A simple model for self organization of bipartite networks

We suggest a minimalistic model for directed networks and suggest an application to injection and merging of magnetic field lines. We obtain a network of connected donor and acceptor vertices with degree distribution $1/s^2$, and with dynamical reconnection events of size $\Delta s$ occurring with frequency that scale as $1/\Delta s^3$. This suggest that the model is in the same universality class as the model for self organization in the solar atmosphere suggested by Hughes et al.(PRL {\bf 90} 131101)

Large Scale Structures, Symmetry, and Universality in Sandpiles

We introduce a sandpile model where, at each unstable site, all grains are transferred randomly to downstream neighbors. The model is local and conservative, but not Abelian. This does not appear to change the universality class for the avalanches in the self-organized critical state. It does, however, introduce long-range spatial correlations within the metastable states. We find large scale networks of occupied sites whose density vanishes in the thermodynamic limit, for d>1.Comment: 4 pages, 5 figures, submitted to Phys. Rev. Let

Comment on "Dynamic Opinion Model and Invasion Percolation"

In J. Shao et al., PRL 103, 108701 (2009) the authors claim that a model with majority rule coarsening exhibits in d=2 a percolation transition in the universality class of invasion percolation with trapping. In the present comment we give compelling evidence, including high statistics simulations on much larger lattices, that this is not correct. and that the model is trivially in the ordinary percolation universality class.Comment: 1 pag

Infinite Hierarchy of Exact Equations in the Bak-Sneppen Model

We derive an infinite hierarchy of exact equations for the Bak-Sneppen model in arbitrary dimensions. These equations relate different moments of temporal duration and spatial size of avalanches. We prove that the exponents of the BS model are the same above and below the critical point and express the universal amplitude ratio of the avalanche spatial size in terms of the critical exponents. The equations uniquely determine the shape of the scaling function of the avalanche distribution. It is suggested that in the BS model there is only one independent critical exponent.Comment: Submitted to PRL, 4 two-column pages (revtex), 1 ps figure included with epsf, g-zipped, uuencode

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 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.Comment: 39 pages (Latex) + 20 Figures and missing Figure 1 (sorry), submitted to J. Math. Eco

A Monte Carlo Renormalization Group Approach to the Bak-Sneppen model

A recent renormalization group approach to a modified Bak-Sneppen model is discussed. We propose a self-consistency condition for the blocking scheme to be essential for a successful RG-method applied to self-organized criticality. A new method realizing the RG-approach to the Bak-Sneppen model is presented. It is based on the Monte-Carlo importance sampling idea. The new technique performs much faster than the original proposal. Using this technique we cross-check and improve previous results.Comment: 11 pages, REVTex, 2 Postscript figures include

Solitons in the one-dimensional forest fire model

Fires in the one-dimensional Bak-Chen-Tang forest fire model propagate as solitons, resembling shocks in Burgers turbulence. The branching of solitons, creating new fires, is balanced by the pair-wise annihilation of oppositely moving solitons. Two distinct, diverging length scales appear in the limit where the growth rate of trees, $p$, vanishes. The width of the solitons, $w$, diverges as a power law, $1/p$, while the average distance between solitons diverges much faster as $d \sim \exp({\pi}^2/12p)$.Comment: 4 pages with 2 figures include

Different hierarchy of avalanches observed in the Bak-Sneppen evolution model

We introduce a new quantity, average fitness, into the Bak-Sneppen evolution model. Through the new quantity, a different hierarchy of avalanches is observed. The gap equation, in terms of the average fitness, is presented to describe the self-organization of the model. It is found that the critical value of the average fitness can be exactly obtained. Based on the simulations, two critical exponents, avalanche distribution and avalanche dimension, of the new avalanches are given.Comment: 5 pages, 3 figure

d_c=4 is the upper critical dimension for the Bak-Sneppen model

Numerical results are presented indicating d_c=4 as the upper critical dimension for the Bak-Sneppen evolution model. This finding agrees with previous theoretical arguments, but contradicts a recent Letter [Phys. Rev. Lett. 80, 5746-5749 (1998)] that placed d_c as high as d=8. In particular, we find that avalanches are compact for all dimensions d<=4, and are fractal for d>4. Under those conditions, scaling arguments predict a d_c=4, where hyperscaling relations hold for d<=4. Other properties of avalanches, studied for 1<=d<=6, corroborate this result. To this end, an improved numerical algorithm is presented that is based on the equivalent branching process.Comment: 4 pages, RevTex4, as to appear in Phys. Rev. Lett., related papers available at http://userwww.service.emory.edu/~sboettc