Study On Some Issues Of Dynamics Of Complex Networks And Its Applications

In the last decade, complex networks have received increasing attention from all fields of the humanities and sciences. Complex networks lie in everywhere in the real word, such as the World Wide Web, social networks and food-webs. Therefore, the study of complex networks is of great significance for us. As for the exploitation of the complex networks, we should study the commonness of different complex networks and the general method in making analysis of com-plex networks. As a special case of complex networks, neural networks have been a focal subject for research due to its wide applications in signal processing, im-age processing, pattern recognition and nonlinear dynamics.In the thesis, the dynamical analysis and application problems are discussed for several classes of complex networks. At first, the stability analysis of a class of neural networks is investigated. Then, some control and synchronization prob-lems of neural networks and complex networks are discussed. Finally, we study the applications of complex networks in image encryption and evolution com-putation. The compendious frame and description of the thesis are given as follows:(1) Stability analysis of a class of discrete-time networks with ran-domly mixed delays based on Bernoulli stochastic variablesThe stability analysis problem for a new class of discrete-time neural net-works with randomly discrete and distributed time-varying delays has been inves-tigated based on the Bernoulli stochastic variables. Compared with the previous work, the distributed delay is assumed to be time-varying. Moreover, the effects of both variation range and probability distribution of mixed time-delays are taken into consideration in the proposed approach. The stochastic disturbances are described in term of a Brownian motion, and the mixed time-varying de-lays are considered by introducing two Bernoulli stochastic variables. By using some new analysis techniques and a novel Lyapunov-Krasovskii function, some delay-distribution-dependent conditions are derived to ensure that the stochastic discrete-time neural network with randomly mixed time-varying delays is stable in mean square. A numerical example is provided to demonstrate the effective-ness and the applicability of the proposed method.(2) Synchronization of continuous-time stochastic neural networks with discrete and distributed delaysThe adaptive scheme to the problem of lag synchronization and param-eters identification for stochastic chaotic neural networks with discrete delay and distributed time-varying delays is investigated in detail. The chaotic neural networks are subjected to stochastic disturbances described in terms of a Brow-nian motion. By adaptive feedback technique, a simple, rigorous and systematic synchronization-based parameters identification scheme is proposed to solve the problem addressed. The method using in this Letter is simple to implement in practice. The variable feedback strength will be automatically adapted to a suitable strength. Moreover, we can adjust the synchronization speed and pa-rameters identification speed by regulating the adaptive gain. In the end, the numerical simulations are given to show the feasibility of the proposed approach. On the other hand, exponential synchronization criteria for stochastic jumping chaotic neural networks (SJCNNs) with mixed delays and sector non-linearities are presented. Based on Lyapunov function and free-weighting matrices, suffi-cient conditions have been developed to ensure the global exponential stability for the error system,and thus the drive system synchronizes with the response system. The activation functions are assumed to be neither monotonic, nor dif-ferentiable, nor bounded. Finally, the numerical example is given to show the effectiveness of the proposed method.(3) Synchronization of complex networks with discrete, distributed delays and jumping stochastically hybrid couplingA general model of an array of Markovian jumping neural networks with stochastically hybrid coupling and mixed time-varying delays is proposed. The stochastically hybrid coupling is composed of constant coupling, time-varying coupling and distributed time-varying time-varying coupling, which are all sub-jected to stochastic disturbances. The network switches from one mode to an-other according to a Markovian chain with known transition probability. Based on adaptive control, some synchronization criteria have been derived to guaran-tee the synchronization of an array of jumping neural networks with stochasti-cally hybrid coupling and mixed time-varying delays in mean square. In the end, the simulations examples are provided to show the effectiveness of the derived results.(4) Pinning control of fractional-order weighted complex networksThe pinning control problem of a class of fractional-order weighted com-plex dynamical networks has been investigated in detail. The general strategy is to apply a feedback control scheme to a small fraction of the network nodes. By utilizing eigenvalue analysis approach and fractional-order stability theory, we employ the numerical algorithms for fractional-order complex networks ad-dressed in this paper to establish some local stability criteria and valid stability regions of such pinned fractional-order networks. The analytical analysis show that the largest eigenvalue of matrix R determines the control of the weighted fractional-order complex networks. By seeking an appropriate R, overall cou-pling strength c and fractional-order q, we are able to achieve our goal. It is surprising to find that a network can realize stabilization under a suitable q, even without controller. In addition, it is interesting to find that the fractional-order q, the control gain matrix D, the tunable weight parameterβ, the overall coupling strength c, the specially pinning of largest nodes will effectively affect the convergence rate of controlling fractional-order complex dynamical networks. Some simulation examples in scale-free networks are exploited to demonstrate the applicability of the proposed results.(5) Image encryption using chaotic coupled map lattices with time-varying delaysThe complex networks theory is applied to image encryption scheme. We have proposed a new image cipher based on the confusion-diffusion structure, which utilizes tent map and coupled map lattices with time-delay. The encryp-tion scheme is related to the plain image, cipher image and chaos-controlled time-varying delay. Thus, different plain-images result in distinct control pa-rameters and key streams. The confusion and diffusion performance has been enhanced and the cryptosystem can resist known-plaintext, chosen-plaintext at-tacks, differential attacks, statistical attacks and brute-force attacks effectively. Both theoretical analysis and experiments have verified that the new cipher technique possesses high security level, therefore having excellent potential for practical image encryption applications.(6) A controllable probabilistic particle swarm optimizationThe complex networks theory is used to design a particle swarm optimiza-tion. Based on the information of fitness value, an evolutionary state function is defined and computed, which offers an effective way to control inertia weight. As illustrated in the benchmark tests, the adaptive control of the inertia weight enables the PSO more efficient, providing an improved convergence speed in terms of FEs to reach acceptable solutions for benchmark functions. A velocity updating equation with Bernoulli stochastic variables is proposed to make the particles learn from different strategies. The learning strategies are automat-ically selected efficiently using a competitive penalized method. Furthermore, an elite local learning approach is developed to lead the swarm to refine con-verging solutions efficiently. The searching radius is switched between different values governed by a Markov chain in ELLA. The substantially improved global solution accuracy as a result of the ESF, PSO with controllable probability and ELLA are verified in both unimodal and multimodal problems. Note that the ESF, PSO with controllable probability and ELLA are easy to set and require no burden to implement. Therefore, the CPPSO is simple and easy to use as the standard PSO, whereas it offers in substantially improved performance in terms of convergence speed and solution accuracy.