$Q$-system Cluster Algebras, Paths and Total Positivity

In the first part of this paper, we provide a concise review of our method of solution of the $A_r$ $Q$-systems in terms of the partition function of paths on a weighted graph. In the second part, we show that it is possible to modify the graphs and transfer matrices so as to provide an explicit connection to the theory of planar networks introduced in the context of totally positive matrices by Fomin and Zelevinsky. As an illustration of the further generality of our method, we apply it to give a simple solution for the rank 2 affine cluster algebras studied by Caldero and Zelevinsky.