Introduction to the Modeling and Analysis of Complex Systems

412CHAPTER 18. DYNAMICAL NETWORKS III: ANALYSIS OF NETWORK DYNAMICSNote that, although α and λ i are both non-negative, H ′ (x s (t)) could be either positive ornegative, so which inequality is more important depends on the nature of the output functionH and the trajectory x s (t) (which is determined by the reaction term R). If H ′ (x s (t))always stays non-negative, then the first inequality is sufficient (since the second inequalitynaturally follows as λ 2 ≤ λ n ), and thus the spectral gap is the only relevant informationto determine the synchronizability of the network. But if not, we need to consider both thespectral gap and the largest eigenvalue of the Laplacian matrix.Here is a simple example. Assume that a bunch of nodes are oscillating in an exponentiallyaccelerating pace:dθ idt = βθ i + α ∑ j∈N i(θ j − θ i ) (18.16)Here, θ i is the phase of node i, and β is the rate of exponential acceleration that homogeneouslyapplies to all nodes. We also assume that the actual values of θ i diffuse to andfrom neighbor nodes through edges. Therefore, R(θ) = βθ and H(θ) = θ in this model.We can analyze the synchronizability of this model as follows. Since H ′ (θ) = 1 > 0,we immediately know that the inequality (18.14) is the only requirement in this case. Also,R ′ (θ) = β, so the condition for synchronization is given byαλ 2 > β, or λ 2 > β α . (18.17)Very easy. Let’s check this analytical result with numerical simulations on the Karate Clubgraph. We know that its spectral gap is 0.4685, so if β/α is below (or above) this value,the synchronization should (or should not) occur. Here is the code for such simulations:Code 18.2: net-sync-analysis.pyimport matplotlibmatplotlib.use(’TkAgg’)from pylab import *import networkx as nxdef initialize():global g, nextgg = nx.karate_club_graph()g.pos = nx.spring_layout(g)for i in g.nodes_iter():g.node[i][’theta’] = random()