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 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 500 and assortative large networks N 500 is observed.

Ising spins put onto a Barabási-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.

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

BackgroundE-communities, social groups interacting online, have recently become an object of interdisciplinary research. As with face-to-face meetings, Internet exchanges may not only include factual information but also emotional information – how participants feel about the subject discussed or other group members. Emotions in turn are known to be important in affecting interaction partners in offline communication in many ways. Could emotions in Internet exchanges affect others and systematically influence quantitative and qualitative aspects of the trajectory of e-communities? The development of automatic sentiment analysis has made large scale emotion detection and analysis possible using text messages collected from the web. However, it is not clear if emotions in e-communities primarily derive from individual group members' personalities or if they result from intra-group interactions, and whether they influence group activities.Methodology/Principal FindingsHere, for the first time, we show the collective character of affective phenomena on a large scale as observed in four million posts downloaded from Blogs, Digg and BBC forums. To test whether the emotions of a community member may influence the emotions of others, posts were grouped into clusters of messages with similar emotional valences. The frequency of long clusters was much higher than it would be if emotions occurred at random. Distributions for cluster lengths can be explained by preferential processes because conditional probabilities for consecutive messages grow as a power law with cluster length. For BBC forum threads, average discussion lengths were higher for larger values of absolute average emotional valence in the first ten comments and the average amount of emotion in messages fell during discussions.Conclusions/SignificanceOverall, our results prove that collective emotional states can be created and modulated via Internet communication and that emotional expressiveness is the fuel that sustains some e-communities.

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 *

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).

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.

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 reactiondiffusion equation, meaning that it has a certain lifetime, which models memory effects, further it can spread out in the community. Within our stochastic approach, the opinion change of the individuals is described 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 which holds 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) can change the ratio between minority and majority, until above a critical external support the supported subpopulation exists always as a majority.Considering spatial effects, we are able to derive two critical "social" temperatures, between which the community exists in a metastable state. That means, fluctuations below a certain critical wave number may result in a spatial opinion separation. We find that the range of metastability is particularly determined by a parameter, which characterizes the individual response to the communication field. In our discussion, we draw analogies to phase transitions in physical systems.