Introduction to the Modeling and Analysis of Complex Systems

406CHAPTER 18. DYNAMICAL NETWORKS III: ANALYSIS OF NETWORK DYNAMICSdx= x 1 − x − xy + xvdtdy= −y + xy − yzdtxydz= −z + yz + udtdu= −u + xu 1 − udtvzdvdt = 1 − vuFigure 18.1: Schematic illustration of how a multidimensional dynamical system canbe viewed as a dynamical network. Left: Actual dynamical equations. Right: Interdependentrelationships among variables, represented as a dynamical network.techniques we discussed before—finding equilibrium points, linearizing dynamics aroundan equilibrium point, analyzing the stability of the system’s state using eigenvalues of aJacobian matrix, etc.—will apply to dynamical network models without any modification.Analysis of dynamical networks is easiest when the model is linear, i.e.,orx t = Ax t−1 , (18.3)dxdt= Ax. (18.4)If this is the case, all you need to do is to find eigenvalues of the coefficient matrix A,identify the dominant eigenvalue(s) λ d (with the largest absolute value for discrete-timecases, or the largest real part for continuous-time cases), and then determine the stabilityof the system’s state around the origin by comparing |λ d | with 1 for discrete-time cases,or Re(λ d ) with 0 for continuous-time cases. The dominant eigenvector(s) that correspondto λ d also tell us the asymptotic state of the network. While this methodology doesn’tapply to other more general nonlinear network models, it is still quite useful, becausemany important network dynamics can be written as linear models. One such example isdiffusion, which we will discuss in the following section in more detail.