Networks have been a fruitful and prosperous research subject in several scientific disciplines, particularly in physics, over the past two decades. The report under review surveys recent progress in the study of networks. There have been a number of books and articles addressing the development of complex networks, with interest in a variety of phenomena and emphasis on various kinds of network models. This report is intended to bring in new research lines, such as weighted networks and spatial networks. A weighted network, with a real number associated to each link, takes into account the fact that in real situations a network is often associated with a large heterogeneity in the capacity and intensity of the connections. A spatial network is concerned with, among other things, the spatial structure or constraint from the geographical embedding or Euclidean distance between nodes.

In addition, a substantial part of this article is devoted to the dynamical behavior of networks, such as collective synchronized dynamics in complex networks. Interest in this topic, according to the authors, is a new trend for research. Related phenomena include the evidence that some brain diseases are the result of an abnormal and sometimes abrupt synchronization of a large number of neural populations. Moreover, a series of topics which currently attract much attention in the science community are also surveyed in Section 7. They include the problem of building manageable algorithms to find community structures, the issue of searching within a complex network, and the modeling of adaptive networks. One example of community structure is the tightly connected groups of nodes in a social network which represents individuals belonging to social communities. Searching within a network is related to how to reach one node from another node in the network. Adaptive networks deal with cases in which the topology of the network is allowed to evolve and adapt in time. This notion is needed in modeling certain phenomena, such as genetic regulatory networks, ecosystems, and financial markets.

This 130-page article covers the structure of complex networks, static and dynamic robustness, spreading processes, synchronization and collective dynamics, and applications, respectively, in Sections 2 through 6. Each of these sections also contains several subsections. For example, Section 2 gives detailed definitions for the structure of complex networks, including node degree, correlations, clustering, and motifs, and introductions to weighted networks and spatial networks, etc. This report is a useful contribution and a valuable reference tool for the mathematics community.