How to characterizethe stochastic features of complex networks is discussed and theirpatterns in dynamic evolution are found out from four aspects:classical models and network characters, empirical studies andmathematical modeling, theoretical methods for analyzing networks,and human behaviors in network society. It is emphasized how agood model of networks describes the evolutionary mechanisms ofactual networks and how various measures characterize thestructural properties of actual networks. Based on stochastic anddynamic features of complex networks, the existing theoreticalmethods are commented and possible developments of analyticmethods are discussed. The modeling of human dynamics and itsinfluence on collective performances in the networked society arefinally reviewed.

In the past ten years,the fast development of complex network theory has provided a goodsupport for the study of complexity and complex systems, sincethey describe clearly the important characteristics of complexsystems, and show bright prospects in theory and applications.This paper presents mainly the application of graph theory tocomplex networks, especially to the synchronization problem ofcomplex networks. First, its application to the estimations ofsmallest nonzero, largest eigenvalues and synchronizability indexof certain graphs are commented, followed by the effects ofsubgraph and graph eigenvector in the estimation ofsynchronizability index. Furthermore, the complexity between therelationships of synchronizability and network structuralparameters are discussed via two simple graphs, and the effects ofcomplementary graph, edge-addition and graph operation on thesynchronization of complex networks are elaborated. Finally, somepossible development directions in complex networks are predictedfrom the viewpoint of graph and control theory.

Complex networks are everywhere in the real world, e.g. the Internet, power grids, business networks, biological neural networks, social networks etc., and all such complex networks have close relationships with our daily life. A further investigation of these complex networks may not only help develop many important science disciplines but also bring on profound transformations of our life styles. Synchronization is a very important type of nonlinear phenomena existing ubiquitously in nature, and there are abundant cases of synchronization in various complex networks. Over the past ten years, scientists from different fields have further explored the synchronization of complex networks from different points of view, and main advances in related studies, including theories, approaches, applications and prospects, are briefly reviewed in the present paper in an attempt to further promote the current research on the synchronization of complex networks.

The spreading andpropagation of viruses, rumors and opinions etc. on differenttypes of real-life networks affect significantly human activitiesin social and economic environments, and thus the study ondynamical behaviors of various spreading phenomena in complexnetworks has become one of the main subjects in the field ofcomplex network theory. This paper briefly surveys the latestprogresses of the study on spreading dynamics on complex dynamicalnetworks in the aspects of adaptive avoiding, response delay,discrete sampling and growing of open-systems, in an attempt toassist the investigations on dynamical processes of epidemicspreading behaviors on complex networks.

Collective behaviorsare important emergent phenomena in complex systems, from whichmacroscopic spatiotemporal or functional structure will emerge viainteractions between microscopic individuals. Previous studiesindicated that the structure of complex network related with theinteraction between individuals had important effects oncollective behaviors in complex systems. Based on the researcheson Ising model of complex networks, recent studies on collectivebehaviors on complex networks are reviewed in this article. First,the critical phenomena emerged in Ising model is presented as wellas the effects of network's structure on phase transition. Second,efforts are focused on the application of Ising model to socialand economic systems, including opinion formation, herd behaviorand segregation phenomenon, with emphasis laid on the effects ofnetwork's structure on collective behaviors. Thirdly, theapproaches to detect community structure of networks based onIsing model are presented, and finally discussions and concludingremarks are given.

Evolutionary gametheory has various applications to social, economic, biologicaland ecological systems, and complex network theory is a newscientific research direction emerging recently to addressconceptual and technical problems in various complex systems. Thecombination of the evolutionary game theory with the complexnetwork theory is a focal research topic that has recentlyattracted lots of attentions and efforts. In this article, we givea brief overview of related achievements and developments in thefield of evolutionary games on complex networks, including modelsof evolutionary games, concerned topics, and the co-evolution ofnetwork structure and game dynamics, and so on. Finally, someremarks are made on future developments.