Journal of Physical Studies 10(4), 247–289 (2006) DOI: https://doi.org/10.30970/jps.10.247 COMPLEX NETWORKS Yu. Holovatch{1,2}, O. Olemskoi{3,4}, C. von Ferber{5,6}, T. Holovatch{7}, O. Mryglod{1,8}, I. Olemskoi{4}, V. Palchykov{1} 1Insitute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 79011 Lviv, Ukraine 2Institut für Theoretische Physik, Johannes Kepler Universität Linz, 4040 Linz, Austria {3}Institute for Applied Physics, National Academy of Sciences of Ukraine, 79011 Sumy, Ukraine {4}Sumy State University, 40007 Sumy, Ukraine {5}Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, United Kingdom {6}Theoretische Polymerphysik, Universität Freiburg, 79104 Freiburg, Germany {7}Ivan Franko National University of Lviv, 79000 Lviv, Ukraine {8}Lviv Polytechnic National University, 79013 Lviv, Ukraine

We review recent results obtained in empirical numerical and theoretical studies of complex networks that characterize many systems in nature and society. Examples are the Internet, the world wide web, and food webs, as well as networks of neurons, of the metabolism of biological cells, of transportation, of distribution, of citations and many more. The empirical and theoretical analysis of general complex networks has only recently been approached by physicists, seminal papers in this field dating from the late 1990s. In this course the perspective has moved from the analysis of single small graphs and properties of individual vertices and edges to the consideration of statistical properties of ensembles of graphs (networks). This induced the need for the introduction of methods as they are provided by statistical physics.

In this review we sketch the evolution of network science and present some natural and man-made networks in detail, their main features and quantitative characteristics. Starting with three basic network models, the Erdös-Renyi random graph, the Watts-Strogatz small world network, and the Barabási-Albert scale free network, we introduce the statistical mechanics of complex networks. We consider phase transitions and critical phenomena on complex networks and, in particular, we elaborate network phenomena that can be described in terms of percolation theory.

PACS number(s): 01.90.+g, 89.75.Hc, 89.75.Da, 05.50.+q, 64.60.Ak

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