create an incidence matrix with restrictions in r (i.graph) - r

Question

I would like to create a (N*M)-Incidence Matrix for a bipartite graph (N=M=200).
However, the following restrictions have to be considered:
Each column i ( 1 , ... , 200 ) has a column sum of g = 10
each row has a Row sum of h = 10
no multiedges (The values in the incidence Matrix only take on the values [0:1]
So far I have
M <- 200; # number of rows
N <- 200; # number of colums
g <- 10
I <- matrix(sample(0:1, M*N, repl=T, prob= c(1-g/N,g/N)), M, N);
Does anybody has a solution?

Answer 1

Here's one way to do what you want. First the algorithm idea, then its implementation in R.
Two step Algorithm Idea
You want a matrix of 0's and 1's, with each row adding up to be 10, and each column adding up to be 10.
Step 1: First,create a trivial solution as follows:
The first 10 rows have 1's for the first 10 elements, then 190 zeros.
The second set of ten rows have 1's from the 11th to the 20th element and so on.
In other words, a feasible solution is to have a 200x200 matrix of all 0's, with dense matrices of 10x10 1's embedded diagonally, 20 times.
Step 2: Shuffle entire rows and entire columns.
In this shuffle, the rowSum and columnSums are maintained.
Implementation in R
I use a smaller matrix of 16x16 to demonstrate. In this case, let's say we want each row and each column to add up to 4. (This colsum has to be integer divisible of the larger square matrix dimension.)
n <- 4 #size of the smaller square
i <- c(1,1,1,1) #dense matrix of 1's
z <- c(0,0,0,0) #dense matrix of 0's
#create a feasible solution to start with:
m <- matrix(c(rep(c(i,z,z,z),n),
rep(c(z,i,z,z),n),
rep(c(z,z,i,z),n),
rep(c(z,z,z,i),n)), 16,16)
#shuffle (Run the two lines following as many times as you like)
m <- m[sample(16), ] #shuffle rows
m <- m[ ,sample(16)] #shuffle columns
#verify that the sum conditions are not violated
colSums(m); rowSums(m)
#solution
print(m)
Hope that helps you move forward with your bipartite igraph.