In a resistor network, is there any relation between the shortest path and the maximum electric current path? Consider a shortest path problem between the source $s$ and sink $t$ in an undirected weighted graph. There's a well known algorithm such as Dijkstra's algorithm that solves this problem.

Naturally, this graph can be considered as a resistive network where each edge $e$ with distance (or a cost) $d_{e}$ corresponds to a resistor with resistance $d_{e}$ or conductance $1/d_{e}$. From a physical point of view, assume that we send one unit of current from s to t, say by attaching $s$ to a current source and $t$ to ground. It will induce a certain electric flow in $G$ and this flow is unique. And the approximated solution (or the approximate electric flow) can be obtained in time $\tilde{O}(m)$ where $m$ is the number of edges in the graph.

My question: what is the relation between the shortest path in a graph and the path in an electric network where the max electric current flows? Is it true that two paths are always the same when there exists a unique shortest path from $s$ to $t$?

The intuition: The electric current tends to flow through the path with the least resistance, and the least resistance between $s$ to $t$ may be interpreted as the shortest path between $s$ to $t$. 



updated

8--R--7--R--+
|     |     |
|     R     |
|     |     |
+--R--6--R--+
|     |     |
|     R     |
|     |     |
+--R--5--R--+
|     |     |
|     R     |
|     |     |
+--R--3--R--4
|     |     |
|     R     |
|     |     |
+--R--2     R
|           |
|           |
|           |
+--R--1--R--0


@MarzioDeBiasi: Thank you for the picture and actual current computation. But ironically, your answer only fortifies my conjecture about the shortest path and max current path. (I apologize for not explaining my conjecture clearly if that causes some confusion.)

As you pointed out, we have the current of 33.9 mA from 4 to 0, but that's due to the superimposition of many currents. For example, the current flows 8 - 2 - 3 - 4 - 0, 8 - 3 - 4 - 0, 8 - 5 - 6 - 7 - 4 - 0 and 8 - 7 - 4 - 0 contribute to the current flow from 4 to 0. And none of them are larger than the curent of 25mA on the path of 8 - 1 - 0. On the other hand, there's only one path for the flow 8 - 1 - 0 and it sends the largest amoun of current in the circuit.
Just an extended comment to include the picture: if the following (random) resistor network is valid then the max current flow 33.9mA is not on the shortest path from $s$ (+5V) to $t$ (GND).