An analytical solution to the multicommodity network flow problem with weighted random routing

Abstract We derive an analytical expression for the mean load at each node of arbitrary undirected graph non-uniform multicommodity flow problem under weighted random routing. show node, net its demand and normalized by (weighted) degree, is a constant equal to trace product two matrices: Laplacian matrix generalized inverse Laplacian. For case uniform demand, this reduces sum inverses non-zero eigenvalues note that such closed-form network capacity general has not been reported before, even though routing practical procedure, it enables us (tight) upper bound more standard policies. Using new expression, we compute sample matrices some prototypical networks, including (one unit between all pairs) broadcast source other as destination), finally estimates in asymptotic cases.