Effective Subnetwork Topology for Synchronizing Interconnected Networks of Coupled Phase Oscillators.

Synchronization in Kuramoto networks and the effect of the network inhomogeneity. (A) Dynamics of random networks (ω¯ = 4.5, σω = 0.15) with coupling strengths of (a) K = 0 and (b) K = 10. (B) Dependence of the Kuramoto order parameter r on K for networks of inhomogeneous oscillators. A total of 250 networks are sampled for each condition, and their means are plotted. Shaded error bars represent 95% confidence intervals.

Effect of the network topology in single networks. (A) Schematic illustrations of the four types of network topologies considered in this study. For ease of viewing, the number of nodes N and their degrees k are varied. In the actual calculation, N and k are 50 and 6, respectively. (B,C) Dependence of r on K derived from (B) the numerical simulation and (C) the synchrony alignment function (SAF). The networks are random (Rand; blue circles), SW (orange triangles), SF (green crosses), and super-hub (H; red squares). A total of 250 networks are sampled for each condition, and their means are plotted. Shaded error bars represent 95% confidence intervals.

Synchronization in interconnected networks. (A) Schematic illustration of the interconnected network and the change in the node dynamics after the subnetworks are coupled at 250 s. The upper and lower panels show the evolution of phases and phase velocities, respectively. The topology of each subnetwork is super-hub with N1 = N2 = 50, ω¯ = 4.5, and Kintra = Kinter = 4. (B) Transient change in order parameter r upon the coupling of the subnetworks. A total of 250 networks are sampled and averaged for each topology. Shaded error bars represent 95% confidence intervals. The time average of r plotted here is calculated every 0.5 s instead of every 200 s. (C) The time constant τ of the transient change in r calculated for each topology. The abbreviations are as follows: R, random; SW, small-world; SF, scale-free; H, super-hub. The network parameters are N1 = N2 = 50, Kintra = 4, and Kinter = 10.

Dependence of r on Kinter in interconnected networks. (A) is derived from the numerical simulation, and (B) from the evaluation of the synchrony alignment function (SAF). The network topologies are random (blue), SW (green), SF (orange), and super-hub (red), with N1 = N2 = 50 and Kintra = 4. (C) r-Kinter relationships in large networks (N1 = N2 = 1000, Kintra = 4) with random (blue) and super-hub (red) subnetworks. Plots and solid lines represent the results obtained from the numerical simulation and SAF, respectively. (D–G) Effect of frequency allocation on synchrony in NoNs with (D) random, (E) SW, (F) SF, and (G) super-hub subnetworks (N1 = N2 = 50, Kintra = 4). Natural frequencies are reallocated so that outlying frequencies are placed either at high-degree hub nodes (broken lines) or at low-degree nodes (dotted lines). Solid line represents the default, random allocation. (H) Effect of connector node degree on synchrony. Nodes used to connect two subnetworks are chosen from either the highest-degree (solid lines) or lowest-degree (broken lines) nodes. Natural frequencies are allocated randomly, and Kintra was set to 12. The same color schemes are used for different topologies as in (B). Error bars are removed to aid visualization. (I) Effect of rich-club organization on synchrony of hierarchically modular networks (N1 = N2 = 50, Kintra = 4, k = 6, pin = 0.97). NoNs bearing subnetworks with rich-club organization (α = 10) is compared against those without it (α = 0). The two subnetworks are connected through the highest-degree nodes. A total of 250 networks are sampled for each condition, and their means are plotted. Shaded error bars represent 95% confidence intervals.

Coupling networks of different frequencies. (A) A schematic showing the networks under consideration, and the evolution of phases (upper panel) and phase velocities (lower panel) of the nodes. The average natural frequencies of the two subnetworks are ω¯1=4.3 and ω¯2=4.7. (B) A schematic showing the three-network system and the effect of inter-network coupling on synchronization. A total of 250 networks are sampled for each condition, and their means are plotted. Error bars representing 95% confidence intervals are depicted but are invisible. The network topology of the individual networks is super-hub, with N1 = N2 (= N3) = 50, σω = 0, Kintra = 4, and Kinter = 40.