Complex networks play an increasingly important role in our lives, including power grids, the Internet, transportation, cooperation, gene regulatory and artificial neural networks, to name but a few. In fact, most of the things that drive our lives, like energy, information and even water, are delivered to us by a network structure. These networks are usually large, complex, and structured in such a way that there can be important operational consequences due to the manner in which they behave over time.
The complexity of the 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 (non-linear) 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, synchronization, and chaos control for complex networks, as well as their applications in, for example, communication, and bio-informatics.
The overall aim of this special issue is to bridge the gap between computer mathematics and network dynamics control. There were more than 20 papers submitted to this special issue from control engineers, mathematicians, computer scientists, and biologists. After a rigorous peer review process, 10 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.
Complex network-like structures appear in a wide variety of technological, social and biological systems. Some important examples in computer science include the Internet, peer-to-peer systems, and e-mail networks. An interesting dynamic process taking place in complex networks is the spontaneous spreading of information {via} rumour-like mechanisms. In addition to its relevance to social sciences, such a mechanism also forms the basis of an important class of data dissemination algorithms distributed in computer systems, and these algorithms are generally known as gossip, or epidemic, protocols. In the paper, ‘Dynamics of gossip-like information dissemination in complex computer networks’, by Nekovee and Moreno, the dynamics of a gossip-like process is investigated for information dissemination in complex computer networks. Large-scale Monte Carlo simulations of this process are carried out on top of a scale-free network topology as a prototype model of networks with strongly heterogeneous degree distributions. The results are compared with simulations performed for random graphs that have a homogeneous degree distribution. In addition to these static networks, the spreading process is also investigated on time-dependent networks created by mobile wireless nodes (mobile ad-hoc networks). New insights are provided on how the dissemination dynamics is affected by the complex interplay between network structure, mobility, and the spreading process.
Despite the availability of a large number of mathematical models describing the spreading of human immunodeficiency virus (HIV), a good understanding of the spreading dynamics through numerical analysis is still a major challenge. It is essential to combine epidemiological processes with sociological models and network sciences. The true incidence of the HIV/acquired immuno-deficiency syndrome (AIDS) epidemic is quite uncertain since many people may be unaware of their infection. Moreover, HIV has a very long asymptotic incubation period which makes study of the actual infection spreading a very complicated task. The various routes of infection and the inhomogeneity of the involved population pose additional problems. In the paper, ‘Stochastic simulation of HIV population dynamics through complex network modelling’, by Sloot et al., a parameterized complex network (CN) model is presented to describe the dynamics of HIV spreading. The model takes into consideration all the existing kinds of HIV spreading. All the network parameters are taken from the medical literature and fixed during the numerical experiments. The experiments show a promising correspondence between the model results and real demographic historical epidemiological data. The proposed CN model will be included into a generic system of predictive models for HIV infections and associated drug-ranking strategies.
Recently, particular attention has been focussed on the problem of making a network of coupled dynamical systems synchronize on a common evolution. Many real-world networks are characterized by evolving and adapting coupling gains which vary in time according to different environmental conditions, such as wireless networks of sensors that gather and communicate data to a central base station, control networks of robots when the conditions change unexpectedly, and social insect colonies that have many of the properties of adaptive networks. In the paper, ‘Adaptive synchronization of complex networks’, by De Lellis et al., a set of novel adaptive strategies are proposed and validated numerically for synchronization and consensus of complex networks of dynamical systems. Both centralized and decentralized strategies are presented where the adaptation law is based on, respectively, global and local information at the network nodes. All the proposed adaptive techniques are then validated using computer simulations on ensembles of two types of oscillators: Kuramoto and Rössler oscillators. It is shown that in both cases synchronization can be achieved with the adaptive gains reaching asymptotically finite steady-state values.
There are several computational challenges associated with empirical network analysis and network simulations. Networks are typically very sparse, high-dimensional structures. Modern computers are characterized by a striking contrast between the processing power of the CPU and the latency of main memory accesses. If the data processed are both large compared with processor caches and sparse or high-dimensional in nature, as is commonly the case in complex network research, the main memory latency can become a performance bottleneck. In the paper, ‘Efficient data structures for sparse network representation’, by Hyvönen et al., a cache efficient data structure, which is a variant of a linear probing hash table, is presented in order to represent edge sets of such networks. The performance benchmarks show that it is quite superior to its commonly used counterparts in the application. In addition, its memory footprint only exceeds the absolute minimum by a small constant factor. The practical usability of the proposed approach is well demonstrated in the study of very large real-world networks.
Networked control systems (NCS) have been an attractive research area for more than a decade. However, systematic investigations into the interactions among network components and the complex dynamics of network systems did not occur until recently. With much effort in this area, some basic concepts and approaches of complex networks have been developed in recent years to describe the connectivity, structure, and dynamics of complex systems. In the paper, ‘Dealing with network complexity in real-time networked control’, by Tian and Levy, a general framework is developed to deal with the complexity in real-time networked control applications. Such a framework consists of four main steps: (1) apply a real-time queuing protocol to achieve predictable network-induced delay; (2) configure the worst-case communication delay; (3) activate packet loss compensation when a packet is lost; and (4) compensate for the predictable network-induced delay in control design. In the proposed framework, co-design of network and control is emphasized as an effective way to simplify the network behaviour and consequently to maximize the performance of the overall NCSs.
In recent years, complex dynamical networks (CDNs) have attracted more and more attention from many researchers for their theoretical and practical values, and one of the most significant dynamical properties of CDN is the synchronization motion of its dynamical elements. In practice, the information flow in CDN is not instantaneous. As a result of the finite speed of signal transmission, time delays are often unavoidable in complex networks. These time delays may lead to oscillation, divergence and even instability. In the paper, ‘Further improvement on synchronization stability of complex networks with coupling delays’, by Mou et al., the synchronization problem is revisited for complex delayed networks. With the idea of delay fractioning, a novel Lyapunov–Krasovskii functional is constructed to obtain a new criterion, which is formulated in the form of a linear matrix inequality (LMI). The results prove to be less conservative as delay fractioning becomes thinner. Furthermore, this criterion can be extended to synchronization stability analysis for CDN with time-varying structured uncertainties. Numerical examples are given to demonstrate the effectiveness and merits of the proposed method.
Cellular neural networks (CNNs) have found significant applications in many areas such as image processing, pattern recognition, and solving partial differential equations. Since time delays inevitably exist in the neural networks (NNs) and are frequently important sources of oscillation and instability, various efforts have been made in the past few years for the analysis of CNNs with various time delays. Nevertheless, there is much room to reduce the possible conservatism. In the paper, ‘Improved global robust delay-dependent stability criteria for delayed cellular neural networks’, by Yue et al., the global robust asymptotic stability is studied for a class of continuous-time CNNs with time-varying delay and norm-bounded parameter uncertainties. Some new stability criteria are presented by using the Lyapunov functional method and the convexity of the matrix equation. Two numerical examples are presented to show the effectiveness and applicability of the proposed approach. It can be seen from the comparison results given in the examples that the proposed method can lead to much less conservative results than those by the existing methods.
Exploring network topology is crucial for understanding and predicting network dynamics. An important topic in pinning control of complex dynamical networks is to study the relationship between network topology and effective pinning strategy. It has been shown for a class of undirected dynamical networks that applying local linear feedback injections to a small fraction of the most highly connected vertices is much more effective than pinning the same number of randomly selected vertices. So far, however, vertex degree as a general index has exploited only little information of the relationships among vertices. In the paper, ‘Pinning control of directed dynamical networks based on ControlRank’, by Lu and Wang, a new effective pinning scheme is developed for directed networks. The ControlRank (CR) is a vertex centrality index exploring the link structure of the directed networks. Simulation study is given on a scale-free directed dynamical network, which demonstrates that it is much more effective to pin the vertex with largest CR than pinning the vertex with largest out-degree.
One of the main objectives of traffic analysis is to identify predictive models. These models are important because they make available practical tools (such as traffic simulators) that are widely required by TelCos (telecommunication companies) in core areas such as network engineering and marketing. In addition, those models can supply mathematical and physical insights into the phenomenon's nature, giving a better understanding, providing generalist views and thus making possible valuable innovation. In the paper, ‘Modelling user's activity in a real-world complex network’, by Pellicer-Lostao et al., a new statistical user model for Internet access is presented. Real traces of Internet traffic in a heterogeneous campus network are analysed. Three clearly different styles of an individual user's behaviour are found and their common features in three clusters are studied. A probabilistic mixture model is then built to implement the expected global behaviour for the different types of users. The implications of this emergent phenomenology are discussed in the field of multi-agent complex systems.
The study of synchronization problems between dynamical systems has long been an active field of research in many scientific and technical disciplines. Starting in the 1980s and following the development of the theory of deterministic chaos, the notion of synchronization has been extended to the case of more complex systems; for example, the large-scale and complex networks of chaotic oscillators, coupled systems exhibiting spatio-temporal chaos and autowaves, and the array of coupled neural networks with or without delays. In the paper, ‘On synchronization of coupled neural networks with discrete and unbounded distributed delays’, by Liu et al., the synchronization problem is dealt with for an array of linearly coupled neural networks with simultaneous presence of both the discrete and unbounded distributed time-delays. Different from the commonly used matrix norm theories (such as the M-matrix method), a unified linear matrix inequality (LMI) approach is developed to establish sufficient conditions for the coupled neural networks to be globally synchronized, where the LMIs can be easily solved by using the Matlab LMI toolbox and no tuning of parameters is required. It is also shown that the synchronization of coupled neural networks with bounded distributed delays is just a special case of the main results. A numerical example is provided to show the usefulness of the proposed global synchronization condition.
This special issue is a timely reflection of the research progress in the area of complex networks. Finally, we would like to acknowledge all authors for their efforts in submitting high-quality papers. We are also very grateful to the reviewers for their thorough and on-time reviews of the papers. Last, but not least, our deepest gratitude goes to Professor E.H. Twizell, Editor-in-Chief of the International Journal of Computer Mathematics for his consideration, encouragement, and advice to publish this special issue.