Introduction to the Modeling and Analysis of Complex Systems

414CHAPTER 18. DYNAMICAL NETWORKS III: ANALYSIS OF NETWORK DYNAMICS0.4685 < β/α = 0.5. Therefore, it is predicted that the nodes won’t get synchronized.And, indeed, the simulation result confirms this prediction (Fig. 18.2(a)), where the nodesinitially came close to each other in their phases but, as their oscillation speed becamefaster and faster, they eventually got dispersed again and the network failed to achievesynchronization. However, if beta is slightly lowered to 0.9 so that λ 2 = 0.4685 > β/α =0.45, the synchronized state becomes stable, which is confirmed again in the numericalsimulation (Fig. 18.2(b)). It is interesting that such a slight change in the parameter valuecan cause a major difference in the global dynamics of the network. Also, it is rathersurprising that the linear stability analysis can predict this shift so precisely. Mathematicalanalysis rocks!Exercise 18.3 Randomize the topology of the Karate Club graph and measureits spectral gap. Analytically determine the synchronizability of the acceleratingoscillators model discussed above with α = 2, β = 1 on the randomized network.Then confirm your prediction by numerical simulations. You can also try severalother network topologies.Exercise 18.4 The following is a model of coupled harmonic oscillators wherecomplex node states are used to represent harmonic oscillation in a single-variabledifferential equation:dx idt = iωx i + α ∑ (xγj − )xγ ij∈N i(18.18)Here i is used to denote the imaginary unit to avoid confusion with node indexi. Since the states are complex, you will need to use Re(·) on both sides of theinequalities (18.14) and (18.15) to determine the stability.Analyze the synchnonizability of this model on the Karate Club graph, and obtainthe condition for synchronization regarding the output exponent γ. Then confirmyour prediction by numerical simulations.You may have noticed the synchronizability analysis discussed above is somewhatsimilar to the stability analysis of the continuous field models discussed in Chapter 14.Indeed, they are essentially the same analytical technique (although we didn’t cover stabilityanalysis of dynamic trajectories back then). The only difference is whether the spaceis represented as a continuous field or as a discrete network. For the former, the diffusion