How do I solve the "friendship" problem, i.e., transform a 2D friendship/logical array into n dimensions without using loops?

Hi All,

I'm going to call this a "friendship" problem. I want to find friendship groups. I can set up a 2D logical array with each "person" reflected as both a row and column. For example, person 1 is reflected in both row 1 and column 1, person 2 in row 2 and column 2, etc. If the two people are friends, then cells at the intersections of their respective rows/columns are set to true, otherwise the values are false. In this example, people cannot be friends with themselves. Here's an example array:

n = 10; % number of people
friends = false(n, n); % set up logical array
friends(1,2) = true; % 1 and 2 are friends
friends(2,1) = true; % 1 and 2 are friends
friends(3,4) = true; % 3 and 4 are friends
friends(4,3) = true; % 3 and 4 are friends
friends(3,5) = true; % 3 and 5 are friends
friends(5,3) = true; % 3 and 5 are friends
friends(4,5) = true; % 4 and 5 are friends
friends(5,4) = true; % 4 and 5 are friends

So, 1 and 2 are only friends with each other, whereas 3, 4 and 5 form a group of 3 friends. We can search through the logical array to find the group by storing the results in the next dimension:

friends3 = false(n,n,n); % set up 3D logical array

for F1 = 1:n

      for F2 = 1:n
          for F3 = 1:n
              if  friends2(F1,F2) == true && ... % 1 is friends with 2
                  friends2(F1,F3) == true && ... % 1 is friends with 3
                  friends2(F2,F3) == true % 2 is friends with 3

                  friends3(F1,F2,F3) = true;                
              end

          end
      end
  end

[F1, F2, F3] = ind2sub( size(friends3), find(friends3) ); % find group of 3 friends

group3 = [F1 F2 F3]; % store the results

% ----- deal with repeats to return the group only once
for G = 1:size(group3,1)
    group3(R,:) = sort(group3(G,:));
end

group3 = unique(group3, 'rows')

We can look for groups of 4 friends using the same method:

friends4 = false(n,n,n,n); % set up 3D logical array to look for groups of 4

for F1 = 1:n   
    for F2 = 1:n
        for F3 = 1:n
            for F4 = 1:n
                if  friends2(F1,F2) == true && ... % 1 is friends with 2
                    friends2(F1,F3) == true && ... % 1 is friends with 3
                    friends2(F2,F3) == true && ... % 2 is friends with 3
                    friends2(F1,F4) == true && ... % 1 is friends with 4
                    friends2(F2,F4) == true && ... % 2 is friends with 4
                    friends2(F3,F4) == true % 3 is friends with 4
                    friends4(F1,F2,F3,F4) = true;                
                end
            end
        end
    end
end

sum(sum(sum(sum(friends4)))) % there are no matches in this case!

As should be clear, this method becomes extremely inefficient when searching for large groups of friends. Is a better way to extend by 2D friendship matrix into n dimensions without using loops?

Thanks!

D.