Applications to the PowerGrid network. E cl (see text for definition) vs. d, for the Rössler system (see the differential equations in the text). Data in panel (a) [in panel (b)] refer to ensemble averages over 850 (200) different numerical simulations of the PowerGrid network. As in Fig. 4, the legends of both panels set the color code for the curves corresponding to each of the reported clusters Ci and to the Entire Network (EN). Panel (a) reports the case of identical Rössler systems, and the error of 6 specific clusters is plotted (see Table 1 of the Supplementary Information for the composition of each of the 6 clusters Ci). The observed sequence of events perfectly matches the predicted one, with an excellent fit with the values d1, ..., d6. In panel (b) the effects of heterogeneity in the network are reported. Namely, for each node i of the PowerGrid network, the parameter bi in the Rössler equations is randomly sorted from a uniform distribution in the interval [0.1 − ϵ, 0.1 + ϵ]. The curves plotted refer to ϵ = 0.01.

Contexts in source publication

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... values d 1 , ..., d 6 are explicitly calculated in the Supplementary Information, and are marked as filled dots in the horizontal axis of Fig. 5 with the same colors identifying the corresponding clusters. Looking at Fig. 5 (a) one sees that, once again, the observed sequence of events matches the predicted one, with an excellent fit with the values d 1 , ..., d 6 , thus validating ex-post the approximation adopted in our ...

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... values d 1 , ..., d 6 are explicitly calculated in the Supplementary Information, and are marked as filled dots in the horizontal axis of Fig. 5 with the same colors identifying the corresponding clusters. Looking at Fig. 5 (a) one sees that, once again, the observed sequence of events matches the predicted one, with an excellent fit with the values d 1 , ..., d 6 , thus validating ex-post the approximation adopted in our ...

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... also tested how robust is the predicted scenario against possible heterogeneities present in the network. Fig. 5 for ϵ = 0.01, corresponding to 10 % of the value (b = 0.1) which was used for all nodes in the case of identical systems, thus representing a case of a rather large ...

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... error never vanishes exactly in the ensemble. Nonetheless, it is still observed that the values of the normalized synchronization errors fluctuate around zero for some sets (clusters) of network's nodes which, therefore, anticipate the setting of the almost completely synchronized state (wherein all nodes evolve almost in unison). In Fig. 5(b) it is clearly seen that, while the synchronization errors approach zero at values that are obviously different from those predicted in the case of identical systems, the sequence at which the different clusters emerge during the transition is still preserved. Similar scenarios characterize the evolution of the network also when the ...

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... of N = 1, 647 nodes and E = 2, 518 edges [5]) and the ego-Facebook network (a dataset containing N = 2, 888 nodes and E = 2, 981 edges [6]). The results are reported in Fig. 6, and refer to ensemble averages over 100 different numerical simulations of identical Rössler systems (same parameters and initial conditions as in the case reported in Fig. 5) coupled by the structure of connections of the two graphs. For the biological network, it is found that the transition to synchronization includes 188 clusters, and Fig. 6a reports the synchronization error of 4 of them: C 1 and C 2 (which are both 2 nodes clusters), C 3 (a cluster containing 3 nodes), C 4 (a cluster of 7 nodes). For ...

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... 6 clusters that are highlighted in red are those whose synchronization error is reported in Fig. 5 of the main text, for a Rössler chaotic system with parameters and coupling function such that it belongs to Class II, displaying C 1 = {7,8} C 2 = {1,2} C 3 = {5,6} C 4 = {1,2,3} EN = {1,2,3,4,5,6,7,8} [3]. (b) Our Sn algorithm correctly predicts cluster synchronization on clusters {7, 8}, {1, 2} and {5, 6}, and {1, 2, 3}, in that ...