How to find final stored amount in each sink in a flow network?

Question

I have a flow network with single source and multiple sinks. If I flow 1 gallon of water from the source then all water split at nodes and finally store in different sinks. Here each edge weight c(u,v) represents the water splitting ratio from u. How to find final stored amount of water in each sink?

An example with manual calculation:

Answer 1

This is an example of an absorbing Markov chain.

Let Q[j,i] be the proportion of flow from the j^th non-sink node into the i^th non-sink node.

Let R[j,i] be the proportion of flow from the j^th non-sink node into the i^th sink node.

Then let matrix N=inv(I-Q), and matrix B=N*R.

The final amount of flow in the i^th sink node is given by B[0,i].

Python example for your first case:

import numpy as np

# Rows are proportion of flow out of each non-sink vertex
Q = np.zeros((3,3))
Q[0,:] = 0,0.3,0.7 # Flow out of S
Q[1,:] = 0,0,0     # Flow out of a
Q[2,:] = 0,0.5,0   # Flow out of b
# Rows are flow from non-sink to sink nodes
R = np.zeros((3,2))
R[1,:] = 0.1,0.9
R[2,:] = 0,0.5 
N = np.matrix(np.identity(len(Q)) - Q).I
B = N*R
print B[0,:]

prints

[[ 0.065  0.935]]

Explanation

Once we have the matrices set up, we can simulate one time step of flow by starting with a row vector v containing [1,0,0] (representing the one gallon at the start) and an empty row vector e (representing the amount of water in the sinks).

In each time step we compute:

e += v * R # This works out the amount of flow that has reached the end
v = v * Q # This works out the amount of flow that is still in the network

If we simulate this for long enough we will end up with v approaching 0, and e approaching the answer.

However we can write the value for e as:

e = v*R + v*Q*R + v*Q*Q*R + v*Q*Q*Q*R +...
  = v * (I + Q + Q*Q + Q*Q*Q + ...) * R
  = v * inv(I-Q) * R
  = v * N * R
  = v * B
  = [1,0,0] * B
  = first row of B