We study the impact of interaction of nodes in a layer of a multiplex network on the dynamical behavior and cluster synchronization of these nodes in other layers. We find that nodes interactions in one layer affects the cluster synchronizability of another layer in many different ways. While multiplexing with a sparse network enhances the synchronizability multiplexing with a dense network suppresses the cluster synchronizability with the network architecture deciding the impact of the enhancement and suppression. Additionally, at weak couplings the enhancement in the cluster synchronizability due to multiplexing remains of the driven type, while for strong couplings the multiplexing may lead to a transition to the self-organized mechanism. In this Rapid communication, we study dynamical behavior of nodes in a layer upon multiplexing 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 degree distribution of two layers can be different, for instance in a social system a family network can be denser than a counter friendship network. Similarly, the friendship network can be denser than a corresponding business network. In the present paper, we explore the impact of the network architecture on the cluster synchronizability of another layer. We find that nodes interacting in one layer affect the synchronizability of another layer in many different ways.While multiplexing with sparse networks enhances the cluster synchronizability of a layer, impact of multiplex-

We study the role of delay in phase synchronization and phenomena responsible for cluster formation in delayed coupled maps on various networks. Using numerical simulations, we demonstrate that the presence of delay may change the mechanism of unit to unit interaction. At weak coupling values, same parity delays are associated with the same phenomenon of cluster formation and exhibit similar dynamical evolution. Intermediate coupling values yield rich delay-induced driven cluster patterns. A Lyapunov function analysis sheds light on the robustness of the driven clusters observed for delayed bipartite networks. Our results reveal that delay may lead to a completely different relation, between dynamical and structural clusters, than observed for the undelayed case. Studying the impact of network topology on dynamical processes is of fundamental importance for understanding the functioning of many real world complex networks [1]. The dynamical behavior of a system depends on the collective behavior of its individual units. One of the most fascinating emergent behavior of interacting chaotic units is the observation of synchronization [2]. In general, synchronization may lead to more complicated patterns including clusters [3][4][5]. The interplay between underlying network structure and dynamical clusters has been the prime area of focus for the last two decades [6]. Furthermore, communication delay naturally arises in extended systems [7]. A delay gives rise to many new phenomena in dynamical systems such as oscillation death, stabilizing periodic orbits, enhancement or suppression of synchronization, chimera state, etc [8][9][10][11][12][13][14].In this paper, we study the impact of delay on the phenomenon of phase synchronized clusters in coupled map networks. We investigate the formation of clusters on various networks namely, 1-d lattice, small-world, random, scale-free and bipartite networks [15], and provide a Lyapunov function analysis for bipartite networks to explain possible reasons behind the role of a delay on synchronized clusters. So far, studies on delayed coupled dynamical systems mostly concentrated on a global synchronized state, except a few recent studies which have focused on pattern formation or clustered states [4,5,16,17]. These studies have revealed that delay emulates qualitative changes in clustered state, whereas mechanism of delayed unit to unit interactions has not been investigated so far.Previous studies on undelayed coupled systems have identified two different phenomena for synchronization namely, the driven (D) and the self-organized (SO) [3]. * sarika@iiti.ac.in SO (D) synchronization refer to the state when clusters are formed because of intra-cluster (inter-cluster) couplings. Here, we report that a delay can play a crucial role in the formation of clusters as well as the phenomenon behind it. The formation of delay-induced synchronized clusters may be because of inter-cluster couplings, instead of coupling between synchronized units [5,16]. Introduction of a delay may result ...

We study impact of multiplexing on the global phase synchronizability of different layers in the delayed coupled multiplex networks. We find that at strong couplings, the multiplexing induces the global synchronization in sparse networks. The introduction of global synchrony depends on the connection density of the layers being multiplexed, which further depends on the underlying network architecture. Moreover, multiplexing may lead to a transition from a quasiperiodic or chaotic evolution to a periodic evolution. For the periodic case, the multiplexing may lead to a change in the period of the dynamical evolution. Additionally, delay in the couplings may bring upon synchrony to those multiplex networks which do not exhibit synchronization for the undelayed evolution. Using a simple example of two globally connected layers forming a multiplex network, we show how delay brings upon a possibility for the inter layer global synchrony, that is not possible for the undelayed evolution.p-1

We investigate cluster synchronization in coupled map networks in the presence of heterogeneous delays. We find that while the parity of heterogeneous delays plays a crucial role in determining the mechanism of cluster formation, the cluster synchronizability of the network gets affected by the amount of heterogeneity. In addition, heterogeneity in delays induces a rich cluster pattern as compared to homogeneous delays. The complete bipartite network stands as an extreme example of this richness, where robust ideal driven clusters observed for the undelayed and homogeneously delayed cases dismantle, yielding versatile cluster patterns as heterogeneity in the delay is introduced. We provide arguments behind this behavior using a Lyapunov function analysis. Furthermore, the interplay between the number of connections in the network and the amount of heterogeneity plays an important role in deciding the cluster formation.

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