Exact analytic solution to nonlinear dynamic system of equations for information propagation in complex networks
Abstract We present the exact analytic solution to the nonlinear dynamic system describing the information propagation process in complex networks. The method is to switch to the dynamic system of equations for the new variables defined according to the nonlinear terms appearing in the original dynamic system, and then, after combining relevant consistent condition of solution to reduce the dynamic system of equations for the newly defined variables to the knownly solvable nonlinear Bernoulli differential equation. Numerical comparisons between the purely numerical solution and the exact analytic solution confirm the crosscheck and mutual proof for both of the solutions. The presented exact analytic solution, which does not restrict the coefficients in the original dynamic system to merely constants, is thus suitable for the study of information propagation in networks with evolving structure and changing properties.
Abstract In this work we study conditions for guaranteeing the nonnegativity of a discrete-time singular control system. A first approach can be found in the literature for general systems, using the whole coefficient matrices. Also, the particular case of matrices of index 1 has been treated by using a block decomposition and the group-projector of the matrix that gives the singularity to the system. In order to complete this study, an analysis of the nonnegativity of a singular control system for matrices having arbitrary index is done by means of the core–nilpotent decomposition. This technique allows us to reduce the size of the original matrices, improving the results where the whole coefficients are involved.