Self-organized critical forest-fire model: Mean-field theory and simulation results in 1 to 6 dimenisons

The emergence of scale-free fires in Australia

Between 2019 and 2020, during the country's hottest and driest year on record, Australia experienced a dramatic bushfire season, with catastrophic ecological and environmental consequences. Several studies highlighted how such abrupt changes in fire regimes may have been in large part a consequence of climate change and other anthropogenic transformations. Here, we analyze the monthly evolution of the burned area in Australia from 2000 to 2020, obtained via satellite imaging through the MODIS platform. We find that the 2019-2020 peak is associated with signatures typically found near critical points. We introduce a modeling framework based on forest-fire models to study the properties of these emergent fire outbreaks, showing that the behavior observed during the 2019-2020 fire season matches the one of a percolation transition, where system-size outbreaks appear. Our model also highlights the existence of an absorbing phase transition that might be eventually crossed, after which the vegetation cannot recover.

A short tutorial on the Janus-faced logic of differentiation waves and differentiation trees and their evolution

Differentiation waves offer a different perspective on causality in embryogenesis from that of molecular developmental biology. Janus-faced cybernetic logic, with global and local top down/bottom up dynamics, eschews reductionism, is distinct from emergence, and outlines the process theoretically. Most aspects of differentiation waves require further investigation.

Avalanche size distribution of an integrate-and-fire neural model on complex networks

We considered the neural avalanche dynamics of a modified integrate-and-fire model on complex networks, as well as the neural dynamics in a fully connected network, random network, small-world network, and scale-free network. We observed the self-organized criticality of the neural model on complex networks. The probability distribution of the avalanche size and lifetime follow the power law at the critical synaptic strength. Neuronal dynamics on a complex network are not universal. The critical exponents of the avalanche dynamics depend on the structure of the complex network. We observed that the critical exponents deviate from the mean-field value.

Cascading failures in scale-free interdependent networks

Large cascades are a common occurrence in many natural and engineered complex systems. In this paper we explore the propagation of cascades across networks using realistic network topologies, such as heterogeneous degree distributions, as well as intra- and interlayer degree correlations. We find that three properties, scale-free degree distribution, internal network assortativity, and cross-network hub-to-hub connections, are all necessary components to significantly reduce the size of large cascades in the Bak-Tang-Wiesenfeld sandpile model. We demonstrate that correlations present in the structure of the multilayer network influence the dynamical cascading process and can prevent failures from spreading across connected layers. These findings highlight the importance of internal and cross-network topology in optimizing robustness of interconnected systems.

Measuring neuronal avalanches in disordered systems with absorbing states

Power-law-shaped avalanche-size distributions are widely used to probe for critical behavior in many different systems, particularly in neural networks. The definition of avalanche is ambiguous. Usually, theoretical avalanches are defined as the activity between a stimulus and the relaxation to an inactive absorbing state. On the other hand, experimental neuronal avalanches are defined by the activity between consecutive silent states. We claim that the latter definition may be extended to some theoretical models to characterize their power-law avalanches and critical behavior. We study a system in which the separation of driving and relaxation time scales emerges from its structure. We apply both definitions of avalanche to our model. Both yield power-law-distributed avalanches that scale with system size in the critical point as expected. Nevertheless, we find restricted power-law-distributed avalanches outside of the critical region within the experimental procedure, which is not expected by the standard theoretical definition. We remark that these results are dependent on the model details.

Self-organization of dragon king failures

The mechanisms underlying cascading failures are often modeled via the paradigm of self-organized criticality. Here we introduce a simple network model where nodes self-organize to be either weakly or strongly protected against failure in a manner that captures the trade-off between degradation and reinforcement of nodes inherent in many network systems. If strong nodes cannot fail, any failure is contained to a single, isolated cluster of weak nodes and the model produces power-law distributions of failure sizes. We classify the large, rare events that involve the failure of only a single cluster as "black swans." In contrast, if strong nodes fail once a sufficient fraction of their neighbors fail, then failure can cascade across multiple clusters of weak nodes. If over 99.9% of the nodes fail due to this cluster hopping mechanism, we classify this as a "dragon king," which are massive failures caused by mechanisms distinct from smaller failures. The dragon kings observed are self-organized, existing over a wide range of reinforcement rates and system sizes. We find that once an initial cluster of failing weak nodes is above a critical size, the dragon king mechanism kicks in, leading to piggybacking system-wide failures. We demonstrate that the size of the initial failed weak cluster predicts the likelihood of a dragon king event with high accuracy and we develop a simple control strategy that can dramatically reduce dragon kings and other large failures.

Urban retail location: Insights from percolation theory and spatial interaction modeling

Characterising road networks has been the focus of a large body of research due to it being the main driver of activities in an urban ecosystem and the structuring factor in the dynamics of the city. One of these activities, and one with the largest economical impact in a city, is retail dynamics and its evolution. Therefore, the mathematical modeling of the location of retail activities and of the emergence of clustering in retail centers has as well generated a large number of works. Despite these two interwoven components strongly depending on one another and their fundamental importance in understanding cities, little work has been done in order to compare their local and global properties. Here we compare the road network's hierarchical structure, unveiled through a percolation analysis of the network, with the retail location distribution defined by exploiting a gravity-based retail model. We interpret the great agreement in the city's organizations as it emerges from both methodologies as new evidence of the interdependence of these two crucial dimensions of a city's life.

The angular nature of road networks

Road networks are characterised by several structural and geometrical properties. The topological structure determines partially the hierarchical arrangement of roads, but since these are networks that are spatially constrained, geometrical properties play a fundamental role in determining the network’s behaviour, characterising the influence of each of the street segments on the system. In this work, we apply percolation theory to the UK’s road network using the relative angle between street segments as the occupation probability. The appearance of the spanning cluster is marked by a phase transition, indicating that the system behaves in a critical way. Computing Shannon’s entropy of the cluster sizes, different stages of the percolation process can be discerned, and these indicate that roads integrate to the giant cluster in a hierarchical manner. This is used to construct a hierarchical index that serves to classify roads in terms of their importance. The obtained classification is in very good correspondence with the official designations of roads. This methodology hence provides a framework to consistently extract the main skeleton of an urban system and to further classify each road in terms of its hierarchical importance within the system.

Labyrinthine clustering in a spatial rock-paper-scissors ecosystem

The spatial rock-paper-scissors ecosystem, where three species interact cyclically, is a model example of how spatial structure can maintain biodiversity. We here consider such a system for a broad range of interaction rates. When one species grows very slowly, this species and its prey dominate the system by self-organizing into a labyrinthine configuration in which the third species propagates. The cluster size distributions of the two dominating species have heavy tails and the configuration is stabilized through a complex spatial feedback loop. We introduce a statistical measure that quantifies the amount of clustering in the spatial system by comparison with its mean-field approximation. Hereby, we are able to quantitatively explain how the labyrinthine configuration slows down the dynamics and stabilizes the system.

Dynamics of the one-dimensional self-organized forest-fire model

We examine the dynamical evolution of the one-dimensional self-organized forest-fire model (FFM), when the system is far from its statistically steady state. In particular, we investigate situations in which conditions change on a time scale that is faster than, or of the order of the typical time needed for relaxation. An analytical approach is introduced based on a hierarchy of first-order nonlinear differential equations. This hierarchy can be closed at any level, yielding a sequence of successively more accurate descriptions of the dynamics. It is found that our approximate description can yield a faithful description of the FFM dynamics, even when a low order truncation is used. Employing both full simulations of the FFM and our approximate descriptions, we examine the time scales and cluster-size-dependent dynamics of relaxation to the statistical equilibrium. As an example of changing external conditions in a natural forest, the effects of a time-dependent lightning frequency are considered.

Homeostasis of neuronal avalanches during postnatal cortex development in vitro

Cortical networks in vivo and in vitro are spontaneously active in the absence of inputs, generating highly variable bursts of neuronal activity separated by up to seconds of quiescence. Previous measurements in adult rat cortex revealed an intriguing underlying organization of these dynamics, termed neuronal avalanches, which is indicative of a critical network state. Here we demonstrate that neuronal avalanches persist throughout development in cortical slice cultures from newborn rats. More specifically, we find that in spite of large variations of average rate in activity, spontaneous bursts occur with power-law distributed sizes (exponent -1.5) and a critical branching parameter close to 1. Our findings suggest that cortical networks homeostatically regulate a critical state during postnatal maturation.

Critical scaling in standard biased random walks

The spatial coverage produced by a single discrete-time random walk, with an asymmetric jump probability p not equal 1/2 and nonuniform steps, moving on an infinite one-dimensional lattice is investigated. Analytical calculations are complemented with Monte Carlo simulations. We show that, for appropriate step sizes, the model displays a critical phenomenon, at p=p(c). Its scaling properties as well as the main features of the fragmented coverage occurring in the vicinity of the critical point are shown. In particular, in the limit p-->p(c), the distribution of fragment lengths is scale-free, with nontrivial exponents. Moreover, the spatial distribution of cracks (unvisited sites) defines a fractal set over the spanned interval. Thus, from the perspective of the covered territory, a very rich critical phenomenology is revealed in a simple one-dimensional standard model.

Scaling and universality in proportional elections

A most debated topic of the last years is whether simple statistical physics models can explain collective features of social dynamics. A necessary step in this line of endeavor is to find regularities in data referring to large-scale social phenomena, such as scaling and universality. We show that, in proportional elections, the distribution of the number of votes received by candidates is a universal scaling function, identical in different countries and years. This finding reveals the existence in the voting process of a general microscopic dynamics that does not depend on the historical, political, and/or economical context where voters operate. A simple dynamical model for the behavior of voters, similar to a branching process, reproduces the universal distribution.

Topology regulates the distribution pattern of excitations in excitable dynamics on graphs

We study the average excitation density in a simple model of excitable dynamics on graphs and find that this density strongly depends on certain topological features of the graph, namely connectivity and degree correlations, but to a lesser extent on the degree distribution. Remarkably, the average excitation density is changed via the distribution pattern of excitations: An increase in connectivity induces a transition from globally to locally organized excitations and, as a result, leads to an increase in the excitation density. A similar transition can be induced by increasing the rate of spontaneous excitations while keeping the graph architecture constant.

Investigation of the forest-fire model on a small-world network

It is shown that the forest-fire model of Bak et al. run on a square lattice network with additional long-range interactions in the spirit of a small-world network results in a scale-free system reminiscent of self-organized criticality without recourse to fine tuning. As the number of these long-range interactions is increased, the cluster size distribution exponent is found to decrease in magnitude as the small-world regime is entered, indicating a change in its universality class. It is suggested that such a model could have applicability in the study of disease spreading in human populations.

Synchronization and coarsening (without self-organized criticality) in a forest-fire model

We study the long-time dynamics of a forest-fire model with deterministic tree growth and instantaneous burning of entire forests by stochastic lightning strikes. Asymptotically the system organizes into a coarsening self-similar mosaic of synchronized patches within which trees regrow and burn simultaneously. We show that the average patch length <L> grows linearly with time as t--> infinity. The number density of patches of length L, N(L,t), scales as <L>-2N(L/<L>), and within a mean-field rate equation description we find that this scaling function decays as N(x) approximately e(-1/x) for x-->0, and as e(-x) for x--> infinity. In one dimension, we develop an event-driven cluster algorithm to study the asymptotic behavior of large systems. Our numerical results are consistent with mean-field predictions for patch coarsening.

Type of self-organized criticality model based on neural networks

Based on the standard self-organizing map neural network model, we introduce a kind of coupled map lattice system to investigate self-organized criticality (SOC) in the activity of model neural populations. Our system is simulated by a more detailed integrate-and-fire mechanism and a kind of local perturbation driving rule; it can display SOC behavior in a certain range of system parameters, even with period boundary condition. More importantly, when the influence of synaptic plasticity is adequately considered, we can find that our system's learning process plays a promotive role in the emergence of SOC behavior.

Self-organized critical forest-fire model on large scales

We discuss the scaling behavior of the self-organized critical forest-fire model on large length scales. As indicated in earlier publications, the forest-fire model does not show conventional critical scaling, but has two qualitatively different types of fires that superimpose to give the effective exponents typically measured in simulations. We show that this explains not only why the exponent characterizing the fire-size distribution changes with increasing correlation length, but allows us also to predict its asymptotic value. We support our arguments by computer simulations of a coarse-grained model, by scaling arguments and by analyzing states that are created artificially by superimposing the two types of fires.

Irreversible phase transitions driven by an oscillatory parameter in a far-from-equilibrium system

The dynamic response of a forest-fire model to the harmonic variation of an external parameter is studied by means of numerical simulations. Second-order irreversible phase transitions driven by the harmonic input are reported. The location of such transitions depends on both the amplitude and period of the input signal. By means of epidemic studies the relevant critical exponents can be determined, which allow us to place the reported transitions in the universality class of directed percolation. This conclusion is also supported by a field theoretical calculation.

Power laws, highly optimized tolerance, and generalized source coding

We introduce a family of robust design problems for complex systems in uncertain environments which are based on tradeoffs between resource allocations and losses. Optimized solutions yield the "robust, yet fragile" features of highly optimized tolerance and exhibit power law tails in the distributions of events for all but the special case of Shannon coding for data compression. In addition to data compression, we construct specific solutions for world wide web traffic and forest fires, and obtain excellent agreement with measured data.

Corrections to scaling in the forest-fire model

We present a systematic study of corrections to scaling in the self-organized critical forest-fire model. The analysis of the steady-state condition for the density of trees allows us to pinpoint the presence of these corrections, which take the form of subdominant exponents modifying the standard finite-size scaling form. Applying an extended version of the moment analysis technique, we find the scaling region of the model and compute nontrivial corrections to scaling.

On the critical behaviour of simple epidemics

We show how ideas and models which were originally introduced to gain an understanding of critical phenomena can be used to interpret the dynamics of epidemics of communicable disease in real populations. Specifically, we present an analysis of the dynamics of disease outbreaks for three common communicable infections from a small isolated island population. The strongly fluctuating nature of the temporal incidence of disease is captured by the model, and comparisons between exponents calculated from the data and from simulations are made. A forest-fire model with sparks is used to classify the observed scaling dynamics of the epidemics and provides a unified picture of the epidemiology which conventional epidemiological analysis is unable to reproduce. This study suggests that power-law scaling can emerge in natural systems when they are driven on widely separated time-scales, in accordance with recent analytic renormalization group calculations.

A scaling analysis of measles epidemics in a small population

We present a detailed analysis of the pattern of measles outbreaks in the small isolated community of the Faroe Islands. Measles outbreaks in this population are characterized by frequent fade-out of infection resulting in long intervals when the disease is absent from the islands. Using an analysis of the distribution of epidemic sizes and epidemic durations we propose that the dynamical structure observed in the measles case returns reflects the existence of an underlying scaling mechanism. Consequently the dynamics are not as purely stochastic as is usually thought for epidemiological systems of this sort. We use a lattice-based epidemic model to provide a theoretical estimate of the scaling exponents and show that a conventional compartmental SEIR model is unable to reproduce this result. The methods discussed in this paper are general and represent a novel way to consider the dynamics of any other communicable disease where there is frequent fade-out in the case returns.