Maya Paczuski

Price variations in a stock market with many agents

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 others 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 and the behavior of other traders, 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.

Avalanche dynamics in evolution, growth, and depinning models.

Dynamics of complex systems in nature often occurs in terms of punctuations, or avalanches, rather than following a smooth, gradual path. Theory of avalanche dynamics in models of growth, interface depinning, and evolution is presented. Specifically, we include the Bak-Sneppen evolution model, Sneppen interface depinning model, 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/{ital f} noise, diffusion with anomalous Hurst exponents, L{acute e}vy flights, and punctuated equilibria can all be related to the same underlying avalanche dynamics. This dynamics can be represented as a fractal in {ital d} spatial plus one temporal dimension. The complex state can be reached either by tuning a parameter, or it can be self-organized. We present two {ital 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 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 {gamma} governing the approach tomorexa0» 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 {eta}=0. The exponent {pi} for the L{acute e}vy flight jumps between subsequent active sites can be related to other critical exponents through a study of {open_quote}{open_quote}backward avalanches.{close_quote}{close_quote} 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. (Abstract Truncated)«xa0less

Emergent traffic jams.

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 {ital P}({ital t}){similar_to}{ital t}{sup {minus}3/2} of lifetimes {ital t}. On varying the vehicle density in a closed system, this critical state separates lamellar and jammed regimes and exhibits 1/{ital 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.

Scale-free networks of earthquakes and aftershocks

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.

Self-organized networks of competing boolean agents

A model of Boolean agents competing in a market is presented where each agent bases his action on information obtained from a small group of other agents. The agents play a competitive game that rewards those in the minority. After a long time interval, the poorest players strategy is changed randomly, and the process is repeated. Eventually the network evolves to a stationary but intermittent state where random mutation of the worst strategy can change the behavior of the entire network, often causing a switch in the dynamics between attractors of vastly different lengths.

Analysis of the spatial distribution between successive earthquakes

Spatial distances between subsequent earthquakes in southern California exhibit scale-free statistics, with a critical exponent

Complex networks of earthquakes and aftershocks

delta approx 0.6

Intensity thresholds and the statistics of the temporal occurrence of solar flares.

, as well as finite size scaling. The statistics are independent of the threshold magnitude as long as the catalog is complete, but depend strongly on the temporal ordering of events, rather than the geometry of the spatial epicenter distribution. Nevertheless, the spatial distance and waiting time between subsequent earthquakes are uncorrelated with each other. These observations contradict the theory of aftershock zone scaling with main shock magnitude.

Interoccurrence Times in the Bak-Tang-Wiesenfeld Sandpile Model: A Comparison with the Observed Statistics of Solar Flares

Abstract. We invoke a metric to quantify the correlation between any two earthquakes. This provides a simple and straightforward alternative to using space-time windows to detect aftershock sequences and obviates the need to distinguish main shocks from aftershocks. Directed networks of earthquakes are constructed by placing a link, directed from the past to the future, between pairs of events that are strongly correlated. Each link has a weight giving the relative strength of correlation such that the sum over the incoming links to any node equals unity for aftershocks, or zero if the event had no correlated predecessors. A correlation threshold is set to drastically reduce the size of the data set without losing significant information. Events can be aftershocks of many previous events, and also generate many aftershocks. The probability distribution for the number of incoming and outgoing links are both scale free, and the networks are highly clustered. The Omori law holds for aftershock rates up to a decorrelation time that scales with the magnitude, m , of the initiating shock as t cutoff ~10 β m with β~-3/4. Another scaling law relates distances between earthquakes and their aftershocks to the magnitude of the initiating shock. Our results are inconsistent with the hypothesis of finite aftershock zones. We also find evidence that seismicity is dominantly triggered by small earthquakes. Our approach, using concepts from the modern theory of complex networks, together with a metric to estimate correlations, opens up new avenues of research, as well as new tools to understand seismicity.

Universality in Sandpiles, Interface Depinning, and Earthquake Models.

Introducing thresholds to analyze time series of emission from the Sun enables a new and simple definition of solar flare events and their interoccurrence times. Rescaling time by the rate of events, the waiting and quiet time distributions both conform to scaling functions that are independent of the intensity threshold over a wide range. The scaling functions are well-described by a two-parameter function, with parameters that depend on the phase of the solar cycle. For flares identified according to the current, standard definition, similar behavior is found.