Scaling of internode distances in weighted complex networks

We extend the previously observed scaling equation connecting the internode distances and nodes' degrees onto the case of weighted networks. We show that the scaling takes a similar form in the empirical data obtained from networks characterized by different relations between node's strength and its degree. In the case of explicit equation for s(k) (e.g. linear or scale-free), the new coefficients of scaling equation can be easily obtained. We support our analysis with numerical simulations for Erdos-Renyi random graphs with different weight distributions.Comment: 9 pages, 4 figures, submitted to International Journal of Modern Physics

Anomalous oscillations of average transient lifetimes near crises

It is common that the average length of chaotic transients appearing as a consequence of crises in dynamical systems obeys a power low of scaling with the distance from the crisis point. It is, however, only a rough trend; in some cases considerable oscillations can be superimposed on it. In this letter we report anomalous oscillations due to the intertwined structure of basins of attraction. We also present a simple geometrical model that gives an estimate of the period and amplitude of these oscillations. The results obtained within the model coincide with those yielded by computer simulations of a kicked spin model and the Henon map.Comment: 5 pages, 4 figure

How to calculate the main characteristics of random uncorrelated networks

We present an analytic formalism describing structural properties of random uncorrelated networks with arbitrary degree distributions. The formalism allows to calculate the main network characteristics like: the position of the phase transition at which a giant component first forms, the mean component size below the phase transition, the size of the giant component and the average path length above the phase transition. We apply the approach to classical random graphs of Erdos and Renyi, single-scale networks with exponential degree distributions and scale-free networks with arbitrary scaling exponents and structural cut-offs. In all the cases we obtain a very good agreement between results of numerical simulations and our analytical predictions.Comment: AIP conference proceedings format, 17 pages, 6 figure

Hierarchical Cont-Bouchaud model

We extend the well-known Cont-Bouchaud model to include a hierarchical topology of agent's interactions. The influence of hierarchy on system dynamics is investigated by two models. The first one is based on a multi-level, nested Erdos-Renyi random graph and individual decisions by agents according to Potts dynamics. This approach does not lead to a broad return distribution outside a parameter regime close to the original Cont-Bouchaud model. In the second model we introduce a limited hierarchical Erdos-Renyi graph, where merging of clusters at a level h+1 involves only clusters that have merged at the previous level h and we use the original Cont-Bouchaud agent dynamics on resulting clusters. The second model leads to a heavy-tail distribution of cluster sizes and relative price changes in a wide range of connection densities, not only close to the percolation threshold.Comment: 10 pages, 6 figure