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. Finally, we provide a characterization of the location of critical nodes (for infinitesimally-small balls) as those points which belong to the nodal set of the second eigenfunction of the Laplacian operator.