Overview: PCA Models and Issues

Ripple effect in the supply chain network: Forward and backward disruption propagation, network health and firm vulnerability

“…The transition probability of each node is determined by the current status of itself and all of its neighbors’ current status as Eq (3.1) and Eq (3.2) . To formulate such a problem, the model is set up as locally interacting Markov chains, also known as probabilistic cellular automata (PCA) ( Fernández, Louis, & Nardi, 2018 ). The state space of such a model is the tensor product of the statuses of all the local Markov chains, which is huge in our context.…”

“…This might work for a small sized network, but it will eventually become infeasible when the size or complexity of the network grows ( Garvey & Carnovale, 2020 ). In fact, the area of PCA acknowledges its complexity and suggests that it is used as a flexible modeling, such as agent-based modeling, and simulation framework in an applied context ( Fernández et al, 2018 ).…”

Density classification performance and ergodicity of the Gacs-Kurdyumov-Levin cellular automaton model IV

“…The behavior displayed by N → /L and τ(L, α) in Figures 6, 8 and 9 supports this idea. It should be remarked that while the ergodicity of one-dimensional deterministic CA is in general undecidable, most PCA are believed to be ergodic, with the notable exception of Gacs' very complicated (and still controversial) counterexample [22][23][24][50][51][52][53][54]. We established an upper bound on the critical level of noise of GKL-IV above which it becomes ergodic.…”

“…In other words, for GKL-IV model under noise level α, at every time step the probability of writing the new state to a cell according to the rules is (1 − α) + 1 4 α, while the probability of doing it incorrectly is 3 4 α. A PCA is ergodic if it eventually forgets its initial state, meaning that it has a unique invariant measure-a unique probability distribution of states over the configuration space of the model that does not change under the dynamics [12,[50][51][52][53][54]. Remarkably, GKL found by means of numerical experiments evidence that GKL-IV may be nonergodic below a certain small level of noise α * ≈ 0.05 [2].…”

Coupling from the past for exponentially ergodic one-dimensional probabilistic cellular automata

“…Probabilistic Cellular Automata (PCA) form a class of discrete-time Markov processes on spaces of the form A L , where L is a lattice (typically, L = Z d for some d ≥ 1), and A is a finite set called an alphabet (see e.g. [6] for a seminal reference, and [2] for a recent survey). In the present paper, we consider one-dimensional PCAs, that is, L = Z.…”