Examples of coupled networks showing all three phases. All figures are plotted for scale-free networks, generated by using the BA algorithm (Ref. 19 ) of size and (a) average degree 2 (phase 1), (b) average degree 6, (c) average degree 10, (d) average degree 20. The axis represents the coupling strength and the axis gives the synchronization measure (엯) and the homogeneity measure (●) for the whole network. The largest Lyapunov exponent is plotted as a function of the coupling strength (see inset). Panels (a'), (b'), (c'), and (d') show exact values of the transition probability for different nodes as a function of the coupling strength. For clarity we plot only a few arbitrarily selected nodes.
The global measure of coupled dynamics from the local symbolic dynamics. We take various networks having coupled dynamics in the phase two. The axis gives the coupling strength and the axis depicts (-) (the largest Lyapunov exponent for the coupled dynamics) as well as ( ) (transition probability for a randomly selected node). (a) For a nearest neighbor coupled network of size ; (b) for a three-nearest neighbor coupled network, ; (c) and (d) are for random and scale-free networks, respectively, with average degree 10 and .
The measure of homogeneity as a function of the connectivity ratio .
The ratio of the number phase synchronized clusters to the maximum possible clusters, and the transition probability for the coupled logistic map as a function of the coupling strength , (a) for a nearest neighbor coupled network with average degree 20 and , (b) for a scale-free network with average degree 10 and , (c) for a random network with average degree 10 and , (d) for a nearest neighbor coupled network with average degree 6 and , and the tent map as the local chaotic function.
An illustration of the choice of the threshold as the point where sharply drops to near zero.
The transition probability measure for coupled Hénon maps. The axis displays the coupling strength and the axis shows the different transition probabilities and the measure of synchronization . (a) is for two coupled nodes and plots for the symbolic sequence of length 2. (b) and (c) are plotted for globally coupled networks with . (b) plots and (c) plots transition probabilities for symbolic sequences of length 3, namely (◻), (●), and (엯). The synchronized state is detected when all the transition probabilities are equal to those of the uncoupled map; i.e., the transition probabilities at the zero coupling strength. (d), (e), and (f) show the standard deviation (solid thick line) and (vertical dashed line) for these three transition probabilities of an arbitrary selected node with respect to the transition probabilities of the uncoupled function (solid line), i.e., for . is calculated for 20 simulations for the dynamics with different sets of random initial conditions. (d) For a globally connected network with ; (e) and (f) for a random network with , average degrees 10 and 2, respectively. The last panel is plotted to show the behavior of the transition probabilities when we do not get global synchrony even at large coupling strengths.
The measure of homogeneity as a function of connection ratio , with the Hénon map as the local dynamical function.
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