Introduction to the Modeling and Analysis of Complex Systems

336 CHAPTER 16. DYNAMICAL NETWORKS I: MODELINGillustrative examples of the differential equation version of dynamics on networks models,i.e., the diffusion model and the coupled oscillator model. They are both extensivelystudied in network science.Diffusion on a network can be a generalization of spatial diffusion models into nonregular,non-homogeneous spatial topologies. Each node represents a local site wheresome “stuff” can be accumulated, and each symmetric edge represents a channel throughwhich the stuff can be transported, one way or the other, driven by the gradient of its concentration.This can be a useful model of the migration of species between geographicallysemi-isolated habitats, flow of currency between cities across a nation, dissemination oforganizational culture within a firm, and so on. The basic assumption is that the flow ofthe stuff is determined by the difference in its concentration across the edge:dc idt = α ∑ j∈N i(c j − c i ) (16.3)Here c i is the concentration of the stuff on node i, α is the diffusion constant, and N i isthe set of node i’s neighbors. Inside the parentheses (c j − c i ) represents the differencein the concentration between node j and node i across the edge (i, j). If neighbor j hasmore stuff than node i, there is an influx from j to i, causing a positive effect on dc i /dt. Orif neighbor j has less than node i, there is an outflux from i to j, causing a negative effecton dc i /dt. This makes sense.Note that the equation above is a linear dynamical system. So, if we represent theentire list of node states by a state vector c = (c 1 c 2 . . . c n ) T , Eq. (16.3) can be written asdcdt= −αLc, (16.4)where L is what is called a Laplacian matrix of the network, which is defined asL = D − A, (16.5)where A is the adjacency matrix of the network, and D is the degree matrix of the network,i.e., a matrix whose i-th diagonal component is the degree of node i while all othercomponents are 0. An example of those matrices is shown in Fig. 16.4.Wait a minute. We already heard “Laplacian” many times when we discussed PDEs.The Laplacian operator (∇ 2 ) appeared in spatial diffusion equations, while this new Laplacianmatrix thing also appears in a network-based diffusion equation. Are they related?The answer is yes. In fact, they are (almost) identical linear operators in terms of theirrole. Remember that when we discretized Laplacian operators in PDEs to simulate usingCA (Eq. (13.35)), we learned that a discrete version of a Laplacian can be calculated bythe following principle: