Introduction to the Modeling and Analysis of Complex Systems

342 CHAPTER 16. DYNAMICAL NETWORKS I: MODELINGRepresenting oscillatory behavior naturally requires two or more dynamical variables,as we learned earlier. But purely harmonic oscillation can be reduced to a simple linearmotion with constant velocity, if the behavior is interpreted as a projection of uniform circularmotion and if the state is described in terms of angle θ. This turns the reaction termin Eq. (16.11) into just a constant angular velocity ω i . In the meantime, this representationalso affects the diffusion term, because the difference between two θ’s is no longercomputable by simple arithmetic subtraction like θ j − θ i , because there are infinitely manyθ’s that are equivalent (e.g., 0, 2π, −2π, 4π, −4π, etc.). So, the difference between the twostates in the diffusion term has to be modified to focus only on the actual “phase difference”between the two nodes. You can imagine marching soldiers on a circular track as avisual example of this model (Fig. 16.7). Two soldiers who are far apart in θ may actuallybe close to each other in physical space (i.e., oscillation phase). The model should be setup in such a way that they try to come closer to each other in the physical space, not inthe angular space.To represent this type of phase-based interaction among coupled oscillators, a Japanesephysicist Yoshiki Kuramoto proposed the following very simple, elegant mathematicalmodel in the 1970s [70]:∑dθ idt = ω j∈Ni + αisin(θ j − θ i )(16.12)|N i |The angular difference part was replaced by a sine function of angular difference, whichbecomes positive if j is physically ahead of i, or negative if j is physically behind i onthe circular track (because adding or subtracting 2π inside sin won’t affect the result).These forces that soldier i receives from his or her neighbors will be averaged and usedto determine the adjustment of his or her movement. The parameter α determines thestrength of the couplings among the soldiers. Note that the original Kuramoto model useda fully connected network topology, but here we are considering the same dynamics on anetwork of a nontrivial topology.Let’s do some simulations to see if our networked soldiers can self-organize to marchin sync. We can take the previous code for network diffusion (Code 16.7) and revise itfor this Kuramoto model. There are several changes we need to implement. First, eachnode needs not only its dynamic state (θ i ) but also its static preferred velocity (ω i ), thelatter of which should be initialized so that there are some variations among the nodes.Second, the visualization function should convert the states into some bounded values,since angular position θ i continuously increases toward infinity. We can use sin(θ i ) forthis visualization purpose. Third, the updating rule needs to be revised, of course, toimplement Eq. (16.12).