## Introduction

Networks surround us; we are built from networks and form part of networks. In general, a network is understood to be a set of nodes related to each other by certain links. However, beside this abstract structural description, the appearance of networks is amazingly diverse. One can look at networks of railways and electricity lines connecting power stations, food Webs and networks of neuronal cells, semantic Webs and networks of trade relationships^{1}.

The interest of science in networks has a long tradition. More recent examples are the foundation of random graph theory in mathematics (Erdös, Rényi) starting in the 1930s (Bollobás, 1985) and the emergence of social network analysis as a field in sociology in the 1960s and 1970s (Scott, 2001). Graph theory has long since been described as part of system theory (Laue, 1970). The last decade witnessed an additional increase in interest in describing and analyzing networks. Several books have appeared about the networked character of nature and society (Barabási, 2002; Bornholdt & Schuster, 2003; Buchanan, 2002; Dorogovtsev & Mendes, 2003; Huberman, 2001; Watts, 1999). Interestingly, many of these books are written or edited by physicists.

Once again, physics seems to be taking the initiative in discovering general laws in quite different phenomena. Like the theories of self-organization in the 1970s and 1980s and the complexity theories in the 1990s, network theories are used to establish a bridge in the explanations of different phenomena in society and nature. One can see this development as part of a more general approach of complexity theory^{2}. Physics contributes to the interdisciplinary field of complexity research. As I will discuss in this article, complex networks theory is an emerging specialty inside statistical and non-linear physics. Its aim, however, is to describe phenomena far outside physical processes. In talking about a bridge in the explanation of nature and society, one would expect to observe a diffusion of methods between social and natural sciences. The mutual reception of methods and concepts around complex social networks between physics and sociology is still in its early stages. Although social networks have been studied with methods from complex network theory (Barabási, Jeong, Néda, Ravasz, Schubert & Vicsek, 2002; Newman, 2001a, 2001b; Redner, 1998; Watts, 1999), few papers discuss concepts and methods of social network analysis on the one hand and (physical) complex network theory on the other (Watts, 1999; Watts, Dodds & Newman, 2002). I find the framework of models and concepts developed so far inside physics impressive enough to be worthy of the attention of scientists outside of physics. One aim of this paper is therefore to expose the findings of the physics community in the area of complex networks to a wider audience.

This paper was initially motivated by the sudden increase in the number of papers in physics journals dealing with the Internet and the Web. This raises the question of what makes the Web so interesting to a theoretical physicist. In fact, this development turned out to be an indication of the emergence of a new specialty inside statistical physics that best can be described as complex networks theory. It will be seen that the concept of connectivity is specifically shaped in this literature. By presenting some details of theories, methods and empirical studies in complex networks theory, it will possible to explore intersections between quantitative and qualitative research.

As previously mentioned, the Web plays a prominent role in this paper, since it served as a prominent example for network studies. The development of the Internet and the Web served as a trigger in the increase in the number of papers about complex networks, which is examined via a bibliometric analysis.

The main part of the paper is devoted to a review of results from the emerging field of complex networks theory. It contains sections about the topology and statistics of networks, growth and the evolution of networks, and dynamic processes on networks. The paper uses examples of the application of this knowledge to the Web and the Internet.