A stochastic cellular automaton (SCA), also known as a probabilistic cellular automaton (PCA), is a type of computational model. It consists of a grid of cells, where each cell has a particular state (e.g., "on" or "off"). The states of all cells evolve in discrete time steps according to a set of rules.
Unlike a standard cellular automaton where the rules are deterministic (fixed), the rules in a stochastic cellular automaton are probabilistic. This means a cell's next state is determined by chance, according to a set of probabilities that depend on the states of neighboring cells.[1]
Despite the simple, local, and random nature of the rules, these models can produce complex global patterns through processes like emergence and self-organization. They are used to model a wide variety of real-world phenomena where randomness is a factor, such as the spread of forest fires, the dynamics of disease epidemics, or the simulation of ferromagnetism in physics (see Ising model).
As a mathematical object, a stochastic cellular automaton is a discrete-time random dynamical system. It is often analyzed within the frameworks of interacting particle systems and Markov chains, where it may be called a system of locally interacting Markov chains.[2][3] See [4] for a more detailed introduction.
Formal definition
From the perspective of probability theory, a stochastic cellular automaton is a discrete-time Markov process. The configuration of all cells at a given time is a state in a product space. Here, is a graph representing the grid of cells (e.g., ), and each is the finite set of possible states for the cell (e.g., ).
The transition probability, which defines the dynamics, has a product form:
where is the next configuration and is a probability distribution on .
Locality is a key requirement, meaning the probability of a cell changing its state depends only on the states of its neighbors. This is expressed as , where is a finite neighborhood of cell and are the states of the cells in that neighborhood. See [5] for a more detailed introduction from this point of view.
↑ Toom, A. L. (1978), Locally Interacting Systems and their Application in Biology: Proceedings of the School-Seminar on Markov Interaction Processes in Biology, held in Pushchino, March 1976, Lecture Notes in Mathematics, vol.653, Springer-Verlag, Berlin-New York, ISBN978-3-540-08450-1, MR0479791
↑ Boas, Sonja E. M.; Jiang, Yi; Merks, Roeland M. H.; Prokopiou, Sotiris A.; Rens, Elisabeth G. (2018). "Chapter 18: Cellular Potts Model: Applications to Vasculogenesis and Angiogenesis". In Louis, P.-Y.; Nardi, F. R. (eds.). Probabilistic Cellular Automata. Springer. doi:10.1007/978-3-319-65558-1_18. hdl:1887/69811. ISBN9783319655581.
Further reading
Almeida, R. M.; Macau, E. E. N. (2010), "Stochastic cellular automata model for wildland fire spread dynamics", 9th Brazilian Conference on Dynamics, Control and their Applications, June 7–11, 2010, vol.285, p.012038, doi:10.1088/1742-6596/285/1/012038.
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