Variations in graph energy: A measure for network resilience

There are many models and metrics developed to study the resilience of networks. Eigenvalues are the roots of the characteristic polynomial for a given graph and are mathematically rigorous compared to a statistical measure such as degree distribution. The graph energy is the sum of absolute values of eigenvalues; there is a subtle difference between the adjacency, Laplacian, and normalized Laplacian graph energy calculations. Our primary objective in this paper is to understand what different graph energy mean from a network resilience point of view. We calculate the adjacency, Laplacian, and normalized Laplacian graph energies on four backbone networks under targeted node and link attack scenarios. While adjacency and Laplacian graph energy decrease with node and link attacks, the normalized Laplacian energy increases with link attacks converging to a maximum value equal to the network order. The structural similarities of physical-level topologies is revealed by the close values of adjacency and Laplacian energies.