Janusz A. Hołyst

Statistical analysis of 22 public transport networks in Poland

Public transport systems in 22 Polish cities have been analyzed. The sizes of these networks range from N = 152 to 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 path lengths 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 approximately < or = 500 and assortative large networks N approximately > or = 500 is observed.

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 …

Ferromagnetic phase transition in Barabási–Albert networks

Ising spins put onto a Barabasi–Albert scale-free network show an effective phase transition from ferromagnetism to paramagnetism upon heating, with an effective critical temperature increasing as the logarithm of the system size. Starting with all spins up and upon equilibration pinning the few most-connected spins down nucleates the phase with most of the spins down.

Negative emotions boost user activity at BBC forum

We present an empirical study of user activity in online BBC discussion forums, measured by the number of posts written by individual debaters and the average sentiment of these posts. Nearly 2.5 million posts from over 18 thousand users were investigated. Scale-free distributions were observed for activity in individual discussion threads as well as for overall activity. The number of unique users in a thread normalized by the thread length decays with thread length, suggesting that thread life is sustained by mutual discussions rather than by independent comments. Automatic sentiment analysis shows that most posts contain negative emotions and the most active users in individual threads express predominantly negative sentiments. It follows that the average emotion of longer threads is more negative and that threads can be sustained by negative comments. An agent-based computer simulation model has been used to reproduce several essential characteristics of the analyzed system. The model stresses the role of discussions between users, especially emotionally laden quarrels between supporters of opposite opinions, and represents many observed statistics of the forum.

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 heat 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.

Modelling collective opinion formation by means of active Brownian particles

Abstract:The concept of active Brownian particles is used to model a collective opinion formation process. It is assumed that individuals in community create a two-component communication field that influences the change of opinions of other persons and/or can induce their migration. The communication field is described by a reaction-diffusion equation, the opinion change of the individuals is given by a master equation, while the migration is described by a set of Langevin equations, coupled by the communication field. In the mean-field limit holding for fast communication we derive a critical population size, above which the community separates into a majority and a minority with opposite opinions. The existence of external support (e.g. from mass media) changes the ratio between minority and majority, until above a critical external support the supported subpopulation exists always as a majority. Spatial effects lead to two critical “social” temperatures, between which the community exists in a metastable state, thus fluctuations below a certain critical wave number may result in a spatial opinion separation. The range of metastability is particularly determined by a parameter characterizing the individual response to the communication field. In our discussion, we draw analogies to phase transitions in physical systems.

Chaos control in economical model by time-delayed feedback method

A two-dimensional map describing chaotic behaviour of an economic model has been stabilized on various periodic orbits by the use of Pyragas time-delayed feedback control. The method avoids fancy data processing used in the Ott–Grebogi–Yorke approach and is based solely on the plain measurement and time lag of a scalar signal which in our case is a value of sales of a firm following an active investment strategy (Behrens–Feichtinger model). We show that the application of this control method is very straightforward and one can easily switch from a chaotic trajectory to a regular periodic orbit and simultaneously improve the systems economic properties.

Phase transitions in social impact models of opinion formation

We study phase transitions in models of opinion formation which are based on the social impact theory. Two different models are discussed: (i) a cellular-automata-based model of a finite group with a strong leader where persons can change their opinions but not their spatial positions, and (ii) a model with persons treated as active Brownian particles interacting via a communication field. In the first model, two stable phases are possible: a cluster around the leader, and a state of social unification. The transition into the second state occurs for a large leader strength and/or for a high level of social noise. In the second model, we find three stable phases, which correspond either to a “paramagnetic” phase (for high noise and strong diffusion), a “ferromagnetic” phase (for small nose and weak diffusion), or a phase with spatially separated “domains” (for intermediate conditions).

Correlations in commodity markets

In this paper we analyzed dependencies in commodity markets, investigating correlations of future contracts for commodities over the period 1998.09.01–2007.12.14. We constructed a minimal spanning tree based on the correlation matrix. The tree provides evidence for sector clusterization of investigated contracts. We also studied dynamical properties of commodity dependencies. It turned out that the market was constantly getting more correlated within the investigated period, although the increase of correlation was distributed non-uniformly among all contracts, and depended on contracts branches.

Majority model on a network with communities.

We focus on the majority model in a topology consisting of two coupled fully connected networks, thereby mimicking the existence of communities in social networks. We show that a transition takes place at a value of the interconnectivity parameter. Above this value, only symmetric solutions prevail, where both communities agree with each other and reach consensus. Below this value, in contrast, the communities can reach opposite opinions and an asymmetric state is attained. The importance of the interface between the subnetworks is shown.