Introduction to the Modeling and Analysis of Complex Systems

416CHAPTER 18. DYNAMICAL NETWORKS III: ANALYSIS OF NETWORK DYNAMICSis represented by the Laplacian operator ∇ 2 , while for the latter, it is represented by theLaplacian matrix L (note again that their signs are opposite for historical misfortune!). Networkmodels allows us to study more complicated, nontrivial spatial structures, but therearen’t any fundamentally different aspects between these two modeling frameworks. Thisis why the same analytical approach works for both.Note that the synchronizability analysis we covered in this section is still quite limitedin its applicability to more complex dynamical network models. It relies on the assumptionthat dynamical nodes are homogeneous and that they are linearly coupled, so the analysiscan’t generalize to the behaviors of heterogeneous dynamical networks with nonlinearcouplings, such as the Kuramoto model discussed in Section 16.2 in which nodes oscillatein different frequencies and their couplings are nonlinear. Analysis of such networks willneed more advanced nonlinear analytical techniques, which is beyond the scope of thistextbook.18.4 Mean-Field Approximation of Discrete-State NetworksAnalyzing the dynamics of discrete-state network models requires a different approach,because the assumption of smooth, continuous state space, on which the linear stabilityanalysis is based on, no longer applies. This difference is similar to the difference betweencontinuous field models and cellular automata (CA). In Section 12.3, we analyzed CAmodels using mean-field approximation. Since CA are just a special case of discretestatedynamical networks, we should be able to apply the same analysis to dynamicalnetworks as well.In fact, mean-field approximation works on dynamical networks almost in the sameway as on CA. But one important issue we need to consider is how to deal with heterogeneoussizes of the neighborhoods. In CA, every cell has the same number of neighbors,so the mean-field approximation is very easy. But this is no longer the case on networksin which nodes can have any number of neighbors. There are a few different ways to dealwith this issue.In what follows, we will work on a simple binary-state example, the Susceptible-Infected-Susceptible (SIS) model, which we discussed in Section 16.2. As you may remember, thestate transition rules of this model are fairly simple: A susceptible node can get infectedby an infected neighbor node with infection probability p i (per infected neighbor), whilean infected node can recover to a susceptible node with recovery probability p r . In theprevious chapter, we used asynchronous updating in simulations of the SIS model, but