Cluster synchronization in multiplex networks

Synchronization is a universal phenomenon observed in a range of systems [1]. It is known to be important for the proper functioning of many complex systems. For example in the brain, the phase synchronization at different locations is responsible for action, motion, vision and sensation [2]. Recently, cluster synchronization has gained tremendous attention due to its occurrence and importance in many real-world systems [3–5]. The cluster synchronization refers to the case when the nodes of a system divide into several synchronized groups such that nodes in the same group synchronize with each other while do not synchronize with the nodes of the other groups. So far the studies on the cluster synchronization have mainly focused on complex systems being represented as isolated networks [6–8].

However, a complex system may consist of a superposition of a number of interacting networks [9,10], such as a social system which is composed of different sub-networks consisting of family, friends, colleagues, work collaborators and hence forming a multiplex network. The realization that multiplex networks present a more realistic representation of real-world interactions, has led to a spurt in the activities of modeling real-world complex systems [9,11–13] under this framework.

Most of the studies on multiplex networks have concentrated on the investigation of various structural properties or on the emergence of spectral properties [14,15]. Few works considering dynamical properties report that multiplexing reduces the rate of the global synchronization and may lead to an explosive synchronization [16]. Though, such understanding for the cluster synchronization of the multiplex networks is lacking.

In this letter, we investigate the dynamical behavior of nodes in a layer upon multiplexing it with another layer. Particularly, we investigate the impact of nodes interactions in one layer on the cluster synchronization of the same nodes in the other layer.

In a realistic situation, the connection density as well as the network architecture of two layers can be different, for instance in a social system a family network can be denser than a counter friendship network as well as can have a different network architecture. Similarly, the friendship network can be denser than a corresponding business network. In this work we consider model networks studied widely which are known to capture many properties of the real-world systems, for instance of social networks. One of the most important property captured by the scale-free (SF) network is the power law degree distribution of many social networks.

We find that nodes interactions in one layer affect the synchronizability of the other layer in many different ways. While multiplexing with sparse networks enhances the cluster synchronizability of a layer, the impact of multiplexing with dense networks depend on the network architecture. The enhancement in the cluster synchronization is referred to the case when the number of the nodes forming synchronized clusters increases. Furthermore, we report that the change in the density of connections in a layer of the multiplex network may bring a change in the mechanism of the cluster synchronization in the other layer. Previous studies on the dynamical behavior of isolated networks have identified two different mechanisms of cluster synchronization, namely, the driven (D) and the self-organized (SO) [4]. The D and SO synchronization correspond to the synchronization due to the inter- and intra-cluster couplings, respectively.

We consider the well-known coupled maps model [17] to investigate the phase-synchronized clusters in the multiplex networks. Furthermore, we consider phase synchronization instead of the exact synchronization ($x_i(t)=x_j(t)\;\forall\; i,j\in N$ and t > t₀) as for sparse networks number of nodes displaying the exact synchronization is very less and with an increase in the connection density there is a transition to the globally synchronized state spanning all the nodes [9], whereas the prime motive of the current work is to study the cluster synchronization. The phase synchronization reveals interesting cluster patterns as well as the dependence of the mechanism of cluster formation in one layer on the network structure of the other layers. Let each node of the network be assigned a dynamical variable $x_i, i = 1, 2,\ldots, N$. The dynamical evolution is defined by

$$\begin{equation} x_i(t+1)=(1-\varepsilon) f(x_i(t)) + \frac{\varepsilon}{k_i} \sum_{j=1}^{mN} A_{ij} g(x_j(t)). \end{equation} \tag{ 1 }$$

Here, A is the adjacency matrix with elements A_ij taking values 1 and 0 depending upon whether there is a connection between i and j or not, ε is the overall coupling constant, N is total number of nodes in a layer and m is the number of layers in the multiplex network. We consider the simplest multiplex network with two layers and N nodes in each layer (fig. 1), the adjacency matrix A for this two-layer multiplex network can be given as

$$\begin{equation} A=\left(\begin{array}{c@{\quad}c} A^1 & I \\ I & A^2 \end{array}\right), \end{equation} \tag{ 2 }$$

where, A¹ and A² are the adjacency matrices corresponding to the layer 1 and layer 2. $k_i=(\sum_{j=1}^{N} A^l_{ij})+1$ is the degree of the i-th node in the l-th layer of the multiplex network.

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The average degree of the two layers may be different and is indicated as $\langle k_1 \rangle$ and $\langle k_2 \rangle$. The function f(x) defines a local nonlinear map, whereas g(x) defines the nature of coupling between the nodes. The local nonlinear map f(x) is considered as the logistic map ($f(x) = \mu x(1-x)$, $\mu=4$) and the coupling function is taken as $g(x) = f(x)$.

We consider phase synchronization defined as follows [4,18]. Let n_i and n_j denote the number of times when the variables $x_i(t)$ and $x_j(t)$, $t=1, 2, \ldots T$ for the nodes i and j exhibit local minima (maxima) during the time interval T. Let n_ij denote the number of times these local minima (maxima) match with each other. The phase distance between two nodes i and j is then given as $d_{ij} = 1 - 2n_{ij}/(n_i + n_j)$. The nodes i and j are phase synchronized if $d_{ij}=0$. All the pairs of nodes in a cluster are phase synchronized¹ . Further, we define cluster synchronizability of a network in terms of the number of nodes participating in the clusters. In order to measure the cluster synchronizability, we define the measure $f_{\textit{clus}}=N_{\textit{clus}}/N$, where $N_{\textit{clus}}$ is the total number of nodes participating in all the clusters. A network exhibits better synchronizability if $f_{\textit{clus}}$ of this is more than the $f_{\textit{clus}}$ of another network in a larger range of coupling.

Furthermore, to investigate the mechanism behind the cluster formation, we use $f_{\textit{intra}}$ and $f_{\textit{inter}}$ as a measure for the intra- and inter-cluster couplings, respectively [4], which are defined as; $f_{\textit{intra}}= N_{\textit{intra}}/N_c$ and $f_{\textit{inter}}= N_{\textit{inter}}/ N_c$. where $N_ {\textit{intra}}$ and $N_{\textit{inter}}$ are the numbers of intra-cluster and inter-cluster couplings, respectively. In $N_{\textit{inter}}$, couplings between two isolated nodes are not included. The ideal D cluster state corresponds to $f_{\textit{intra}}=0$, whereas for the ideal SO cluster state $f_{\textit{inter}}$ can take a very small value but cannot attain the zero value as at least there is one connection between different clusters for a connected network. Note that, while calculating the values of $f_{\textit{intra}}$ and $f_{\textit{inter}}$ for a particular layer, we do not consider the inter-layer connections. Further, if $f_{\textit{intra}}> f_{\textit{inter}}\ (f_{\textit{inter}}>f_{\textit{intra}})$, clusters are considered here to be of the dominant SO (dominant D) type.

We study the impact of multiplexing on the cluster synchronizability of the random as well as the regular networks. The 1-d lattices used in the simulation have circular boundary conditions with each node having k nearest neighbors. The scale-free (SF) and random networks are obtained by using BA and ER models, respectively [19]. This letter focuses on the cluster synchronizability of networks and hence we consider sparse networks representing the first layer as denser networks that tend to form the global synchronized state spanning all the nodes.

We evolve eq. (1) starting from a set of random initial conditions and consider the phase-synchronized clusters after an initial transient. For the two-layers multiplex network, the first layer can be represented by a regular or a random network, similarly the second layer can also be represented by a regular or random network. Here, we present results for all the possible combinations, such as random-random, random-regular, regular-regular and regular-random. First, we discuss the cluster synchronizability of a regular network represented by a 1-d lattice upon multiplexing with a ER random network. The isolated sparse 1-d lattice at weak couplings leads to the phase-synchronized clusters with all the nodes participating in the clusters, whereas strong couplings lead to a very few nodes forming clusters (fig. 2(a)). The multiplexing with a sparse ER network enhances the cluster synchronizability of the 1-d lattice at all the couplings as well as maintains the good cluster synchronizability exhibited by the isolated networks at the weak coupling range $(0.16\overset{\sim}{<} \varepsilon \overset{\sim}{<}0.2)$. The multiplexing with a denser ER network while enhances the cluster synchronizability of 1-d lattice at weak couplings, it leaves the cluster synchronizability unchanged with few nodes keep forming synchronized clusters at intermediate and strong couplings. For example, fig. 2(b) demonstrates that cluster synchronizability of 1-d lattice enhances at weak couplings irrespective of the value of $\langle k_2 \rangle$, whereas at the intermediate couplings, for $\langle k_2 \rangle = 4$, there is an enhancement in the cluster synchronization, which for the higher values of $\langle k_2 \rangle$ gets vanished.

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At strong couplings, the cluster synchronization enhances up to a certain value of $\langle k_2 \rangle$ as depicted in fig. 2(b). For a 1-d lattice of $N = 250$ and $\langle k_1 \rangle=4$, synchronization enhances up to $\langle k_2 \rangle \overset{\sim}{<} 10$ (fig. 2(b)). The denser connections in the second layer do not enhance the cluster synchronizability of the first layer with sparser connections. Note that the synchronizability of the second layer, represented as the ER random network, increases with an increase in the average degree as observed for the isolated networks.

Next we discuss the cluster synchronizability of the 1-d lattice upon multiplexing with various other network architectures. At weak couplings, multiplexing with the SF networks also leads to an enhancement in the cluster synchronizability (fig. 2(c)), as observed for the multiplexing with the ER random network. Whereas at intermediate and strong couplings, there is an enhancement in the cluster synchronizability as observed for the multiplexing with the random networks, but the connection density of the second layer for which this enhancement occurs becomes lower (fig. 2(b)). For example, a 1-d lattice with $\langle k_1 \rangle = 4$, exhibits an enhancement in the cluster synchronization upon multiplexing with the SF networks of $\langle k_2 \rangle \overset{\sim}{<} 4$ (fig. 2(c)).

The multiplexing of the 1-d lattice with another 1-d lattice does not enhance the cluster synchronizability of the 1-d lattice at intermediate and strong couplings (fig. 2(d)). Thus, the enhancement in the cluster synchronizability of the 1-d lattice is least favorable when it is multiplexed with the 1-d lattice and most favorable for multiplexing with the ER random networks. A very high diameter of the 1-d lattice has been attributed for its poor synchronizability [20,21]. Multiplexing of the 1-d lattice with the other 1-d lattice keeps the diameter of the network almost same as for the isolated network, whereas multiplexing with the random networks leads to a sharp reduction in the diameter of the entire network as well as of the layer represented with the 1-d lattice as any pair of the distant nodes get connected with a shorter path via the layer represented by the ER random network. For example, the diameter of the 1-d lattice with N = 250 and $\langle k \rangle = 4$ is 63. Upon multiplexing with the other 1-d lattice (N = 250, $\langle k \rangle = 4$) the diameter becomes 64. Hence, the synchronizability remains unaffected after the multiplexing. While multiplexing with the ER random network of the same average degree, the diameter of the entire multiplex network reduces to 8.

Further, we investigate the cluster synchronizability of the ER random networks upon multiplexing with a various network architecture. The isolated sparse ER random networks demonstrates a better cluster synchronizability as compared to the sparse regular networks (fig. 3(a)). First, we discuss the impact on the cluster synchronizability of ER random networks upon the multiplexing with the 1-d lattice. At weak couplings, the multiplexing with a 1-d lattice as well with other ER random network enhances the cluster synchronization of the ER network. The enhancement is higher for the multiplexing with the ER random network. For example, the isolated ER random networks with $\langle k_1 \rangle = 4$ and $N = 250$ lead to the participation of about 50% nodes in the cluster formation. After multiplexing with the 1-d lattice, 65% nodes keep participating in the cluster formation as shown in fig. 3(b), whereas multiplexing with the ER random networks leads to the participation of 80% nodes in the cluster formation (fig. 3(c)). The range of coupling in which enhancement occurs keeps on increasing with the increase in the density of the second layer (fig. 3(b), (c)). At intermediate and strong couplings, there is an enhancement in the synchronization for multiplexing with the sparse networks whereas the impact of multiplexing with the dense networks largely depends on the network architecture as we discussed for the 1-d lattice. For example, the cluster synchronizability of ER random networks with $\langle k_1 \rangle = 4$ enhances upon multiplexing with the 1-d lattice and ER random networks up to a certain limit of $\langle k_2 \rangle$. The multiplexing with the 1-d lattice enhances the cluster synchronizability for $\langle k_2 \rangle \overset{\sim}{<} 20$ (fig. 3(b)), while in the case of multiplexing with the ER random networks the enhancement occurs for $\langle k_2 \rangle \overset{\sim}{<} 10$ (fig. 3(c)). The higher connection density in the second layer leads to the suppression in the cluster synchronizability of random networks (fig. 3(a), (b) and (c)). Infact multiplexing of the sparse ER random network with the globally connected network also leads to a suppression in the cluster synchronizability of the random network, as well as it destroys the global synchrony in the globally connected layer. Note that as a layer becomes denser, it shows an enhancement in the cluster synchronizability, but there is a reduction in the cluster synchronization as compared to the corresponding isolated networks. Thus, the cluster synchronizability of the first layer mainly depends on its own network architecture and on the network architecture of the second layer.

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In order to demonstrate the generality of the enhancement in the cluster synchronizability of a sparse network upon multiplexing with the other sparse network, we present values of $f_{\textit{clus}}$ with error bars in fig. 4. It reflects a clear enhancement in the cluster synchronization after multiplexing.

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Further, in order to understand the suppression in the cluster synchronizability of a sparse network upon multiplexing with the denser networks, we examine the degree-degree correlation of the mirror nodes. We calculate the Pearson coefficient for the degree-degree correlation of the mirror nodes, given as follows [22]: $r = \frac{\sum_{jk} jk(e_{jk} - q_{j} q_{k})}{\sigma_{q}^2}$, where e_jk is the fraction of the j-degree node connected with k-degree nodes. q_j is the fraction of all the nodes connected to the j-th degree, yielding $\sum_{jk}e_{jk} = 1$, $\sum_j e_{j}= q_{k}$. Figure 5(a) plotting the variation of r with $\langle k_2 \rangle$ demonstrates that the multiplexing of a sparse network with a denser network corresponds to the strong disassortivity [22] in the degree-degree correlations of the mirror nodes. For r > 0.85, irrespective of the network architecture, the cluster synchronizability of the sparse network is suppressed, while the denser network loses the global synchrony, which is indicated in the isolated case at the strong couplings. Thus, the strong disassortivity in the degree of the mirror nodes is associated with the suppression in the synchronizability of both the networks. Additionally, the multiplex network yields the chaotic dynamics for almost all the coupling values for the layers being represented by sparse networks (fig. 5(b)).

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The network parameters for which the multiplexing imposes a change in the dynamical behavior may change with an increase in the network size but the observation that there is an enhancement in the synchronization of a sparse network upon multiplexing with the other sparse networks, and the suppression due to the multiplexing with the dense networks remain the same. For example, the limit of $\langle k_2 \rangle$, for which there is an enhancement in the synchronization at strong couplings, depends on the network properties such as the average degree and the size of the network. For the 1-d lattice of N = 100 and $\langle k_1 \rangle = 4$ the enhancement occurs for $\langle k_2 \rangle < 10$ (fig. 6(a)) upon multiplexing with the random network, while for the random network with $N = 500$ and $\langle k_1 \rangle = 6$, an enhancement in the synchronization occurs up to $\langle k_2 \rangle < 50$ (fig. 6(b)) upon multiplexing. Furthermore, the fraction of the nodes in the largest cluster $(f_{NL})$ also exhbits an enhancement in cluster synchronization upon multiplexing with the sparse networks (fig. 6(a) and (b)) and a suppression upon the multiplexing with the denser networks (fig. 6(b)).

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Further, we investigate the mechanism behind the cluster formation in multiplex networks. We find that at weak couplings, the enhancement in the cluster synchronizability due to multiplexing, corresponds to an enhancement in the D mechanism indicating that the D synchronization remains the prime mechanism behind the cluster formation even after multiplexing, as observed for the isolated networks [4]. For example, the cluster synchronizability of a 1-d lattice of N = 50 and $\langle k_1 \rangle = 2$ is enhanced due to the multiplexing with ER random networks of $\langle k_2 \rangle = 4$ and clusters are of dominant D type (fig. 7(a), (b)). At intermediate and strong couplings, the multiplexing may lead to a change in the mechanism behind the cluster formation. The 1-d lattice, which manifests a very poor cluster synchronizability at intermediate and strong couplings, upon multiplexing exhibits an enhanced cluster synchronizability with the dominant SO mechanism. For example, the 1-d lattice of N = 50 and $\langle k_1 \rangle = 2$ does not indicate any significant cluster formation at strong couplings (fig. 7(c)). After multiplexing with the ER random network $(\langle k_2 \rangle = 4)$, the 1-d lattice starts displaying the dominant SO clusters (fig. 7(d)). The isolated random as well as SF sparse networks exhibit a transition from the ideal D or mixed cluster state to the dominant SO cluster state. For example, the random network of N = 50 and $\langle k_1 \rangle = 2$ exhibits a transition from the mixed cluster state (fig. 7(e)) to the dominant SO cluster state (fig. 7(f)) and the SF network shows a transition from the mixed and the dominant D cluster state to the dominant SO cluster state (fig. 8(a) and (b)) upon its multiplexing with the 1-d lattice. Moreover, note that multiplexing of the sparse networks with the dense networks leads to a suppression in the cluster synchronization at strong couplings, which also corresponds to the suppression in the SO mechanism, while the contribution of the D mechanism remains the same as observed for the isolated network (fig. 8(a), (b)).

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To conclude, we have studied the impact of multiplexing on the cluster synchronizability as well as the mechanism behind the synchronization of a layer represented by the sparse networks in the simplest multiplex network consisting of two layers. We find that at weak couplings, the multiplexing enhances the cluster synchronizability, while at strong couplings this enhancement depends on the architecture as well as the connection density of the other layer. The cluster synchronizability of a layer is enhanced when the other layer has a moderate connection density. Moreover, multiplexing favors the enhancement in the cluster synchronization of a regular network if it is multiplexed with another random network. The enhancement in the cluster synchronization is also associated with the change in the mechanism of the cluster formation.

Our work demonstrates that in a multiplex network, the activity in one layer (sub-network) is significantly influenced by the structural properties of the other layer (sub-network). If the connection density in one layer increases above a certain limit, it may spoil the synchronization in the other layer. The results presented here about the dynamical behavior of multiplex networks may provide a guidance for the construction of a better model network with multiplex architecture, such as the airport networks composing different airline companies [23]. Furthermore, as the largest eigenvalue of the adjacency matrix defines the threshold of the epidemic spread [24] as well as the global synchronizability [25] of a network, suppression in the synchronizability of the denser networks, upon multiplexing, can be used to control epidemic spread [14] by introducing a different kind of relation among the same set of nodes, for instance establishing some health awareness camp [26], which can act as the other layer consisting of connections between the nodes which are the participants of that camp.

SJ acknowledges DST (EMR/2014/000368) and CSIR (25(0205)/12/EMR-II) project grants for the financial support. AS thanks complex systems lab members for useful discussions.