The dynamics of complex systems in nature often occurs in terms of punctuations, or avalanches, rather than following a smooth, gradual path. A comprehensive theory of avalanche dynamics in models of growth, interface depinning, and evolution is presented. Specifically, we include the BakSneppen evolution model, the Sneppen interface depinning model, the Zaitsev flux creep model, invasion percolation, and several other depinning models into a unified treatment encompassing a large class of far from equilibrium processes. The formation of fractal structures, the appearance of 1/f noise, diffusion with anomalous Hurst exponents, Levy flights, and punctuated equilibria can all be related to the same underlying avalanche dynamics. This dynamics can be represented as a fractal in d spatial plus one temporal dimension. The complex state can be reached either by tuning a parameter, or it can be self-organized (SOC). We present two exact equations for the avalanche behavior in the latter case. (1) The slow approach to the critical attractor, i.e. the process of self-organization, is governed by a "gap" equation for the divergence of avalanche sizes.(2) The hierarchical structure of avalanches is described by an equation for the average number of sites covered by an avalanche. The exponent γ governing the approach to the critical state appears as a constant rather than as a critical exponent. In addition, the conservation of activity in the stationary state manifests itself through the superuniversal result η = 0. The exponent π for the Levy flight jumps between subsequent active sites can be related to other critical exponents through a study of "backward avalanches." We develop a scaling theory that relates many of the critical exponents in this broad category of extremal models, representing different universality classes, to two basic exponents characterizing the fractal attractor. The exact equations and the derived set of scaling relations are consistent with numerical simulations of the above mentioned models.

We consider percolation on interdependent locally treelike networks, recently introduced by Buldyrev et al., Nature 464, 1025, and demonstrate that the problem can be simplified conceptually by deleting all references to cascades of failures. Such cascades do exist, but their explicit treatment just complicates the theory -which is a straightforward extension of the usual epidemic spreading theory on a single network. Our method has the added benefits that it is directly formulated in terms of an order parameter and its modular structure can be easily extended to other problems, e.g. to any number of interdependent networks, or to networks with dependency links.

Large variations in stock prices happen with sufficient frequency to raise doubts about existing models, which all fail to account for non-Gaussian statistics. We construct simple models of a stock market, and argue that the large variations may be due to a crowd effect, where agents imitate each other's behavior. The variations over different time scales can be related to each other in a systematic way, similar to the Levy stable distribution proposed by Mandelbrot to describe real market indices.In the simplest, least realistic case, exact results for the statistics of the variations are derived by mapping onto a model of diffusing and annihilating particles, which has been solved by quantum field theory methods. When the agents imitate each other and respond to recent market volatility, different scaling behavior is obtained. In this case the statistics of price variations is consistent with empirical observations. The interplay between "rational" traders whose behavior is derived from fundamental analysis of the stock, including dividends, and "noise traders", whose behavior is governed solely by studying the market dynamics, is investigated. When the relative number of rational traders is small, "bubbles" often occur, where the market price moves outside the range justified by fundamental market analysis. When the number of rational traders is larger, the market price is generally locked within the price range they define.

We propose a metric to quantify correlations between earthquakes. The metric consists of a product involving the time interval and spatial distance between two events, as well as the magnitude of the first one. According to this metric, events typically are strongly correlated to only one or a few preceding ones. Thus a classification of events as foreshocks, main shocks, or aftershocks emerges automatically without imposing predetermined space-time windows. In the simplest network construction, each earthquake receives an incoming link from its most correlated predecessor. The number of aftershocks for any event, identified by its outgoing links, is found to be scale free with exponent gamma=2.0(1). The original Omori law with p=1 emerges as a robust feature of seismicity, holding up to years even for aftershock sequences initiated by intermediate magnitude events. The broad distribution of distances between earthquakes and their linked aftershocks suggests that aftershock collection with fixed space windows is not appropriate.

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We study a single-lane traffic model that is based on human driving behavior. The outflow from a traffic jam self-organizes to a critical state of maximum throughput. Small perturbations of the outflow far downstream create emergent traffic jams with a power law distribution P (t) ∼ t −3/2 of lifetimes, t. On varying the vehicle density in a closed system, this critical state separates lamellar and jammed regimes, and exhibits 1/f noise in the power spectrum. Using random walk arguments, in conjunction with a cascade equation, we develop a phenomenological theory that predicts the critical exponents for this transition and explains the self-organizing behavior. These predictions are consistent with all of our numerical results.

We study four Achlioptas type processes with "explosive" percolation transitions. All transitions are clearly continuous, but their finite size scaling functions are not entire holomorphic. The distributions of the order parameter, the relative size smax/N of the largest cluster, are doublehumped. But -in contrast to first order phase transitions -the distance between the two peaks decreases with system size N as N −η with η > 0. We find different positive values of β (defined via smax/N ∼ (p − pc) β for infinite systems) for each model, showing that they are all in different universality classes. In contrast, the exponent Θ (defined such that observables are homogeneous functions of (p − pc)N Θ ) is close to -or even equal to -1/2 for all models. Percolation is a pervasive concept in statistical physics and probability theory and has been studied in extenso in the past. It came thus as a surprise to many, when Achlioptas et al. [1] claimed that a seemingly mild modification of standard percolation models leads to a discontinuous phase transition -named "explosive percolation" (EP) by them -in contrast to the continuous phase transition seen in ordinary percolation. Following [1] there appeared a flood of papers [2-20] studying various aspects and generalizations of EP. In all cases, with one exception [20], the authors agreed that the transition is discontinuous: the "order parameter", defined as the fraction of vertices/sites in the largest cluster, makes a discrete jump at the percolation transition. In the present paper we join the dissenting minority and add further convincing evidence that the EP transition is continuous in all models, but with unusual finite size behavior.From the physical point of view, the model seems somewhat unnatural, since it involves non-local control (there is a 'supervisor' who has to compare distant pairs of nodes to chose the actual bonds to be established [21]). Also, notwithstanding [8], no realistic applications have been proposed. It is well known that the usual concept of universality classes in critical phenomena is invalidated by the presence of long range interactions. Thus it is not surprising that a percolation model with global control can show completely different behavior [22].Usually, e.g. in thermal equilibrium systems, discontinuous phase transitions are identified with "first order" transitions, while continuous transitions are called "second order". This notation is also often applied to percolative transitions. But EP lacks most attributes -except possibly for the discontinuous order parameter jump -considered essential for first order transitions. None of these other attributes (cooperativity, phase coexistence, and nucleation) is observed in Achlioptas type processes, although they are observed in other percolationtype transitions [23]. Thus EP should never have been viewed as a first order transition, and it is gratifying that it is also not discontinuous.Apart from the behavior of the average value m of the order parameter m, phase transitions can also b...

Complexity originates from the tendency of large dynamical systems to organize themselves into a critical state, with avalanches or "punctuations" of all sizes. In the critical state, events which would otherwise be uncoupled become correlated. The apparent, historical corntingency in many sciences, including geology, biology, and economics, finds a natural interpretation as a self-organized critical phenomenon. These ideas are discussed in the context of simple mathematical models of sandpiles and biological evolution. Insights are gained not only from numerical simulations but also from rigorous mathematical analysis.