Motivated by broad applications in various fields of engineering, we study network resource allocation problems where the goal is to optimally allocate a fixed portion of resources over a network of nodes. In these problems, due to the large scale of the network and complicated interconnections between nodes, any solution must be implemented in parallel and based only on local data resulting in a need for distributed algorithms. In this paper, we propose a distributed Lagrangian method, which requires only local computation and communication. Our focus is to understand the performance of this algorithm on the underlying network topology. Specifically, we obtain an upper bound on the rate of convergence of the algorithm as a function of the size and the topology of the underlying network. The effectiveness and applicability of the proposed method is demonstrated by its use in solving the important economic dispatch problem in power systems, specifically on the benchmark IEEE-14 and IEEE-118 bus test systems.