Exactly soluble hierarchical clustering model: inverse cascades, self-similarity, and scaling

Emergence of patterns in random processes. III. Clustering in higher dimensions

Newman et al. [Phys. Rev. E 86, 026103 (2012)10.1103/PhysRevE.86.026103] showed that points uniformly distributed as independent and identically distributed random variables with nearest-neighbor interactions form clusters with a mean number of three points in each. Here, we extend our analysis to higher dimensions, ultimately going to infinite dimensions, and we show that the mean number of points per cluster rises monotonically with a limiting value of four.

Entropy rates for Horton self-similar trees

In this paper, we examine finite unlabeled rooted planted binary plane trees with no edge length. First, we provide an exact formula for the number of trees with given Horton-Strahler numbers. Then, using the notion of entropy, we examine the structural complexity of random trees with N vertices. Finally, we quantify the complexity of the tree's structural properties as tree is allowed to grow in size, by evaluating the entropy rate for trees with N vertices and for trees that satisfy Horton Law with Horton exponent R .

Tokunaga self-similarity arises naturally from time invariance

The Tokunaga condition is an algebraic rule that provides a detailed description of the branching structure in a self-similar tree. Despite a solid empirical validation and practical convenience, the Tokunaga condition lacks a theoretical justification. Such a justification is suggested in this work. We define a geometric branching process G(s) that generates self-similar rooted trees. The main result establishes the equivalence between the invariance of G(s) with respect to a time shift and a one-parametric version of the Tokunaga condition. In the parameter region where the process satisfies the Tokunaga condition (and hence is time invariant), G(s) enjoys many of the symmetries observed in a critical binary Galton-Watson branching process and reproduces the latter for a particular parameter value.

Implications of an inverse branching aftershock sequence model

The branching aftershock sequence (BASS) model is a self-similar statistical model for earthquake aftershock sequences. A prescribed parent earthquake generates a first generation of daughter aftershocks. The magnitudes and times of occurrence of the daughters are obtained from statistical distributions. The first generation daughter aftershocks then become parent earthquakes that generate second generation aftershocks. The process is then extended to higher generations. The key parameter in the BASS model is the magnitude difference Deltam* between the parent earthquake and the largest expected daughter earthquake. In the application of the BASS model to aftershocks Deltam* is positive, the largest expected daughter event is smaller than the parent, and the sequence of events (aftershocks) usually dies out, but an exponential growth in the number of events with time is also possible. In this paper we explore this behavior of the BASS model as Deltam* varies, including when Deltam* is negative and the largest expected daughter event is larger than the parent. The applications of this self-similar branching process to biology and other fields are discussed.

An inverse cascade explanation for the power-law frequency-area statistics of earthquakes, landslides and wildfires

Frequency-magnitude statistics for natural hazards can greatly help in probabilistic hazard assessments. An example is the case of earthquakes, where the generality of a power-law (fractal) frequency-rupture area correlation is a major feature in seismic risk mapping. Other examples of this power-law frequency-size behaviour are landslides and wildfires. In previous studies, authors have made the potential association of the hazard statistics with a simple cellular-automata model that also has robust power-law statistics: earthquakes with slider-block models, landslides with sandpile models, and wildfires with forest-fire models. A potential explanation for the robust power-law behaviour of both the models and natural hazards can be made in terms of an inverse-cascade of metastable regions. A metastable region is the region over which an ‘avalanche’ spreads once triggered. Clusters grow primarily by coalescence. Growth dominates over losses except for the very largest clusters. The cascade of cluster growth is self-similar and the frequency of cluster areas exhibits power-law scaling. We show how the power-law exponent of the frequency-area distribution of clusters is related to the fractal dimension of cluster shapes.

Inverse cascade in a percolation model: hierarchical description of time-dependent scaling

The dynamics of a two-dimensional site percolation model on a square lattice is studied using the hierarchical approach introduced by Gabrielov [Phys. Rev. E 60, 5293 (1999)]. The key elements of the approach are the tree representation of clusters and their coalescence, and the Horton-Strahler scheme for cluster ranking. Accordingly, the evolution of the percolation model is considered as a hierarchical inverse cascade of cluster aggregation. A three-exponent time-dependent scaling for the cluster rank distribution is derived using the Tokunaga branching constraint and classical results on percolation in terms of cluster masses. Deviations from the pure scaling are described. An empirical constraint on the dynamics of a rank population is reported based on numerical simulations.

Cluster size distributions: signatures of self-organization in spatial ecologies

Three different lattice-based models for antagonistic ecological interactions, both nonlinear and stochastic, exhibit similar power-law scalings in the geometry of clusters. Specifically, cluster size distributions and perimeter-area curves follow power-law scalings. In the coexistence regime, these patterns are robust: their exponents, and therefore the associated Korcak exponent characterizing patchiness, depend only weakly on the parameters of the systems. These distributions, in particular the values of their exponents, are close to those reported in the literature for systems associated with self-organized criticality (SOC) such as forest-fire models; however, the typical assumptions of SOC need not apply. Our results demonstrate that power-law scalings in cluster size distributions are not restricted to systems for antagonistic interactions in which a clear separation of time-scales holds. The patterns are characteristic of processes of growth and inhibition in space, such as those in predator-prey and disturbance-recovery dynamics. Inversions of these patterns, that is, scalings with a positive slope as described for plankton distributions, would therefore require spatial forcing by environmental variability.

Self-organization, the cascade model, and natural hazards

We consider the frequency-size statistics of two natural hazards, forest fires and landslides. Both appear to satisfy power-law (fractal) distributions to a good approximation under a wide variety of conditions. Two simple cellular-automata models have been proposed as analogs for this observed behavior, the forest fire model for forest fires and the sand pile model for landslides. The behavior of these models can be understood in terms of a self-similar inverse cascade. For the forest fire model the cascade consists of the coalescence of clusters of trees; for the sand pile model the cascade consists of the coalescence of metastable regions.

Power law distributions of burst duration and interburst interval in the solar wind: turbulence or dissipative self-organized criticality?

We calculate the probability density functions P of burst energy e, duration T, and interburst interval tau for a known turbulent system in nature. Bursts in the Earth-Sun component of the Poynting flux at 1 AU in the solar wind were measured using the MFI and SWE experiments on the NASA WIND spacecraft. We find P(e) and P(T) to be power laws, consistent with self-organized criticality (SOC). We find also a power-law form for P(tau) that distinguishes this turbulent cascade from the exponential P(tau) of ideal SOC, but not from some other SOC-like sandpile models. We discuss the implications for the relation between SOC and turbulence.

Inverse cascade via Burgers equation

Burgers equation is employed as a pedagogical device for analytically demonstrating the emergence of a form of inverse cascade to the lowest wavenumber in a flow. The transition from highly nonlinear mode-mode coupling to an ordered preference for large scale structure is shown, both analytically (revealing the presence of a global attractor) and via a numerical example. (c) 2000 American Institute of Physics.