Optimization Problems on Complex Networks: Method of Gravitational Potentials

We study the problem of building an efficient service network on a complex map of potential customers' locations and develop an iterative algorithm producing nearly optimal service center configurations. We propose a probabilistic model of customers' behavior when they choose service centers. A service network of a given size is optimized according to several criteria that ensure an even distribution of the load on the network, predictability of the number of requests to each center, and convenient location of service centers for customers. This optimization problem is NP-hard, but we propose an efficient greedy algorithm based on the spectral properties of the graph Laplacian. The concept of gravitational potential for a particular service network configuration is introduced. The gravitational potential allows us to calculate customer preferences and metrics that measure network efficiency. In addition, the gravitational potential is the basis of our greedy algorithm, that rebuilds the network at each iteration step to achieve better efficiency. Our research was inspired by a recent work of Steinberger [1] on the graph's shortest path problem.