Statistical analysis of 22 public transport networks in Poland

Public transport systems in 22 Polish cities have been analyzed. Sizes of these networks range from N=152 to N=2881. Depending on the assumed definition of network topology the degree distribution can follow a power law or can be described by an exponential function. Distributions of paths in all considered networks are given by asymmetric, unimodal functions. Clustering, assortativity and betweenness are studied. All considered networks exhibit small world behavior and are hierarchically organized. A transition between dissortative small networks N=500 is observed.Comment: 11 pages, 17 figures, 2 tables, REVTEX4 forma

Universal scaling of distances in complex networks

Universal scaling of distances between vertices of Erdos-Renyi random graphs, scale-free Barabasi-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 =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 _nn calculated for the nearest neighbors and on network clustering coefficients.Comment: 4 pages, 3 figures, 1 tabl

Estimation of a Noise Level Using Coarse-Grained Entropy of Experimental Time Series of Internal Pressure in a Combustion Engine

We report our results on non-periodic experimental time series of pressure in a single cylinder spark ignition engine. The experiments were performed for different levels of loading. We estimate the noise level in internal pressure calculating the coarse-grained entropy from variations of maximal pressures in successive cycles. The results show that the dynamics of the combustion is a nonlinear multidimensional process mediated by noise. Our results show that so defined level of noise in internal pressure is not monotonous function of loading.Comment: 12 pages, 6 figure

Supremacy distribution in evolving networks

We study a supremacy distribution in evolving Barabasi-Albert networks. The supremacy $s_i$ of a node $i$ is defined as a total number of all nodes that are younger than $i$ and can be connected to it by a directed path. For a network with a characteristic parameter $m=1,2,3,...$ the supremacy of an individual node increases with the network age as $t^{(1+m)/2}$ in an appropriate scaling region. It follows that there is a relation $s(k) \sim k^{m+1}$ between a node degree $k$ and its supremacy $s$ and the supremacy distribution $P(s)$ scales as $s^{-1-2/(1+m)}$. Analytic calculations basing on a continuum theory of supremacy evolution and on a corresponding rate equation have been confirmed by numerical simulations.Comment: 4 pages, 4 figure

Log-periodic oscillations due to discrete effects in complex networks

We show that discretization of internode distribution in complex networks affects internode distances l_ij calculated as a function of degrees (k_i k_j) and an average path length as function of network size N. For dense networks there are log-periodic oscillations of above quantities. We present real-world examples of such a behavior as well as we derive analytical expressions and compare them to numerical simulations. We consider a simple case of network optimization problem, arguing that discrete effects can lead to a nontrivial solution.Comment: 5 pages, 5 figures, REVTE

Ferromagnetic fluid as a model of social impact

The paper proposes a new model of spin dynamics which can be treated as a model of sociological coupling between individuals. Our approach takes into account two different human features: gregariousness and individuality. We will show how they affect a psychological distance between individuals and how the distance changes the opinion formation in a social group. Apart from its sociological aplications the model displays the variety of other interesting phenomena like self-organizing ferromagnetic state or a second order phase transition and can be studied from different points of view, e.g. as a model of ferromagnetic fluid, complex evolving network or multiplicative random process.Comment: 8 pages, 5 figure

Thermodynamic forces, flows, and Onsager coefficients in complex networks

We present Onsager formalism applied to random networks with arbitrary degree distribution. Using the well-known methods of non-equilibrium thermodynamics we identify thermodynamic forces and their conjugated flows induced in networks as a result of single node degree perturbation. The forces and the flows can be understood as a response of the system to events, such as random removal of nodes or intentional attacks on them. Finally, we show that cross effects (such as thermodiffusion, or thermoelectric phenomena), in which one force may not only give rise to its own corresponding flow, but to many other flows, can be observed also in complex networks.Comment: 4 pages, 2 figure

Scaling of human behavior during portal browsing

We investigate transitions of portals users between different subpages. A weighted network of portals subpages is reconstructed where edge weights are numbers of corresponding transitions. Distributions of link weights and node strengths follow power laws over several decades. Node strength increases faster than linearly with node degree. The distribution of time spent by the user at one subpage decays as power law with exponent around 1.3. Distribution of numbers P(z) of unique subpages during one visit is exponential. We find a square root dependence between the average z and the total number of transitions n during a single visit. Individual path of portal user resembles of self-attracting walk on the weighted network. Analytical model is developed to recover in part the collected data.Comment: 6 pages, 7 figure

Volatility clustering and scaling for financial time series due to attractor bubbling

A microscopic model of financial markets is considered, consisting of many interacting agents (spins) with global coupling and discrete-time thermal bath dynamics, similar to random Ising systems. The interactions between agents change randomly in time. In the thermodynamic limit the obtained time series of price returns show chaotic bursts resulting from the emergence of attractor bubbling or on-off intermittency, resembling the empirical financial time series with volatility clustering. For a proper choice of the model parameters the probability distributions of returns exhibit power-law tails with scaling exponents close to the empirical ones.Comment: For related publications see http://www.helbing.or

Mean-field theory for clustering coefficients in Barabasi-Albert networks

We applied a mean field approach to study clustering coefficients in Barabasi-Albert 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.Comment: 8 pages, 3 figure