Abstract: A communication network having a multiplicity of nodes provides efficient exchange of messages between the nodes. The messages may be originated and received by the computers of a parallel computer system, the processors and associated memories of which are connected to each node. Each node includes a routing system which results in efficient system performance for the parallel computer system associated with the nodes. The messages have control information (a message tag) to which the routing system at each node is responsive. The tag contains data identifying the destination node of the message in the network and prioritization data which, when the message is generated, is initialized to have a value (weight) corresponding to the length (number of links) of the minimal path from the source node where the message originates to the destination node of the message. The routing system utilizes the weights to establish message priority.