Introduction to the Modeling and Analysis of Complex Systems

18.2. DIFFUSION ON NETWORKS 40718.2 Diffusion on NetworksMany important dynamical network models can be formulated as a linear dynamical system.The first example is the diffusion equation on a network that we discussed in Chapter16:dcdt= −αLc (18.5)This is a continuous-time version, but you can also write a discrete-time equivalent. Aswe discussed before, L = D − A is the Laplacian matrix of the network. It is a symmetricmatrix in which diagonal components are all non-negative (representing node degrees)while other components are all non-positive. This matrix has some interesting, usefuldynamical properties:A Laplacian matrix of an undirected network has the following properties:1. At least one of its eigenvalues is zero.2. All the other eigenvalues are either zero or positive.3. The number of its zero eigenvalues corresponds to the number of connectedcomponents in the network.4. If the network is connected, the dominant eigenvector is a homogeneity vectorh = (1 1 . . . 1) T a .5. The smallest non-zero eigenvalue is called the spectral gap of the network,which determines how quickly the diffusion takes place on the network.a Even if the network is not connected, you can still take the homogeneity vector as one of the basesof its dominant eigenspace.The first property is easy to show, because Lh = (D − A)h = d − d = 0, where d is thevector made of node degrees. This means that h can be taken as the eigenvector thatcorresponds to eigenvalue 0. The second property comes from the fact that the Laplacianmatrix is positive-semidefinite, because it can be obtained by L = M T M where M is thesigned incidence matrix of the network (not detailed in this textbook).To understand the rest of the properties, we need to consider how to interpret theeigenvalues of the Laplacian matrix. The actual coefficient matrix of the diffusion equationis −αL, and its eigenvalues are {−αλ i }, where {λ i } are L’s eigenvalues. According to