Identification of Critical Nodes in Large-Scale Spatial Networks

The notion of network connectivity is used to characterize the robustness and failure tolerance of networks, with high connectivity being a desirable feature. In this paper, we develop a novel gradient flow-based approach to the problem of identifying critical nodes in large-scale networks, with algebraic connectivity (the second smallest eigenvalue of the graph Laplacian) as the chosen metric. Employing a graph-embedding technique, we reduce the class of considered weight-balanced graphs to spatial networks with uniformly distributed nodes and nearest-neighbors communication topologies. Through a continuum approximation, we consider the Laplace operator on a manifold (with the Neumann boundary condition) as the limiting case of the graph Laplacian. We then reduce the critical node set identification problem to that of finding a ball of fixed radius, whose removal minimizes the second (Neumann) eigenvalue of the Laplace operator on the residual domain. This leads us to consider two functional and nested optimization problems. Resorting to the min-max theorem, we first treat the problem of determining the second smallest eigenvalue for a fixed domain by minimizing an energy functional. We then obtain a closed-form expression for a projected gradient flow that converges to the set of eigenfunctions of the Laplace operator and prove that the only locally asymptotically stable critical point is the second eigenfunction. Building on these results, we consider the critical ball identification problem and define novel dynamics to converge asymptotically to these points. Finally, we provide a characterization of the location of critical nodes (for infinitesimally small balls) as those points that belong to the nodal set of the second eigenfunction of the Laplacian operator.