Self-organized critical models of earthquakes

In this paper we briefly review few self-organized critical (SOC) models of the phenomenon of earthquakes. For example, the two-dimensional non-conservative SOC model of Olami, Feder and Christensen (OFC) has been described. It is known that the effect of the fixed boundary on this model is very strong. It has been recently observed that imposition of a moving boundary condition helps to remove the strong non-uniformity originated from the fixed boundary. A generalized spatio-temporal scaling for the recurrence time distribution was proposed by Bak et al. which was later confirmed by Corral. We studied the same scalings on the conservative OFC model with moving boundary condition.     

Dynamical evolution of clustering in complex network of earthquakes

The network approach plays a distinguished role in contemporary science of complex systems/phenomena. Such an approach has been introduced into seismology in a recent work [S. Abe, N. Suzuki, Europhys. Lett. 65, 581 (2004)]. Here, we discuss the dynamical property of the earthquake network constructed in California and report the discovery that the values of the clustering coefficient remain stationary before main shocks, suddenly jump up at the main shocks, and then slowly decay following a power law to become stationary again. Thus, the dynamical network approach characterizes main shocks in a peculiar manner.     

Olami-Feder-Christensen model on different networks

We investigate numerically the Self Organized Criticality (SOC) properties of the dissipative Olami-Feder-Christensen model on small-world and scale-free networks. We find that the small-world OFC model exhibits self-organized criticality. Indeed, in this case we observe power law behavior of earthquakes size distribution with finite size scaling for the cut-off region. In the scale-free OFC model, instead, the strength of disorder hinders synchronization and does not allow to reach a critical state.     

Damage spreading in 2-dimensional isotropic and anisotropic Bak-Sneppen models

We implement the damage spreading technique on 2-dimensional isotropic and anisotropic Bak-Sneppen models. Our extensive numerical simulations show that there exists a power-law sensitivity to the initial conditions at the statistically stationary state (self-organized critical state). Corresponding growth exponent α for the Hamming distance and the dynamical exponent z are calculated. These values allow us to observe a clear data collapse of the finite size scaling for both versions of the Bak-Sneppen model. Moreover, it is shown that the growth exponent of the distance in the isotropic and anisotropic Bak-Sneppen models is strongly affected by the choice of the transient time.     

Scaling relation for earthquake networks

The scaling relation, 2γ−δ=1, for the exponents of the power-law connectivity distribution, γ, and the power-law eigenvalue distribution of the adjacency matrix, δ, is theoretically predicted to be fulfilled by a locally treelike scale-free network in the “effective medium approximation” (i.e., an analog of the mean field approximation). Here, it is shown that such a relation holds well for the reduced simple earthquake networks (i.e., the network without tadpole-loops and multiple edges) constructed from the seismic data taken from California and Japan. This validates the goodness of the effective medium approximation in the earthquake networks and is consistent with the hierarchical organization of the networks. The present result may be useful for modeling seismicity on complex networks.     

Convergence dynamics of 2-dimensional isotropic and anisotropic Bak-Sneppen models

The conventional Hamming distance measurement captures only short-time dynamics of the displacement between uncorrelated random configurations. The minimum difference technique introduced by Tirnakli and Lyra [U. Tirnakli, M.L. Lyra. Int. J. Mod. Phys. C 14 (2003) 805] is used to study short-time and long-time dynamics of the two distinct random configurations of isotropic and anisotropic Bak-Sneppen models on a square lattice. Similar to a 1-dimensional case, the time evolution of the displacement is intermittent. The scaling behavior of the jump activity rate and waiting time distribution reveal the absence of typical spatial-temporal scales in the mechanism of displacement jumps used to quantify convergence dynamics.     

Hypocenter interval statistics between successive earthquakes in the two-dimensional Burridge-Knopoff model

We study statistical properties of spatial distances between successive earthquakes, the so-called hypocenter intervals, produced by a two-dimensional (2D) Burridge-Knopoff model involving stick-slip behavior. It is found that cumulative distributions of hypocenter intervals can be described by the q-exponential distributions with q<1, which is also observed in nature. The statistics depend on a friction and stiffness parameters characterizing the model and a threshold of magnitude. The conjecture which states that q_t+q_r∼2, where q_t and q_r are an entropy index of time intervals and spatial intervals, respectively, can be reproduced semi-quantitatively. It is concluded that we provide a new perspective on the Burridge-Knopoff model which addresses that the model can be recognized as a realistic one in view of the reproduction of the spatio-temporal interval statistics of earthquakes on the basis of nonextensive statistical mechanics.     

The Weibull-log Weibull transition of the interoccurrence time statistics in the two-dimensional Burridge-Knopoff Earthquake model

In analyzing synthetic earthquake catalogs created by a two-dimensional Burridge-Knopoff model, we have found that a probability distribution of the interoccurrence times, the time intervals between successive events, can be described clearly by the superposition of the Weibull distribution and the log-Weibull distribution. In addition, the interoccurrence time statistics depend on frictional properties and stiffness of a fault and exhibit the Weibull-log Weibull transition, which states that the distribution function changes from the log-Weibull regime to the Weibull regime when the threshold of magnitude is increased. We reinforce a new insight into this model; the model can be recognized as a mechanical model providing a framework of the Weibull-log Weibull transition.     

On the scaling of probability density functions with apparent power-law exponents less than unity

We derive general properties of the finite-size scaling of probability density functions and show that when the apparent exponent of a probability density is less than 1, the associated finite-size scaling ansatz has a scaling exponent τ equal to 1, provided that the fraction of events in the universal scaling part of the probability density function is non-vanishing in the thermodynamic limit. We find the general result that τ≥1 and . Moreover, we show that if the scaling function approaches a non-zero constant for small arguments, , then . However, if the scaling function vanishes for small arguments, , then τ= 1, again assuming a non-vanishing fraction of universal events. Finally, we apply the formalism developed to examples from the literature, including some where misunderstandings of the theory of scaling have led to erroneous conclusions.     

A deterministic sandpile automaton revisited

The Bak-Tang-Wiesenfeld (BTW) sandpile model is a cellular automaton which has been intensively studied during the last years as a paradigm for self-organized criticality. In this paper, we reconsider a deterministic version of the BTW model introduced by Wiesenfeld, Theiler and McNamara, where sand grains are added always to one fixed site on the square lattice. Using the Abelian sandpile formalism we discuss the static properties of the system. We present numerical evidence that the deterministic model is only in the BTW universality class if the initial conditions and the geometric form of the boundaries do not respect the full symmetry of the square lattice. Received 19 August 1999     

Self-organized branching process for a one-dimensional rice-pile model

A self-organized branching process is introduced to describe one-dimensional rice-pile model with stochastic topplings. Although the branching processes are generally expected to describe well high-dimensional systems, our modification highlights some of the peculiarities present in one dimension. We find analytically that the crossover behavior from the trivial one-dimensional BTW behaviour to self-organized criticality is characterised by a power-law distribution of avalanches. The finite-size effects, which are crucial to the crossover, are calculated. Received 21 June 2001 and Received in final form 14 November 2001     

The short-time critical behaviour of the Ginzburg-Landau model with long-range interaction

The renormalisation group approach is applied to the study of the short-time critical behaviour of the d-dimensional Ginzburg-Landau model with long-range interaction of the form in momentum space. Firstly the system is quenched from a high temperature to the critical temperature and then relaxes to equilibrium within the model A dynamics. The asymptotic scaling laws and the initial slip exponents and of the order parameter and the response function respectively, are calculated to the second order in . Received 9 June 2000 and Received in final form 2 August 2000     

Self organization of interacting polya urns

We introduce a simple model which shows non-trivial self organized critical properties. The model describes a system of interacting units, modelled by Polya urns, subject to perturbations and which occasionally break down. Three equivalent formulations - stochastic, quenched and deterministic - are shown to reproduce the same dynamics. Among the novel features of the model are a non-homogeneous stationary state, the presence of a non-stationary critical phase and non-trivial exponents even in mean field. We discuss simple interpretations in term of biological evolution and earthquake dynamics and we report on extensive numerical simulations in dimensions d=1,2 as well as in the random neighbors limit. Received: 18 February 1998 / Revised: 20 March 1998 / Accepted: 29 March 1998     

Randomness and a step-like distribution of pile heights in avalanche models

This paper considers a one-parameter family of sand-piles. The family exhibits the crossover between the models with deterministic and stochastic relaxation. The mean pile height is used to describe the crossover. The height densities corresponding to the models with relaxation of both types approach one another as the parameter increases. Relaxation is supposed to deal with the local losses of grains by a fixed amount. In that case the densities show a step-like behaviour in contrast to the peaked shape found in the models with the local loss of grains down to a fixed level [S. Lübeck, Phys. Rev. E 62, 6149 (2000)]. A spectral approach based on the long-run properties of the pile height considers the models with deterministic and random relaxation more accurately and distinguishes between the two cases for admissible parameter values.     

Limited ability driven phase transitions in the coevolution process in Axelrod's model

We study the coevolution process in Axelrod's model by taking into account of agents' abilities to access information, which is described by a parameter α to control the geographical range of communication. We observe two kinds of phase transitions in both cultural domains and network fragments, which depend on the parameter α. By simulation, we find that not all rewiring processes pervade the dissemination of culture, that is, a very limited ability to access information constrains the cultural dissemination, while an exceptional ability to access information aids the dissemination of culture. Furthermore, by analyzing the network characteristics at the frozen states, we find that there exists a stage at which the network develops to be a small-world network with community structures.     

An evolving network model in polymer melts with community structures

In order to describe the entangled network structure in polymer melts visually, we propose an evolving network model with community structure. This network model grows according to the inner-community and inter-community preferential mechanisms of both community sizes and node degrees. Numerical simulation results indicate that the cumulative distribution of community size and node degree distribution follow power-law distributions P(S≥s)∼s^-υ and P(k)∼k^-γ respectively, with the exponents of υ≥1 and .     

Non-extensive trends in the size distribution of coding and non-coding DNA sequences in the human genome

We study the primary DNA structure of four of the most completely sequenced human chromosomes (including chromosome 19 which is the most dense in coding), using non-extensive statistics. We show that the exponents governing the spatial decay of the coding size distributions vary between 5.2 ≤r ≤5.7 for the short scales and 1.45 ≤q ≤1.50 for the large scales. On the contrary, the exponents governing the spatial decay of the non-coding size distributions in these four chromosomes, take the values 2.4 ≤r ≤3.2 for the short scales and 1.50 ≤q ≤1.72 for the large scales. These results, in particular the values of the tail exponent q, indicate the existence of correlations in the coding and non-coding size distributions with tendency for higher correlations in the non-coding DNA.