Session W22: Applications to Networks and Organized Systems

Predicting large events in power-law distributed avalanches: implications for earthquake forecast.

It is a common idea that power law distributed avalanches are inherently unpredictable. It mostly comes from the concept of Self-organized criticality. Nevertheless, we have found in classical simulations and experiments where the slowly addition of energy drives the system into a state of power law distributed avalanches, clear signs of both long and short term prediction. The simulations consist of a more realistic modification of the Olami-Feder-Christensen earthquake model where criticality and periodicity coexist. The experiment shows a clear power law behaviour for almost three decades in the avalanche size distribution of moving grains in a ``sandpile'' setup. Both systems display characteristic waiting times between large avalanches, and their internal structure suffer continuous variation preceding a large event. Monitoring those variations it is possible to predict large avalanches in a system if the slope of its pdf is larger than 1 in absolute value.   

Space-Time Clustering and Correlations of Earthquakes

Earthquake occurrence in nature is thought to result from correlated elastic stresses, leading to clustering in space and time. We show that occurrence of major earthquakes correlates with time intervals when fluctuations in small earthquakes are suppressed relative to the long term average and estimate a probability of less than 1\% that this coincidence is due to random clustering. Furthermore, we show that an order parameter can be defined to characterize these fluctuations and that a generalized Ginzburg criterion can be established to measuring the relative importance of fluctuations in the parameter.   

Numerical simulations of the 2-dimensional Robin-Hood model

The Robin Hood, or Zaitsev model [1] has been successfully used to model depinning of interfaces, friction, dislocation motion and flux creep, because it is one of the simplest extremal models for self-organized criticallity Until now, its properties have been well understood theoretically in one dimension and its scaling laws numerically verified. It is important to extend the range of validity of these laws into higher dimensions, to find precise values for the scaling exponents, and to investigate how they depend on the details of the model (like anisotropy). The case of two dimensions is of particular importance when studying surface friction [2]. Here, we numerically evaluate high precision scaling exponents for the avalanche size distribution, the avalanche fractal dimension, and the Levy flight-like distribution of the jumps between extremal active sites. [1] S.I. Zaitsev , Physica \textbf{A 189}, 411 (1992). [2] S. Buldyrev, J. Ferrante and F. Zypman Phys. Rev E (accepted)   

Self-organized criticality of elastic networks

We consider a model of elastic network self-organization inspired by studies of covalent glasses [1,2]. In the model, networks self-organize by avoiding stress whenever possible, but otherwise are random. Instead of a single rigidity percolation transition, with percolation always absent below a certain bond concentration and always present above, we find that the percolating rigid cluster exists with a probability between 0 and 1 in a finite range of bond concentrations, the {\it intermediate phase}. A power-law distribution of non-percolating cluster sizes, normally observed at a single critical point in percolation transitions, is seen everywhere in the intermediate phase. There is also a finite probability of percolation appearing and disappearing upon the application of a microscopic perturbation (addition or removal of a single bond). These properties indicate that in this phase the network maintains itself in a critical state on the verge of rigidity, a signature of self-organized criticality, but in a system at equilibrium.\\ \noindent [1] M.V. Chubynsky, M.-A. Bri\`ere and N. Mousseau, Phys. Rev. E 74, 016116 (2006)\\ \noindent [2] M.-A. Bri\`ere, M.V. Chubynsky and N. Mousseau, cond-mat/0610557   

Transport in Weighted Networks: Partition into Superhighways and Roads

Transport in weighted networks is dominated by the minimum spanning tree (MST), the tree connecting all nodes with the minimum total weight. We find that the MST can be partitioned into two distinct components, having significantly different transport properties, characterized by centrality—the number of times a node (or link) is used by transport paths. One component, superhighways, is the infinite incipient percolation cluster, for which we find that nodes (or links) with high centrality dominate. For the other component, roads, which includes the remaining nodes, low centrality nodes dominate. We find also that the distribution of the centrality for the infinite incipient percolation cluster satisfies a power law, with an exponent smaller than that for the entire MST. The significance of this finding is that one can improve significantly the global transport by improving a tiny fraction of the network, the superhighways.   

Geometric properties of minimal-cost spanning trees

The minimal-cost spanning tree (MST) problem is one of the oldest combinatorial optimization problems of computer science: given a graph with a unique cost associated with each edge, the MST is the subset of edges that will connect all vertices of the graph to each other at lowest total cost. While the MST is easy to compute (i.e., of polynomial complexity), it is of interest both intrinsically and as a heuristic approximation to harder questions in optimization, such as the Steiner tree and Traveling Salesman problems. Using techniques of statistical field theory, we study the random MST where the edge costs are independent, identically distributed, random variables. We develop a mean field theory by solving the MST exactly on the Bethe lattice using the relationship between bond percolation and Kruskal's greedy algorithm for the MST. These considerations carry over to finite dimensional lattices and the field theory for percolation in $6- \epsilon$ dimensions. From this we find that the critical dimension $d_c = 6$ for the MST problem, contrary to the result $d_c = 8$ previously suggested by Newman and Stein. Finally we calculate to order $\epsilon$ the Hausdorff (fractal) dimension of the unique path on the MST connecting two widely separated points.   

Ridge Network of Crumpled Paper

The work presented has investigated the network formed by the complete sets of ridges from samples of crumpled paper. Sheets of paper were crumpled, and their height profiles measured by a laser profilometer. From these data lines of high curvature were identified as ridges. Intersections between ridges were considered as nodes, and the ridges as links between these nodes. The emerging networks have been investigated using network theory. Properties such as the degree distribution, degree correlation and clustering coefficient are reported. These are compared to comparable random networks and networks formed by the Voronoi diagrams. Spatial properties such as the ridge length, domain area and vertex distributions have also been investigated.   

Physics of curling ribbons

Curling decorative ribbons by dragging it past one's thumb and the blade of a scissor is a well known technique used frequently. However a quantitative understanding of this apparently simple phenomenon is still lacking. We present results from recent experimental and theoretical investigations of this problem. Using the insights gained from this we propose a method of generating novel shapes by differential stretching and subsequent selective stress relief for thin sheets. We discuss the implications of this mechanism for the formation of ribbon like structures in biological systems.   

Griffiths singularities and algebraic order in the exact solution of an Ising model on a modular network

We use an exact renormalization-group transformation to study the Ising model on a modular network composed of tightly-knit clusters with a scale-free distribution of cluster sizes. By varying the ratio $K/J$ of inter-cluster to intra-cluster interaction strengths, we obtain an unusual phase diagram: at high temperatures or small $K/J$ the system exhibits a disordered phase with a Griffiths-like singularity in the free energy as a function of magnetic field, due to the effects of rare large clusters. As the temperature is lowered, true long-range order is not seen, but there is a transition to an algebraically ordered phase, with thermodynamic characteristics reminiscent of the XY model, but in a different universality class. The transition is infinite-order at small $K/J$, and becomes second-order above a threshold value $(K/J)^\ast$. We investigate the nature of pair correlations in the model, allowing us to test recent predictions on the relationship between network topology and correlations in cooperative systems. Despite the absence of magnetization in the low-temperature phase, we find that a subset of spin pairs (vanishingly small in the thermodynamic limit) remains strongly correlated at arbitrarily large distances.   

Nonstationary Increments, Scaling Distributions, and Variable Diffusion Processes in Financial Markets

Arguably the most important problem in quantitative finance is to understand the nature of stochastic processes that underlie market dynamics. One aspect of the solution to this problem involves determining characteristics of the distribution of fluctuations in returns. Empirical studies conducted over the last decade have reported that they are non-Gaussian, scale in time, and have power-law (or fat) tails. However, these studies implicitly assume that the underlying process has stationary increments. We explicitly show that this assumption is not valid for the Euro-Dollar exchange rate between 1999-2004. In addition, we find that fluctuations in returns of the exchange rate are uncorrelated and scale as power-laws for certain time intervals during each day. This behavior is consistent with a diffusive process with a diffusion coefficient that depends both on the time and the price change. Within scaling regions, we find that sliding interval methods can generate fat-tailed distributions as an artifact.   

Parameter Inference in the Ornstein-Uhlenbeck Process

The Ornstein-Uhlenbeck process has been proposed as a model for the spontaneous activity of a neuron. In this model, the firing of the neuron corresponds to the first passage of the process to a constant boundary, or threshold. While the Laplace transform of the first passage time distribution is available, the real density has not been obtained in any tractable form. We address the problem of estimating the parameters of the process when the only available data from a neuron are the interspike intervals, or the times between firings. In particular, we give an algorithm for computing maximum likelihood estimates (MLEs) and their corresponding confidence regions for three of the five model parameters by numerically inverting the Laplace transform. We also provide an analysis on the reliability of the estimates and their confidence regions when simulated data is used to generate the first passage sample.