Better synchronizability predicted by crossed double cycle

Abstract

In this brief report, we propose a network model named crossed double cycles, which are completely symmetrical and can be considered as the extensions of nearest-neighboring lattices. The synchronizability, measured by eigenratio R, can be sharply enhanced by adjusting the only parameter, crossed length m. The eigenratio R is shown very sensitive to the average distance L, and the smaller average distance will lead to better synchronizability. Furthermore, we find that, in a wide interval, the eigenratio R approximately obeys a power-law form as R ∼ L 1.5 .PACS numbers: 89.75, 05.45.Xt Synchronization is observed in a variety of natural, social, physical and biological systems [1], and has found applications in a variety of field including communications, optics, neural networks and geophysics [2,3,4,5,6,7]. The large networks of coupled dynamical systems that exhibit synchronized state are subjects of great interest. In the early stage, the corresponding studies are restricted to either the regular networks [8,9], or the random ones [10,11]. However, recent empirical studies have demonstrated that many real-life networks can not be treated as regular or random networks. The most important two of their common statistic characteristics are called small-world effect [12] and scale-free property [13]. Therefore, very recently, most of the studies about network synchronization focus on complex networks, and find that the networks of small-world effect and scalefree property may be easier to synchronize than regular lattices [14,15,16]. One of the ultimate goals in studying network synchronization is to understand how the network topology affects the synchronizability. In the simplest case (see below), the network synchronizability can be well measured by the eigenratio R [17,18,19,20], thus the above question degenerates to understanding the relationship between network structure and its eigenvalues. Since there are countless topological characters for networks, a natural question is addressed: what is the most important factor by which the synchroizability of the system is mainly determined? Some previous works indicated the average distance L[21] is one of the key factors. However, the consistent conclusion have not been achieved [19,22,23,24,25]. Another extensively studied one is the network heterogeneity, which can be measured by the variance of degree distribution or betweenness distribution [26,27]. Some detailed comparisons among various networks have been done, indicating the network synchronizability will be better with smaller heterogeneity [24,28,29]. However, a well-known counterexample is the regular networks with homogeneous structure while displaying very poor synchronizability. Because of the networks used for comparing in previous studies are of both varying average distances and degree variances, the strict and clear conclusions can not be achieved. In addition, some researchers deem that the more intrinsic ingredient leading to better synchronizability is the randomicity [30], that is to say, ...

Enhanced synchronizability via age-based coupling

“…With the partial knowledge of relations between network structure and its synchronizability [4,5,6,7,8], scientists have proposed many methods to enhance the network synchronizability [9,10,11,12,13,14,15,16,17,18]. Generally speaking, these methods can be divided into two classes, one is to modify the network structure [9,10,11], the other is to regulate the coupling pattern [12,13,14,15,16,17,18]. In the former class, networks are modified either to shorten the average distance [10] or to eliminate the maximal betweenness [9,11].…”

Enhancing the network synchronizability

“…Based on this observation, Zhou et al [40] investigated the synchronizability of a network model named crossed double cycles (CDCs), not only testifying the relationship between the average distance and the network synchronizability but also providing a possible way to make network more synchronizable.…”

Control for a synchronization-desynchronization switch

“…According to previous studies, the topology of networks plays an important role in synchronization, and the degree-heterogeneous networks are usually difficult to synchronize [14][15][16]. Many methods have been proposed to enhance network synchronizability, such as (i) slightly modifying the network structure, including removing nodes with maximal betweenness [17], dividing hub nodes [18], deleting overload edges [19], rewiring network edges [20,21], and shortening the average distance [22]; or (ii) properly introducing weight and directionality to each link (edge) [23][24][25][26]. In addition, the so-called pinning control, which usually refers to a local feedback method by inputting an external signal and stabilizing the network to a synchronous state (or a homogeneous steady state), has been verified to be very effective [27][28][29][30][31].…”