Analytic solution for the average path length in a large class of random graphs is found. We apply the approach to classical random graphs of Erdös and Rényi (ER) and to scale-free networks of Barabási and Albert (BA). In both cases our results confirm previous observations: small world behavior in classical random graphs lER ∼ ln N and ultra small world effect characterizing scale-free BA networks lBA ∼ ln N/ ln ln N . In the case of scale-free random graphs with power law degree distributions we observed the saturation of the average path length in the limit of N → ∞ for systems with the scaling exponent 2 < α < 3 and the small-world behaviour for systems with α > 3. 05.50.+q

We study the biased random-walk process in random uncorrelated networks with arbitrary degree distributions. In our model, the bias is defined by the preferential transition probability, which, in recent years, has been commonly used to study the efficiency of different routing protocols in communication networks. We derive exact expressions for the stationary occupation probability and for the mean transit time between two nodes. The effect of the cyclic search on transit times is also explored. Results presented in this paper provide the basis for a theoretical treatment of transport-related problems in complex networks, including quantitative estimation of the critical value of the packet generation rate.

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Analyzing real data on international trade covering the time interval 1950-2000, we show that in each year over the analyzed period the network is a typical representative of the ensemble of maximally random weighted networks, whose directed connections (bilateral trade volumes) are only characterized by the product of the trading countries' GDPs. It means that time evolution of this network may be considered as a continuous sequence of equilibrium states, i.e., a quasistatic process. This, in turn, allows one to apply the linear response theory to make (and also verify) simple predictions about the network. In particular, we show that bilateral trade fulfills a fluctuation-response theorem, which states that the average relative change in imports (exports) between two countries is a sum of the relative changes in their GDPs. Yearly changes in trade volumes prove that the theorem is valid.