We consider a network of processors that cooperate to solve a problem by exchanging messages across communication channels in a non-synchronized manner. Such a network is naturally modelled as a connected, undirected graph with processors represented as vertices and communication channels represented as edges. The measure of the communication cost of an asynchronous distributed algorithm for such a network typically is the total number of messages sent. This measure assumes that the cost of sending a message along any link is equal to one. In a network of asynchronous processors, the cost to send a message can differ significantly from one communication link to another. In this presentation, we assume that associated with each link is a positive {\em weight\/} representing the cost of sending one message along the link and the cost of an algorithm executed on a {\em weighted\/} network is the sum of the costs of all messages sent during its execution. This, so called ``cost-sensitive" analysis, has been introduced by Awerbuch, Baratz and Peleg in 1990. We present optimal distributed algorithms for Leader Election on a ring topology and Minimum Cost Spanning tree. As we show, both algorithms are optimal with respect to this cost measure. This is joint work with Lisa Higham, University of Calgary.