Introduction to the Modeling and Analysis of Complex Systems

408CHAPTER 18. DYNAMICAL NETWORKS III: ANALYSIS OF NETWORK DYNAMICSthe first two properties discussed above, the coefficient matrix of the equation has atleast one eigenvalue 0, and all the eigenvalues are 0 or negative. This means that thezero eigenvalue is actually the dominant one in this case, and its corresponding dominanteigenvector (h, or eigenspace, if the network is not connected) will tell us the asymptoticstate of the network.Let’s review the other properties. The third property arises intuitively because, ifthe network is made of multiple connected components, each of those components behavesas a separate network and shows diffusion independently, converging to a differentasymptotic value, and therefore, the asymptotic state of the whole network should have asmany degrees of freedom as the connected components. This requires that the dominanteigenspace have as many dimensions, which is why there should be as many degeneratedominant eigenvalue 0’s as the connected components in the network. The fourthproperty can be derived by combining this with the argument on property 1 above.And finally, the spectral gap. It is so called because the list of eigenvalues of a matrixis called a matrix spectrum in mathematics. The spectral gap is the smallest non-zeroeigenvalue of L, which corresponds to the largest non-zero eigenvalue of −αL and thusto the mode of the network state that shows the slowest exponential decay over time. If thespectral gap is close to zero, this decay takes a very long time, resulting in slow diffusion.Or if the spectral gap is far above zero, the decay occurs quickly, and so does the diffusion.In this sense, the spectral gap of the Laplacian matrix captures some topological aspectsof the network, i.e., how well the nodes are connected to each other from a dynamicalviewpoint. The spectral gap of a connected graph (or, the second smallest eigenvalue ofa Laplacian matrix in general) is called the algebraic connectivity of a network.Here is how to obtain a Laplacian matrix and a spectral gap in NetworkX:Code 18.1:>>> import networkx as nx>>> g = nx.karate_club_graph()>>> nx.laplacian_matrix(g)>>> nx.laplacian_spectrum(g)array([ 2.84494649e-15, 4.68525227e-01, 9.09247664e-01,1.12501072e+00, 1.25940411e+00, 1.59928308e+00,1.76189862e+00, 1.82605521e+00, 1.95505045e+00,2.00000000e+00, 2.00000000e+00, 2.00000000e+00,2.00000000e+00, 2.00000000e+00, 2.48709173e+00,