Analysis and synchronization of complex networks

Complex networks do exist in our lives. The brain is a neural network. The global economy is a network of national economies. Computer viruses routinely spread through the Internet. Food-webs, ecosystems and metabolic pathways can be represented by networks. Energy is distributed through transportation networks in living organisms, man-made infrastructures and other physical systems. Dynamic behaviours of complex networks, such as stability, periodic oscillation, bifurcation or even chaos, are ubiquitous in the real world and often reconfigurable. Networks have been studied in the context of dynamical systems in a range of disciplines. However, until recently there has been relatively little work that treats dynamics as a function of network structure, where the states of both the nodes and the edges can change, and the topology of the network itself often evolves in time. The complexity of networks poses many challenges for scientists and engineers. In particular, advanced societies have apparently become dependent on large infrastructure networks to such an extent that it is difficult to plan and control these networks securely with our current capabilities. The recent spate of power grid failures and virus attacks on the Internet illustrate the need for research on modelling, analysis of behaviours, systems theory, planning and control in such networks. Numerous fundamental questions have been addressed about the connections between network structure and (nonlinear) dynamic properties including stability, bifurcations, controllability and other observable aspects. However, some major problems have not been fully investigated, such as the behaviour of stability, synchronisation and chaos control for complex networks, as well as their applications in, for example, communication and bioinformatics. Complex networks have already become an ideal research area for control engineer, mathematicians, computer scientists and biologists to manage, analyse and interpret functional information from real-world networks. Sophisticated computer system theories and computing algorithms have been exploited or emerged in the general area of computer mathematics, such as analysis of algorithms, artificial intelligence, automata, computational complexity, computer security, concurrency and parallelism, data structures, knowledge discovery, DNA and quantum computing, randomisation, semantics, symbol manipulation, numerical analysis and mathematical software. This special issue aims to bring together the latest approaches to understanding complex networks from a dynamic system perspective. We have solicited submissions to this special issue from control engineers, mathematicians, physicists and computer scientists. After a rigorous peer review process, seven papers have been selected that provide solutions, or early promises, to manage, analyse and interpret functional information from real-world networks. These papers have covered both the practical and theoretical aspects of complex networks in the broad areas of artificial intelligence, mathematics, statistics, operational research and engineering. The complexity of complex networks lies mainly in two aspects: structure and dynamics, and these two aspects are linked to each other. A mathematical term ‘graph’ has been used to describe the interactions between individuals in complex systems. Besides regular graph topologies (e.g. the k-nearest neighbourhood, complete graph and star-like wiring), complex structures (especially those including randomness and evolution) have attracted much research attention, and there has been a rich body of literature on the synchronisation problem in complex networks of coupled systems. In the paper ‘Synchronization in complex networks of coupled systems with directed topologies’ by Lu and Chen, a cohesive overview of the key approaches is provided for the synchronisation analysis of complex networks with directed, asymmetric and reducible graphs, including continuous-time and discrete-time models, in a unified framework. The methods are introduced, which rely heavily on matrix analysis and algebraic graph theory in dynamical theory, for synchronisation analysis and synchronisability discussions. The reviewed methods and results describe how the interaction structure among individuals affects the global dynamics. Moreover, the influence of coupling delay on synchronous dynamics is studied. As a by-product, it is shown that the pinning