We present an analysis of the statistical properties and growth of the free on-line encyclopedia Wikipedia. By describing topics by vertices and hyperlinks between them as edges, we can represent this encyclopedia as a directed graph. The topological properties of this graph are in close analogy with those of the World Wide Web, despite the very different growth mechanism. In particular, we measure a scale-invariant distribution of the in and out degree and we are able to reproduce these features by means of a simple statistical model. As a major consequence, Wikipedia growth can be described by local rules such as the preferential attachment mechanism, though users, who are responsible of its evolution, can act globally on the network.

We study the partially asymmetric exclusion process with open boundaries. We generalise the matrix approach previously used to solve the special case of total asymmetry and derive exact expressions for the partition sum and currents valid for all values of the asymmetry parameter q. Due to the relationship between the matrix algebra and the q-deformed quantum harmonic oscillator algebra we find that q-Hermite polynomials, along with their orthogonality properties and generating functions, are of great utility. We employ two distinct sets of q-Hermite polynomials, one for q < 1 and the other for q > 1. It turns out that these correspond to two distinct regimes: the previously studied case of forward bias (q < 1) and the regime of reverse bias (q > 1) where the boundaries support a current opposite in direction to the bulk bias. For the forward bias case we confirm the previously proposed phase diagram whereas the case of reverse bias produces a new phase in which the current decreases exponentially with system size.1 r.a.blythe@ed.ac.uk 2 m.r.evans@ed.ac.uk 3 fran@a13.ph.man.ac.uk Present address:

We review the present state of understanding of the Barkhausen effect in soft ferromagnetic materials. Barkhausen noise (BN) is generated by the discontinuous motion of magnetic domains as they interact with impurities and defects. BN is one of the very many examples of crackling noise, arising in a variety of contexts with remarkably similar features, and occurring when a system responds in a jerky manner to a smooth external forcing. Among all crackling system, we focus on BN, where a complete and consistent picture emerges thanks to an exactly solvable model of avalanches dynamics, known as ABBM model, which ultimately describes the system in terms of a Langevin equation for the velocity of the avalanche front. Despite its simplicity the ABBM model is able to accurately reproduce the phenomenology observed in the experiments on a large class of magnetic materials, as long as universal properties are involved. To complete the picture and to understand the longstanding discrepancy between the ABBM theory and the experiments -which otherwise agree exceptionally well -consisting in the puzzling asymmetric shape of the noise pulses, microscopic details must be taken into account, namely the effects of eddy currents retardation. These effects can be incorporated in the model, and result, to a first order approximation, in a negative effective mass associated with the wall. The progress made in understanding BN is potentially relevant for other crackling systems: on one hand, the ABBM model turns out to be a paradigmatic model for the universal behavior of avalanches dynamics; on the other hand, the microscopic explanation of the asymmetry in the noise pulses suggests that inertial effects may also be at the origin of pulses asymmetry observed in other crackling systems.

C rackling noise is a common feature in many dynamic systems 1-9 , the most familiar instance of which is the sound made by a sheet of paper when crumpled into a ball. Although seemingly random, this noise contains fundamental information about the properties of the system in which it occurs. One potential source of such information lies in the asymmetric shape of noise pulses emitted by a diverse range of noisy systems [8][9][10][11][12] , but the cause of this asymmetry has lacked explanation 1 . Here we show that the leftward asymmetry observed in the Barkhausen effect 2 -the noise generated by the jerky motion of domain walls as they interact with impurities in a soft magnet-is a direct consequence of a magnetic domain wall's negative effective mass. As well as providing a means of determining domain-wall effective mass from a magnet's Barkhausen noise, our work suggests an inertial explanation for the origin of avalanche asymmetries in crackling-noise phenomena more generally.Crackling noise is the response of many physical systems to a slow external driving force, characterized by outbursts of activity (avalanches or pulses) spanning a broad range of sizes, separated by quiescent intervals 1 . In condensed matter, notable examples are the magnetization noise emitted along the hysteresis loop in ferromagnets (that is, the Barkhausen effect 2 ), the noise from magnetic vortices in type-II superconductors 3 , ferroelectric materials 4 and driven ionic crystals 5 . In the context of mechanics, examples are the acoustic emission signal in fracture 6 and plasticity 7 and, on a larger scale, seismic activity corresponding to an earthquake 8,9 . Quantitative understanding of crackling noise is of fundamental importance in different applications, from non-destructive material testing to hazard prediction. This goal can be achieved only through the identification of general universal properties common to these systems, irrespective of their differences in the internal dynamics and microstructural details. In this context, the average shape of the individual pulses of which the signal is composed has been proposed as the best tool to characterize these universal features of crackling noise 1 . In analogy with critical phenomena, it is expected that pulses of different durations can be rescaled on a universal function, whose shape would only depend on general features of the physical process underlying the noise. This scenario is supported by the analysis of a variety of models, where pulse shapes are described by universal symmetric scaling functions [12][13][14] . In most experimental data, however, the pulse shape is markedly asymmetric with respect to its midpoint, that is, avalanches start quickly but return to zero more slowly 1,[8][9][10][11][12] . These results are puzzling because the models accurately reproduce several other universal quantities, such as avalanche distributions and power spectra 11,15 .One of the most studied examples of crackling noise is the Barkhausen effect recorded in soft magnetic mate...