it is presented as a graph problem): […]

2. Ajay Ohri Says:

Hmm. How about putting A and L in a double loop – and add in animation using a gif. What do you think?

Cool art!

Ajay Ohri

• xi'an Says:

Thanks, Ajay! My daughter also suggested to move this post to the “Art brut” posts (she is kindly making fun of!. How would you produce this animation in R?

3. Paul Says:

Can’t the igraph package do this for you?

• xi'an Says:

As mentioned on the main post, there must be dozens of good packages doing this. Including igraph indeed. Thanks for the reference. (I did not want to start building a competitor to those packages, just to solve a small puzzle suggested by the original problem.) Again, thanks for pointing out the package. It looks great and I hope I can find there a way to improve display from my circle representation (using the Fruchterman-Reingold layout algorithm)… Actually, igraph showed me that the problem of generating a graph with a fixed number of vertices and of edges is called the Erdös-Rényi problem.

4. pierrejacob Says:

So that explains the question about hcl! Great plots!

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