Enhancing synchronization in growing networks

Most real systems are growing. In order to model the evolution of real systems, many growing network models have been proposed to reproduce some specific topology properties. As the structure strongly influences the network function, designing the function-aimed growing strategy is also a significant task with many potential applications. In this letter, we focus on synchronization in the growing networks. In order to enhance the synchronizability during the network evolution, we propose the Spectral-Based Growing (SBG) strategy. Based on the linear stability analysis of synchronization, we show that our growing mechanism yields better synchronizability than the existing topology-aimed growing strategies starting from both artificial and real-world networks. We also observe that there is an optimal degree of new added nodes, which means adding nodes with neither too large nor too low degree could enhance the synchronizability. Furthermore, some topology measurements are considered in the resultant networks. The results show that the degree and node betweenness centrality from SBG strategy are more homogenous than those from other growing strategies. Our work highlights the importance of the functionaimed growth of the networks and deepens our understanding of it. Introduction. – Networks, despite their simplicity, represent the interaction structure among components in a wide range of real complex systems, from social relationships among individuals, to interactions of proteins in biological systems, and even the interdependence of function calls in large software projects. The network concept has been developed as an important tool for analyzing the relationship of structure and function for many complex systems in the last decades [1–5]. An interesting phenomenon observed in complex networks is synchronization [1]. As a kind of collective behavior, synchronization is often encountered in living systems, such as circadian rhythm, phase locking respiration with mechanical ventilator, phase locking of chicken embryo heart cells with external stimuli [6]. The early works on synchronization were concerned with only a small number of coupled oscillators. However, as the synchronization in many real-world systems is based on a large number of dynamical units interacting with a complex coupling (a)E-mail: [email protected] (b)E-mail: [email protected] structure, the research of synchronization on complex networks has been intensively studied in the past decade. Previous studies have given us an overview over the synchronizability of some well-known network models. In general, random networks have better synchronizability than regular networks, and Strogatz-Watts networks have better synchronizability than scale-free networks [7–9]. Actually, there are many factors that affect the network synchronizability. For example, average shortest distance is an important factor. Besides, the heterogeneity of the network is also one of the most influential factors determining the synchronizability [10,11]. Generally speaking, the less heterogeneous the network is, the better its synchronizability will be. With these understandings, many methods have been proposed to enhance synchronizability in complex networks. There are several main directions including designing strategies for coupling strength [12–17], modifying network structures [18–21], flipping the directionality [22–24] and so on. Each group of methods has its corresponding application field since the properties of real systems may form different operation constraints. Published in which should be cited to refer to this work. ht tp :// do c. re ro .c h Most of the former works on synchronization focus on given size networks. However, real-world networks are growing, which is evidenced by numerous systems including man-made and natural ones. Many growing mechanisms are proposed to reproduce some specific topology properties. For instance, in the pioneering work of Barabási-Albert (BA) network [3], the preferential attachment growing mechanism reproduces the power-law degree distribution observed in many realistic complex systems. However, in some technological networks such as electric power grids, the neural networks and even social networks, the introduction of new nodes are supposed to enhance the network functions [25,26]. In these cases, the topology properties of the network should be by-products during the improvement of the functions [27]. Therefore, the growing mechanisms for network functions are meaningful and have attracted much attention recently [28–30]. In this letter, employing synchronization as the typical function, we propose the Spectral-Based Growing (SBG) strategy. Note that this is only one choice of functional growing strategies and there are perhaps some other growing mechanisms which can yield similar results. We compare the new growing mechanisms with some existing topology-aimed growing mechanisms including the Preferential Attachment (PA), Reversed Preferential Attachment (RPA), Random Attachment (RA) mechanisms. We find that the SBG strategy leads to a better synchronizability starting from both artificial and realworld networks. Moreover, detailed study on some topology measurements of the resultant networks shows that the topology measurements such as the degree and node betweenness centrality in SBG networks are more homogeneous than those in the networks from other growing mechanisms. Interestingly, we observe that the synchronizability enhancement is strongly related to the degree of new added nodes. The synchronizability would weaken if the degree of the new nodes are too large or too low. Actually, there is an optimal degree for each given system and the optimal degree is approximately proportional to the average degree of the original networks. Finally, in order to overcome the drawback of trapping in local optimality of SBG method when a large number of nodes are added, we tested the simulated-annealing–based SBG (SSBG) strategy and found a further synchronizability improvement. Our work deepens our understanding of the function-aimed growth of networks and may have wide potential applications. Spectral-Based Growing (SBG) strategy. – Let us consider a system withN identical oscillators symmetrically coupled through a network. The equations of motion for the oscillator state vector x at each node i are