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Edited by: Carlo Laing, College of Sciences, Massey University, New Zealand

Reviewed by: Marko Gosak, University of Maribor, Slovenia; Changsong Zhou, Hong Kong Baptist University, Hong Kong

†Present Address: Michio Niwano, Kansei Fukushi Research Institute, Tohoku Fukushi University, Sendai, Japan

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

A system consisting of interconnected networks, or a network of networks (NoN), appears diversely in many real-world systems, including the brain. In this study, we consider NoNs consisting of heterogeneous phase oscillators and investigate how the topology of subnetworks affects the global synchrony of the network. The degree of synchrony and the effect of subnetwork topology are evaluated based on the Kuramoto order parameter and the minimum coupling strength necessary for the order parameter to exceed a threshold value, respectively. In contrast to an isolated network in which random connectivity is favorable for achieving synchrony, NoNs synchronize with weaker interconnections when the degree distribution of subnetworks is heterogeneous, suggesting the major role of the high-degree nodes. We also investigate a case in which subnetworks with different average natural frequencies are coupled to show that direct coupling of subnetworks with the largest variation is effective for synchronizing the whole system. In real-world NoNs like the brain, the balance of synchrony and asynchrony is critical for its function at various spatial resolutions. Our work provides novel insights into the topological basis of coordinated dynamics in such networks.

Many biological, social, and technological systems comprise of interacting subsystems and can be modeled as a network of networks (NoN) (Gao et al.,

In the brain, synchronized activity of neurons is essential for the development and computation. Synchronization in NoNs and modular networks has been explored theoretically based several models, including phase oscillators (Arenas et al.,

The synchrony in complex networks is strongly affected by the topology of the networks, such as a regular lattice, random, small-world (SW), or scale-free (SF) structure, which lies as the foundation of diverse dynamics observed in naturally occurring complex networks, such as the central nervous system (Feldt et al.,

In this study, we consider a simple NoN consisting of two coupled subnetworks and examine the effect of the topology of the subnetworks on the global synchrony. The topologies we consider are: random; SW; SF; and “super-hub,” which is characterized by the presence of a few hub nodes that are fully connected to all other nodes. Each node is represented as a Kuramoto phase oscillator, and the synchronization is evaluated by calculating the Kuramoto order parameter,

Each node in a network is modeled as a Kuramoto oscillator. The state of node _{i}, and its dynamics are calculated by the 4th-order Runge-Kutta method with a time step of

where ω_{i} is the natural frequency, _{i} is the set of nodes directly coupled to node _{i} is selected from a Gaussian distribution _{ω} of 0.15, and allocated randomly in the nodes, unless otherwise noted. The coupling strength is normalized using the average degree (Hong et al.,

Each single network or a subnetwork of a NoN is composed of _{ij}, initial phase ϕ_{i}(0), and natural frequency ω_{i} are sampled.

The network topologies that we consider are Erdös–Rényi random networks, Watts–Strogatz SW networks, Barabási–Albert SF networks, and a super-hub network. A random network is constructed by connecting randomly selected pairs of nodes until the necessary amount of connections (_{i}), is given by: _{i} the degree of

The degree of synchrony in the networks is evaluated using the order parameter,

where |⋯| and 〈⋯〉 denote the absolute values and time averages, respectively. In single networks, the first 50 s of the simulation are neglected, and the time average is obtained for the remaining 200 s. In the case of NoNs, the first 50 s after the coupling (250–300 s) are neglected, and the time average is obtained for the remaining 200 s (300–500 s). As discussed later, the time constants of transient phases are mostly in the order of seconds after which the order parameters of networks saturate. Hence the “burn-in” period of 50 s is sufficient to equilibrate the networks.

Analytically, the order parameter of a network is derived using the synchrony alignment function (Skardal et al., _{ij}]:

where _{ij}] is the Laplacian matrix with _{ij} = δ_{ij}_{i} − _{ij} and _{j} is the ^{th} eigenvalue of ^{j} is the normalized eigenvector associated with λ_{j}; and 〈·, ·〉 denotes the inner product. When a network is synchronizable,

We begin with a description of the basic properties of a single network of Kuramoto oscillators. A large portion of the results presented in this Section has already been under thorough investigation and are reviewed in Rodrigues et al. (

Figure _{ω} = 0.15), _{ω} is scaled by a factor of _{ω} = 0.15.

Synchronization in Kuramoto networks and the effect of the network inhomogeneity. _{ω} = 0.15) with coupling strengths of (a)

Next, we examine the effect of the network topology on the synchrony of a single network. We consider four types of topologies: random, SW, SF, and super-hub (Figure

Effect of the network topology in single networks.

The SW networks always exhibit the lowest synchrony among the four topologies (Figures

In summary, random connection is the most efficient strategy for synchronizing a network of inhomogeneous nodes with the minimum coupling strength.

Next, we investigate the effect of the network topology of subnetworks in NoNs on the global synchrony. We consider the simplest case, i.e., a NoN consisting of two subnetworks (Figure _{intra}, is kept constant at _{intra} = 4, and the inter-subnetwork coupling strength, _{inter}, is varied.

Synchronization in interconnected networks. _{1} = _{2} = 50, _{intra} = _{inter} = 4. _{1} = _{2} = 50, _{intra} = 4, and _{inter} = 10.

Figure _{inter} = 10). Prior to the coupling of the two subnetworks, the nodes within each subnetwork are synchronized, as _{intra} is sufficiently high (_{intra} = 4). However, the synchronizing phases of the two subnetworks are independent, yielding a global order parameter of

Analysis of the transient state after coupling the subnetworks (_{0} − _{0})/τ] are summarized in Table

Transient change in the order parameter after the coupling of two subnetworks.

_{0} |
0.98 | 0.93 | 0.99 | 0.99 |

0.36 | 0.30 | 0.38 | 0.36 | |

τ (s) | 4.7 | 10.2 | 3.9 | 4.0 |

_{0} − A exp[−(t − t_{0})/τ]) to derive the parameters r_{0}, A, and τ

We next analyze _{inter}. Despite the fact that the random topology synchronized (

Dependence of _{inter} in interconnected networks. _{1} = _{2} = 50 and _{intra} = 4. _{inter} relationships in large networks (_{1} = _{2} = 1000, _{intra} = 4) with random (blue) and super-hub (red) subnetworks. Plots and solid lines represent the results obtained from the numerical simulation and SAF, respectively. _{1} = _{2} = 50, _{intra} = 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. _{intra} was set to 12. The same color schemes are used for different topologies as in _{1} = _{2} = 50, _{intra} = 4, _{in} = 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.

Since previous works have shown that the number of nodes in a network can dramatically impact the degree of synchrony in modular networks (Oh et al.,

One of the major advantage of the synchrony alignment function is the ability to combine the network structure with the allocation of natural frequencies. The effect of frequency allocation on the synchrony of NoNs (_{1} = _{2} = 50) with different subnetwork topologies is shown in Figures _{i}, are reallocated depending on the node degree, _{i}: ω_{i} is sorted so that _{i}. In the former case, outlying frequencies are allocated at the hub nodes, the first three of which are the connector nodes of the NoN. Contrarily, in the latter case, _{i} and allows them to oscillate at near

We also investigate how the choice of connector nodes influences synchrony in NoNs of various subnetwork topologies. Here the synchrony alignment function is used to analyze _{intra} is increased to 12. The order parameters of the four topologies are summarized in Figure

The properties of the network topologies explored in the current study is not mutually exclusive. For instance, in the neural networks of the brain, SW and SF properties coexist as a result of cost-efficiency trade-off (Chen et al., _{in} × _{in}) × _{in} designates the probability of intra-connections. The two nodes to be connected are chosen with a probability Π_{i} = _{i} is the degree of node

The variation in the effective topology for synchronizing a single network and NoN is the primary finding of the present work. In a single network, the random topology is more robust to node inhomogeneity (section Synchronization in Single Networks). However, in NoNs, the existence of high-degree hub nodes and connection of the subnetworks through the hubs is effective for achieving synchronization. This advantage overwhelms the aforementioned disadvantage in individual networks, and hence, super-hub and SF networks synchronize with weaker coupling strength than random networks. The result is consistent with a recent report on the synchronization in multilayer networks of identical chaotic oscillators, in which synchrony was achieved at weaker inter-layer coupling strength when a network with layers configured under the SF topology rather than random topology (Leyva et al.,

In real systems, the natural frequencies of the nodes are often distributed. In the previous sections, we considered this by using nonzero values for σ_{ω}. Previously, a number of reports have investigated effective connection strategies for synchronizing an isolated single network comprised of inhomogeneous oscillators (Gleiser and Zanette,

When two subnetworks with different average natural frequencies,

Coupling networks of different frequencies. _{1} = _{2} (= _{3}) = 50, σ_{ω} = 0, _{intra} = 4, and _{inter} = 40.

We further investigate whether there is an efficient coupling scheme for a NoN comprising three subnetworks of different _{α} = 50 (α = 1, 2, 3), _{intra} = 4, and _{inter} = 40 and that the total number of inter-subnetwork links is nine. The average natural frequencies of the three subnetworks are _{0} = 4.5. To simplify the discussion, the variation of the natural frequency within each subnetwork (σ_{ω1}, σ_{ω2}, σ_{ω3}) is set as zero. A total of 250 realizations are sampled for each condition.

Three types of inter-subnetwork connections are considered (Figure

The order parameter for Networks (i)–(iii) and its dependence on Δω is shown in Figure

The mechanism of Network (ii) being more synchronizable than Network (iii) can be understood from the previous discussion regarding the steady-state frequency of the coupled subnetworks (Figure

In summary, the direct coupling of dissimilar subnetworks is favorable for achieving a high degree of global synchrony with a limited number of connections.

We investigated the effect of the subnetwork topology on the synchronization of interconnected networks, or NoNs. Although random connection was favorable for synchronizing individual networks, the SF and super-hub topology with high-degree hub nodes exhibited highest synchrony in NoNs. This variation in the optimal topology in single and interconnected networks is the major finding of our study. The use of a phase oscillator model as local node dynamics allowed us to analytically investigate the structure-function relationships in NoNs via the synchrony alignment function. The results provide a first-order approximation to the study of dynamics within complex networks of biologically more plausible neural oscillators, such as the neural mass models (Zhao et al.,

In the brain, synchronized neural activity is critical for its function at various spatial resolutions. At the microscopic scale, it modulates the synaptic weights (Kubota and Kitajima,

The balance of synchrony and asynchrony, and the resulting complexity of the dynamics, has been quantified based on, e.g., the probability distribution of pairwise correlation between nodes (Zhao et al.,

HY, SK, and MN: Conceived and designed the research; HY and FS: Performed the simulations and analyzed the results; AH-I and MN: Supervised the research; HY and SK: Wrote the manuscript; FS, AH-I, and MN: Reviewed and edited the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors thank Mr. Yudai Chida of Tohoku University for his assistance with the numerical analysis.

We show here that the order parameter, _{ω} and _{ω} and _{ω} → _{ω} and _{i}, as

where _{i} is equal to that for ψ_{i}, indicating that:

where, for the simplicity of notation, we denoted the order parameter _{i}~

Next, we consider a rescaling of time _{i} under

Since rescaling of time does not influence a temporally averaged value,

Finally, by using the relationships in Equations (A2) and (A4), we obtain:

meaning that the replacement of σ_{ω} → _{ω} and