Introduction to the Modeling and Analysis of Complex Systems

340 CHAPTER 16. DYNAMICAL NETWORKS I: MODELING• barbell graph• ring-shaped graph (i.e., degree-2 regular graph)Continuous state/time models (2): Coupled oscillator model Now that we have adiffusion model on a network, we can naturally extend it to reaction-diffusion dynamics aswell, just like we did with PDEs. Its mathematical formulation is quite straightforward; youjust need to add a local reaction term to Eq. (16.3), to obtaindc idt = R i(c i ) + α ∑ j∈N i(c j − c i ). (16.11)You can throw in any local dynamics to the reaction term R i . If each node takes vectorvaluedstates (like PDE-based reaction-diffusion systems), then you may also have differentdiffusion constants (α 1 , α 2 , . . .) that correspond to multiple dimensions of the statespace. Some researchers even consider different network topologies for different dimensionsof the state space. Such networks made of superposed network topologies arecalled multiplex networks, which is a very hot topic actively studied right now (as of 2015),but we don’t cover it in this textbook.For simplicity, here we limit our consideration to scalar-valued node states only. Evenwith scalar-valued states, there are some very interesting dynamical network models. Aclassic example is coupled oscillators. Assume you have a bunch of oscillators, each ofwhich tends to oscillate at its own inherent frequency in isolation. This inherent frequencyis slightly different from oscillator to oscillator. But when connected together in a certainnetwork topology, the oscillators begin to influence each other’s oscillation phases. Akey question that can be answered using this model is: When and how do these oscillatorssynchronize? This problem about synchronization of the collective is an interestingproblem that arises in many areas of complex systems [69]: firing patterns of spatiallydistributed fireflies, excitation patterns of neurons, and behavior of traders in financialmarkets. In each of those systems, individual behaviors naturally have some inherentvariations. Yet if the connections among them are strong enough and meet certain conditions,those individuals begin to orchestrate their behaviors and may show a globallysynchronized behavior (Fig.16.6), which may be good or bad, depending on the context.This problem can be studied using network models. In fact, addressing this problem waspart of the original motivation for Duncan Watts’ and Steven Strogatz’s “small-world” paperpublished in the late 1990s [56], one of the landmark papers that helped create thefield of network science.