Numerical Assessment of the Percolation Threshold Using Complement Networks. Models of percolation processes on networks currently assume locally treelike structures at low densities, and are derived exactly only in the thermodynamic limit. Finite size effects and the presence of short loops in real systems however cause a deviation between the empirical percolation threshold (p_c) and its modelpredicted value (pi _c). Here we show the existence of an empirical linear relation between (p_c) and (pi _c) across a large number of real and model networks. Such a putatively universal relation can then be used to correct the estimated value of (pi _c). We further show how to obtain a more precise relation using the concept of the complement graph, by investigating on the connection between the percolation threshold of a network, (p_c), and that of its complement, (barp_c).

The controllability Gramian of lattice graphs Abstract   The control of networked dynamical systems continues to receive much attention across the engineering and scientific research fields. Here, we investigate the control of lattice systems which provides analytical insight into the relationship between network structure and controllability. First we derive the exact expression for the individual elements of the controllability Gramian of infinite lattice systems. Second, we focus on nearest neighbor lattices for which the distance between nodes appears in the expression for the controllability Gramian. We show that common control energy metrics scale exponentially with respect to the maximum of all the distances between driver nodes and target nodes.