Piotr Fronczak

Average path length in random networks

Analytic solution for the average path length in a large class of uncorrelated random networks with hidden variables is found. We apply the approach to classical random graphs of Erdös and Rényi (ER), evolving networks introduced by Barabási and Albert as well as random networks with asymptotic scale-free connectivity distributions characterized by an arbitrary scaling exponent alpha>2. Our result for 2<alpha<3 shows that structural properties of asymptotic scale-free networks including numerous examples of real-world systems are even more intriguing than ultra-small world behavior noticed in pure scale-free structures and for large system sizes N-->infinity there is a saturation effect for the average path length.Analytic solution for the average path length in a large class of uncorrelated random networks with hidden variables is found. We apply the approach to classical random graphs of Erdös and Rényi (ER), evolving networks introduced by Barabási and Albert as well as random networks with asymptotic scale-free connectivity distributions characterized by an arbitrary scaling exponent α > 2. Our result for 2 < α < 3 shows that structural properties of asymptotic scale-free networks including numerous examples of real-world systems are even more intriguing then ultra-small world behavior noticed in pure scale-free structures and for large system sizes N → ∞ there is a saturation effect for the average path length. During the last few years random, evolving networks have become a very popular research domain among physicists [1, 2, 3, 4]. A lot of efforts were put into investigation of such systems, in order to recognize their structure and to analyze emerging complex properties. It was observed that despite network diversity, most of real web-like systems share three prominent structural features: small average path length (AP L), high clustering and scale-free (SF) degree distribution [1, 2, 3, 4, 5]. Several network topology generators have been proposed to embody the fundamental characteris-To find out how the small-world property (i.e. small AP L) arises, the idea of shortcuts has been proposed by Watts and Strogatz [13]. To understand where the ubiquity of scale-free distributions in real networks comes from, the concept of evolving networks basing on preferential attachment has been introduced by Barabási and Albert [6]. Recently Calderelli and coworkers [12] have presented another mechanism that accounts for origins of power-law connectivity distributions. It is interesting that the mechanism is neither related to dynamical properties nor to preferential attachment. Caldarelli et al. have studied a simple static network model in which each vertex i has assigned a tag h i (fitness, hidden variable) randomly drawn from a fixed probability distribution ρ(h). In their fitness model, edges are assigned to pairs of vertices with a given connection probability p ij , depending on the values of the tags h i and h j assigned at the edge end points. Similar models have been also analyzed in several other studies [14, 15, 16]. A generalization of the above-mentioned network models has been recently proposed by Boguñá and Pastor-Satorras [17]. In the cited paper, the authors have argued that such diverse networks like …

Biased random walks in complex networks: the role of local navigation rules.

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.

Limits of time-delayed feedback control

Abstract General features of stability domains for time-delayed feedback control exist, which can be predicted analytically. We clarify, why the control scheme with a single delay term can only stabilise orbits with short periods or small Lyapunov exponents, and derive a quantitative estimate. The limitation can be relaxed by employing multiple delay terms. In particular, the extended time delay autosynchronisation method is investigated in detail. Analytic calculations are in good agreement with results of numerical simulations and with experimental data from a nonlinear diode resonator.

Mean-field theory for clustering coefficients in Barabási-Albert networks.

We applied a mean-field approach to study clustering coefficients in Barabási-Albert (BA) networks. We found that the local clustering in BA networks depends on the node degree. Analytic results have been compared to extensive numerical simulations finding a very good agreement for nodes with low degrees. Clustering coefficient of a whole network calculated from our approach perfectly fits numerical data.

Origins of Taylor's power law for fluctuation scaling in complex systems.

Taylors fluctuation scaling (FS) has been observed in many natural and man-made systems revealing an amazing universality of the law. Here, we give a reliable explanation for the origins and abundance of Taylors FS in different complex systems. The universality of our approach is validated against real world data ranging from bird and insect populations through human chromosomes and traffic intensity in transportation networks to stock market dynamics. Using fundamental principles of statistical physics (both equilibrium and nonequilibrium) we prove that Taylors law results from the well-defined number of states of a system characterized by the same value of a macroscopic parameter (i.e., the number of birds observed in a given area, traffic intensity measured as a number of cars passing trough a given observation point or daily activity in the stock market measured in millions of dollars).

Universal scaling of distances in complex networks

Universal scaling of distances between vertices of Erdos-Rényi random graphs, scale-free Barabási-Albert models, science collaboration networks, biological networks, Internet Autonomous Systems and public transport networks are observed. A mean distance between two nodes of degrees k(i) and k(j) equals to (l(ij)) = A - B log(k(i)k(j)). The scaling is valid over several decades. A simple theory for the appearance of this scaling is presented. Parameters A and B depend on the mean value of a node degree (k)nn calculated for the nearest neighbors and on network clustering coefficients.

Self-organized criticality and coevolution of network structure and dynamics.

An interplay between network structure and self-organized criticality has been studied. We investigated by numerical simulations how the avalanche dynamics of the Bak-Tang-Wiesenfeld (BTW) sandpile model can induce emergence of scale-free (SF) networks and how this emerging structure affects system dynamics. P. Fronczak, A. Fronczak, and J. A. Holyst, Phys. Rev. E 73, 046117 (2006).

Phase transitions in social networks

Abstract. We study a model of network with clustering and desired node degree. The original purpose of the model was to describe optimal structures of scientific collaboration in the European Union. The model belongs to the family of exponential random graphs. We show by numerical simulations and analytical considerations how a very simple Hamiltonian can lead to surprisingly complicated and eventful phase diagram.

Analysis of scientific productivity using maximum entropy principle and fluctuation-dissipation theorem

Using data retrieved from the INSPEC database we have quantitatively discussed a few syndromes of the publish-or-perish phenomenon, including the continuous growth of the rate of scientific productivity, and the continuously decreasing percentage of those scientists who stay in science for a long time. Making use of the maximum entropy principle and fluctuation-dissipation theorem, we have shown that the observed fat-tailed distributions of the total number of papers x authored by scientists may result from the density-of-states function g(x;tau) underlying the scientific community. Although different generations of scientists are characterized by different productivity patterns, the function g(x;tau) is inherent to researchers of a given seniority tau , whereas the publish-or-perish phenomenon is caused only by an external field theta influencing researchers.

International trade network: fractal properties and globalization puzzle

Globalization is one of the central concepts of our age. The common perception of the process is that, due to declining communication and transport costs, distance becomes less and less important. However, the distance coefficient in the gravity model of trade, which grows in time, indicates that the role of distance increases rather than decreases. This, in essence, captures the notion of the globalization puzzle. Here, we show that the fractality of the international trade system (ITS) provides a simple solution for the puzzle. We argue that the distance coefficient corresponds to the fractal dimension of ITS. We provide two independent methods, the box counting method and spatial choice model, which confirm this statement. Our results allow us to conclude that the previous approaches to solving the puzzle misinterpreted the meaning of the distance coefficient in the gravity model of trade.