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

This chapter explores how a “natural” process generates dynamically something that is conceptually similar to a percolation cluster by using the case of forest fires. It first provides an overview of the forest-fire model, which is essentially a probabilistic cellular automata, before discussing its numerical implementation using the Python code. It then describes a representative simulation showing the triggering, growth, and decay of a large fire in a representative forest-fire model simulation on a small 100 x 100 lattice. It also considers the behavior of the forest-fire model as well as its self-organized criticality and concludes with an analysis of the advantages and limitations of wildfire management. The chapter includes exercises and further computational explorations, along with a suggested list of materials for further reading.

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Hypernetwork models based on random hypergraphs

International Journal of Modern Physics C ◽

10.1142/s0129183119500529 ◽

2019 ◽

Vol 30 (08) ◽

pp. 1950052

Author(s):

Feng Hu ◽

Jin-Li Guo ◽

Fa-Xu Li ◽

Hai-Xing Zhao

Keyword(s):

Field Theory ◽

Mean Field Theory ◽

Mean Field ◽

World Systems ◽

Uniform Hypergraphs ◽

Different Types ◽

Powerful Means ◽

Simulation Results ◽

Parameter Values ◽

Random Hypergraphs

Hypernetworks are ubiquitous in real-world systems. They provide a powerful means of accurately depicting networks of different types of entity and will attract more attention from researchers in the future. Most previous hypernetwork research has been focused on the application and modeling of uniform hypernetworks, which are based on uniform hypergraphs. However, random hypernetworks are generally more common, therefore, it is useful to investigate the evolution mechanisms of random hypernetworks. In this paper, we construct three dynamic evolutional models of hypernetworks, namely the equal-probability random hypernetwork model, the Poisson-probability random hypernetwork model and the certain-probability random hypernetwork model. Furthermore, we analyze the hyperdegree distributions of the three models with mean-field theory, and we simulate each model numerically with different parameter values. The simulation results agree well with the results of our theoretical analysis, and the findings indicate that our models could help understand the structure and evolution mechanisms of real systems.

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Universality in the One-Dimensional Self-Organized Critical Forest-Fire Model

Zeitschrift für Naturforschung A ◽

10.1515/zna-1994-0907 ◽

1994 ◽

Vol 49 (9) ◽

pp. 856-860

Author(s):

Barbara Drossel ◽

Siegfried Clar ◽

Franz Schwabl

Keyword(s):

Size Distribution ◽

Forest Fire ◽

Nearest Neighbor ◽

The Self ◽

One Dimension ◽

Fire Model ◽

One Dimensional ◽

Self Organized ◽

Forest Fire Model ◽

The One

Abstract We modify the rules of the self-organized critical forest-fire model in one dimension by allowing the fire to jum p over holes of ≤ k sites. An analytic calculation shows that not only the size distribution of forest clusters but also the size distribution of fires is characterized by the same critical exponent as in the nearest-neighbor model, i.e. the critical behavior of the model is universal. Computer simulations confirm the analytic results.

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Exact results for the one-dimensional self-organized critical forest-fire model

Physical Review Letters ◽

10.1103/physrevlett.71.3739 ◽

1993 ◽

Vol 71 (23) ◽

pp. 3739-3742 ◽

Cited By ~ 56

Author(s):

Barbara Drossel ◽

Siegfried Clar ◽

Franz Schwabl

Keyword(s):

Forest Fire ◽

Exact Results ◽

Fire Model ◽

One Dimensional ◽

Self Organized ◽

Forest Fire Model ◽

The One

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SELF ORGANIZATION IN A FOREST-FIRE MODEL

Fractals ◽

10.1142/s0218348x93001118 ◽

1993 ◽

Vol 01 (04) ◽

pp. 1022-1029 ◽

Cited By ~ 8

Author(s):

B. DROSSEL ◽

F. SCHWABL

Keyword(s):

Forest Fire ◽

Critical Line ◽

Continuous Phase ◽

Excitable Media ◽

Fire Model ◽

Fire Propagation ◽

Self Organized ◽

Forest Fire Model ◽

New Type ◽

Fire Spreading

We generalize the forest-fire model of P. Bak et al., which contains a tree nearest growth probability p and fire spreading to the neighbors, by including a lightning probability f and an immunity g which is the probability that a tree catches no fire although one of its neighbors is burning. The model becomes self-organized critical in the limit f/p→0, provided the time scales of tree growth and burning down of forest clusters are separated. The size distribution of forest clusters obeys a power law. We calculate the critical exponents in one dimension. A continuous phase transition is observed in the general forest-fire model when g reaches its critical value. We determine the critical line gC(p) and show that the critical fire propagation represents a new type of percolation. Finally, we point out similarities between the forest-fire model and excitable media, which comprise such different systems as chemical reactions, spreading of diseases and populations, and propagation of electrical activity in neurons.

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Fractality and Self-Organized Criticality of Wars

Fractals ◽

10.1142/s0218348x98000407 ◽

1998 ◽

Vol 06 (04) ◽

pp. 351-357 ◽

Cited By ~ 90

Author(s):

D. C. Roberts ◽

D. L. Turcotte

Keyword(s):

Forest Fire ◽

Power Law ◽

Forest Fires ◽

Size Dependence ◽

Fire Model ◽

Self Organized ◽

Battle Deaths ◽

Forest Fire Model ◽

Alternative Measures ◽

Self Organized Criticality

This paper considers the frequency-size statistics of wars. Using several alternative measures of the intensity of a war in terms of battle deaths, we find a fractal (power-law) dependence of number on intensity. We show that the frequency-size dependence of forest fires is essentially identical to that of wars. The forest-fire model provides a basis for understanding the distribution of forest firest in terms of self-organized criticality. We extend the analogy to wars in terms of the initial ignition (outbreak of war) and its spread to a group of metastable countries.

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