Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics and function of processes executed on the network. The analysis, discrimination, and synthesis of complex networks rely on the use of measurements capable of expressing the most relevant topological features. This article presents a survey of such measurements. It includes general considerations about complex network characterization, a brief review of the principal models, and the presentation of the main existing measurements organized into classes. Depending on the network and the analysis task one has in mind, a specific set of features may be chosen. It is hoped that the present survey will help the identification of the most suitable measurements. Characterization Representation µ = 1 2 3 M µ µ µ µ Figure 1 : The mappings from a complex network to a feature vector. Generic mappings can be used in order to obtain the characterization of the network in terms of a suitable set of measurements. In case the mapping is invertible, we have a complete representation of the original structure. but also placed special interest on the dynamical evolution of network topology. Supported by the availability of high performance computers and large data collections on real world networks, results like the discovery of the scale-free structure of the Internet [10] and the WWW [11, 12] were of major importance for the increased interest on the new area of complex networks, whose growing importance is substantiated by the large number of related publications which appeared in recent years. A good review of such developments can be obtained from three excellent surveys [13, 14, 15] , introductory papers [16, 17, 18, 19] and books [20, 21, 22] . One of the main reasons why complex networks have become so popular is their flexibility and generality for representing virtually any natural structure, including those undergoing dynamical changes of topology. As a matter of fact, every discrete structure such as lists, trees, or even lattices, can be represented as graphs. It is thus little surprising that many investigations in complex network involve the representation of the structure of interest as a complex network, followed by an analysis of the topological features of the obtained representation performed in terms of a set of informative measurements. Another interesting problem consists of measuring the structural properties of evolving networks in order to characterize how the connectivity of the investigated structures change along the process. Both such activities can be understood as being aimed at the topological characterization of the studied structures. Another related application is to use the obtained measurements in order to discriminate between different classes of structures, which is a problem in the area of pattern recognition. Even when modeling networks, it is often necessary to compare the realizations of the model with real networks, which can be done by comparing respective measurements. Provided the measurements are comprehensive (ideally the representation by the measurements should be one-to-one or invertible), the fact that the simulated networks yield measurements similar to those of the real counterparts supports the validity of the model. Network measurements are therefore essential as a direct or subsidiary resource in many network investigations. Figure 1 shows the mapping of a generic complex network into the feature vector µ, i.e. a vector of related measurements
The analysis of general complex networks is well-described, e.g. in REF .
Characterization of complex networks: A survey of measurements
Advances in Physics
Physics