Probabilistic cellular automata with local transition matrices: synchronization, ergodicity, and inference

TL;DR

This work introduces a probabilistic cellular automaton on a finite cycle where each site's update probability depends linearly on its neighborhood's empirical distribution through a local transition matrix $\\mathbf{T}$. It establishes precise conditions for synchronization (period $K$) and exponential ergodicity (aperiodicity) in terms of $\\mathbf{T}$, and proves a 1-1 correspondence between local and global transition dynamics along with Lipschitz control. The paper then develops least squares estimators to infer $\\mathbf{T}$ from multiple trajectories, long ergodic trajectories, or ensemble data, with rigorous identifiability criteria and asymptotic normality, plus non-asymptotic bounds. These results provide a scalable, data-efficient framework for learning interaction kernels in PCA, with practical implications for predicting synchronization and guiding inference under various data regimes.

Abstract

We introduce a new class of probabilistic cellular automata that are capable of exhibiting rich dynamics such as synchronization and ergodicity and can be easily inferred from data. The system is a finite-state locally interacting Markov chain on a circular graph. Each site's subsequent state is random, with a distribution determined by its neighborhood's empirical distribution multiplied by a local transition matrix. We establish sufficient and necessary conditions on the local transition matrix for synchronization and ergodicity. Also, we introduce novel least squares estimators for inferring the local transition matrix from various types of data, which may consist of either multiple trajectories, a long trajectory, or ensemble sequences without trajectory information. Under suitable identifiability conditions, we show the asymptotic normality of these estimators and provide non-asymptotic bounds for their accuracy.

Figures (3)

• Figure 1: (a) The system of $N= 8$ agents on a graph with an alphabet $\mathcal{A}= \{\text{Yellow, Turquoise, and Blue}\}$ with $K=3$. Each agent's transition depends linearly on the empirical distribution of its nearest neighbors with $n_v=2$ agents on each side. (b) The system moves from deterministic to stochastic dynamics; see Example \ref{['exp:T_move2next']}. (c) The system transitions from stochastic to deterministic dynamics, achieving a synchronization; see Example \ref{['exp:T-permutation']}.
• Figure 2: Mean ratios $\frac{\|\Delta \mathbf{P}\|_{p}}{\|\Delta \mathbf{T}\|_{p} }$ and $\frac{\|\Delta \pi\|_{p}}{\|\Delta \mathbf{T}\|_{p} }$ with $p=1,2$, where each $(N,K)$-pair is computed using 100 random $\mathbf{T}$ with entries sampled from uniform [0,1] followed by a row-normalization. Here the neighborhood has $n_v= \min\{3,\lfloor{N/2}\rfloor\}$. Note that $\frac{\|\Delta \mathbf{P}\|_{1}}{\|\Delta \mathbf{T}\|_{1} } = O(K^N)$, agreeing with Theorem \ref{['thm:Lip_P_T']}. Also, note that $\frac{\|\Delta \pi\|_{1}}{\|\Delta \mathbf{T}\|_{1} } = O(1)$, $\frac{\|\Delta \mathbf{P}\|_{2}}{\|\Delta \mathbf{T}\|_{2} } = O(1)$, and $\frac{\|\Delta \pi\|_{2}}{\|\Delta \mathbf{T}\|_{2} } = O(K^{-N/2})$.
• Figure 3: (a): Box plot of relative errors of the LSE estimators in 100 simulations for a system with $(N, K,n_v)=(8,3,3)$. The estimators converge at the same rate; however, the multi-trajectory LSE is significantly more accurate than the ensemble LSE. (b): A prediction of synchronization for Example \ref{['exp:T-permutation']} with $\mathbf{T}$ estimated from $M=10^3$ trajectories with length $L=100$. The sampling error in LSE-ensemble leads to a system without synchronizations. Here the colors represent the alphabet $\mathcal{A}= [K] = \{\text{Yellow, Turquoise, and Blue}\}$ with $K=3$.

Theorems & Definitions (33)

• Example 2.1: Non-interacting agents
• Example 2.2: Smallest model: $(N,K)=(2,2)$
• Example 2.3: From deterministic to stochastic dynamics
• Example 2.4: Synchronization: from stochastic to deterministic dynamics
• Definition 2.5: Synchronization
• Proposition 2.6
• Theorem 2.7: Synchronization
• Remark 2.8
• Proposition 2.9
• Theorem 2.10