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Owing to the sheer size of large and complex networks, it is necessary to reduce the information to describe essential properties of vertices and edges, regions, or the whole graph. Usually this is done via , i.e., a single number, or a series of numbers, catching the relevant and needed information. In this chapter we will give a list of statistics which are not covered in other chapters of this book, like distance-based and clustering statistics. Based on this collection we are going to classify statistics used in the literature by their basic types, and describe ways of converting the different types into each other.

A fundamental question in comparative network analysis is whether two given networks have the same structure.

The starting point in network analysis is not primarily the mathematically defined object of a graph, but rather almost everything that in ordinary language is called ‘network’. These networks that occur in biology, computer science, economy, physics, or in ordinary life belong to what is often called ‘the real world’. To find suitable models for the real world is the primary goal here. The analyzed real-world networks mostly fall into three categories.

A graph can be associated with several matrices, whose eigenvalues reflect structural properties of the graph. The adjacency matrix, the Laplacian, and the normalized Laplacian are in the main focus of spectral studies. How can the spectrum be used to analyze a graph?

Intuitively, a complex network is if it keeps its basic functionality even under failure of some of its components. The study of robustness in networks is important because a thorough understanding of the behavior of certain classes of networks under failures and attacks may help to protect, for instance, communication networks like the Internet against assaults or to exploit weaknesses of metabolic networks in drug design.