The structure or interconnection pattern of a network can be represented by a graph. This chapter mainly focuses on general properties of graphs that are of interest to the modeling of complex networks.

Around 2000, several remarkable, quite universal phenomena observed in many different complex networks gave birth to a new discipline network science, which encompasses parts of physics, mathematics, engineering, biology, medical sciences, social sciences, and even finance. The most important universal complex network characteristics are (1) a power-law or scale-free degree distribution (first reported via Internet measurements by Faloutsos et al. (1999)), (2) small-world structure proposed by Watts and Strogatz (1998), (3) preferential attachment as a simple driver to explain the power-law degree distribution of scale-free graphs (Barabási and Albert, 1999), (4) clustering and community structure since most complex networks are networks of networks and (5) a high robustness against random failures, but a vulnerability against targeted attacks of mainly the hubs (high-degree nodes).

After the book of Watts (1999), many articles and books on complex networks have followed (see e.g. Strogatz (2001); Barabási (2002); Dorogovtsev and Mendes (2003); Barrat et al. (2008); Dehmer and Emmert-Streib (2009); Newman (2010); Cohen and Havlin (2010); Estrada (2012), and references in these books). Those books overview the current state of the art in network science and present many applications to, for example, the Internet, the World Wide Web, and brain, financial, social and biological networks.