A Spectrum Analysis Approach for the Median Problem in Modern Network Environments

Service facility location is a challenging problem in the emerging dynamic modern network environments of increased number of users, devices, and supported applications. Under this perspective, the median problem is revisited here from a spectrum analysis point of view. More specifically, a median matrix depending on the network topology and the traffic requirements of the network nodes is introduced and studied in this paper. An analytical investigation of its spectral properties reveals that the minimum elements of its principal eigenvector correspond to the solution of the median problem. The smallest eigenvalue is also studied through simulations to demonstrate the effectiveness of the spectrum analysis approach introduced in this paper. Moreover, its relation to a metric related to the robustness of the median solution is investigated, thus shedding further light on the particulars of facility location problems.