matlab - Generating a linear system for popularity calculation -

this half mathematical half computer science question. working on matlab on mathematical problem. want generate linear system calculation of popularity of people (a vector x) in generic social network, given simple matrix of friendship f. if user 1 , user 2 friends, f(1,2)=1 f(2,1)=1 (the friendship bidirectional). transform problem in linear system need popularity value of person (x(i)) , popularity of friends (∀x(j) | f(i,j)=1 , j≠i) linearly dependent. wrote simple formula

x(j) = sum(x(i) x n(i) x f(i,j))  ∀i≠j 

where n(i) number of friends of i.

for size(x)=3 system this

-x(1) + f(1,2) n(2) x(2) + f(1,3) n(3) = 0  f(2,1) n(1) x(1) - x(2) + f(1,3) n(3) x(3) = 0  f(3,1) n(1) x(1) + f(1,2) n(2) x(2) - x(3) = 0  

with specific formula there 2 problems:

1. with no constant term, solution found x(i)=0 ∀i
2. i need positive values of popularity, approach gave negative (and correct!) results.

so need formula doesn't have 2 problems. formula can chosen arbitrarily because problem totally defined me. must take account of friendship relations , number of friends in linear combination (necessary linear system), positive. here code of generation , resolution of system

n=10; f=triu(randi([0,1],n)); f=f-diag(diag(f)); f=f+f'; n=(sum(f))'; a=ones(n);  i=1:n     a(:,i)=diag(n)*f(:,i); end a=a-eye(n); b=zeros(n,1); x=a\b