Introduction to the Modeling and Analysis of Complex Systems

410CHAPTER 18. DYNAMICAL NETWORKS III: ANALYSIS OF NETWORK DYNAMICSHere x i is the state of node i, R is the local reaction term that produces the inherentdynamical behavior of individual nodes, and N i is the neighborhood of node i. We assumethat R is identical for all nodes, and it produces a particular trajectory x s (t) if there is nointeraction with other nodes. Namely, x s (t) is given as the solution of the differentialequation dx/dt = R(x). H is called the output function that homogeneously applies to allnodes. The output function is used to generalize interaction and diffusion among nodes;instead of assuming that the node states themselves are directly visible to others, weassume that a certain aspect of node states (represented by H(x)) is visible and diffusingto other nodes.Eq. (18.6) can be further simplified by using the Laplacian matrix, as follows:⎛dx idt = R(x i) − αL ⎜⎝H(x 1 )H(x 2 ).H(x n )⎞⎟⎠(18.7)Now we want to study whether this network of coupled dynamical nodes can synchronizeor not. Synchronization is possible if and only if the trajectory x i (t) = x s (t) for all i isstable. This is a new concept, i.e., to study the stability of a dynamic trajectory, not of astatic equilibrium state. But we can still adopt the same basic procedure of linear stabilityanalysis: represent the system’s state as the sum of the target state and a small perturbation,and then check if the perturbation grows or shrinks over time. Here we representthe state of each node as follows:x i (t) = x s (t) + ∆x i (t) (18.8)By plugging this new expression into Eq. (18.7), we obtain⎛⎞H(x s + ∆x 1 )H(x s + ∆x 2 )⎟d(x s + ∆x i )dt= R(x s + ∆x i ) − αL ⎜⎝.H(x s + ∆x n )Since ∆x i are very small, we can linearly approximate R and H as follows:⎛⎞H(x s ) + H ′ (x s )∆x 1dx sdt + d∆x i= R(x s ) + R ′ H(x s ) + H ′ (x s )∆x 2(x s )∆x i − αL ⎜⎟dt⎝ . ⎠H(x s ) + H ′ (x s )∆x n⎟⎠(18.9)(18.10)