Operational thresholds of disrupted networks

The operational state of a conductive or convective network consisting of nodes connected by links is discussed in terms of the null space of the underlying graph Laplacian. The number of zero eigenvalues in a disrupted state with a specified percentage of clipped links expresses the population of isolated non-communicating islands. When all links are intact, there is only one zero eigenvalue corresponding to an eigenvector with equal components. When all links are clipped, the number of zero eigenvalues is equal to the number of nodes in the pristine network. Graphs of the number of zero eigenvalues as a function of the percentage of randomly clipped links are presented for square, discoidal, Cartesian, and spherical networks arising from the subdivision of an octahedron or icosahedron. A simple estimate for an operational threshold based on the ratio of nodes to links in the pristine state is proposed and critically assessed for infinite, finite, or closed grids with reference to known percolation thresholds.