I have n pairwise joint cycles in a weighted directed graph. For each edge, we have that:

(where w_ij is the weight of the (i,j) edge) and for each vertex i,

(i.e. the sum of the weights of the outgoing edges is less/equal 1).

If we define the weight of each cycle as the product of the weights of its edges, how can I prove that the sum of the weights of the n cycles is less/equal 1 ?

thanks a lot