A Sensitivity Analysis of Cellular Automata and Heterogeneous Topology Networks: Partially-Local Cellular Automata and Homogeneous Homogeneous Random Boolean Networks

TL;DR

The paper investigates how topological heterogeneity, via PLCA and HHRBN substrates, affects reservoir computing performance on a 5-bit memory benchmark relative to standard ECA. It introduces Temporal Derrida Plots and a defect-collapse framework to quantify sensitivity and attractor stability, revealing that disordered topologies can both enhance sensitivity and promote ordered collapse, yielding a shrinking edge-of-chaos under current conditions. Across Life-Like CA, ECA, and intermediate substrates, CA consistently outperforms heterogeneous substrates on the memory task, while ME reductions, rule dependencies, and network topology critically shape results. The work highlights practical implications for energy-efficient RC implementations and proposes a discrete chaos analogue to better describe computation on intrinsically discrete substrates. In sum, topology and rule-space structure jointly govern the computational utility of CA/RBN reservoirs, with clear hardware-relevant implications for edge AI systems.

Abstract

Elementary Cellular Automata (ECA) are a well-studied computational universe that is, despite its simple configurations, capable of impressive computational variety. Harvesting this computation in a useful way has historically shown itself to be difficult, but if combined with reservoir computing (RC), this becomes much more feasible. Furthermore, RC and ECA enable energy-efficient AI, making the combination a promising concept for Edge AI. In this work, we contrast ECA to substrates of Partially-Local CA (PLCA) and Homogeneous Homogeneous Random Boolean Networks (HHRBN). They are, in comparison, the topological heterogeneous counterparts of ECA. This represents a step from ECA towards more biological-plausible substrates. We analyse these substrates by testing on an RC benchmark (5-bit memory), using Temporal Derrida plots to estimate the sensitivity and assess the defect collapse rate. We find that, counterintuitively, disordered topology does not necessarily mean disordered computation. There are countering computational "forces" of topology imperfections leading to a higher collapse rate (order) and yet, if accounted for, an increased sensitivity to the initial condition. These observations together suggest a shrinking critical range.

Figures (25)

• Figure 1: There is a big difference between CA and BNN. The difference can be viewed as a series of discrete steps between substrates, but even between RBN and CA, there are many discrete steps. This figure illustrates the different substrates as a direct path, but note that the steps from CA to RBN could have been done differently than illustrated. Also note, that this is a simplified imperfect model of the substrate space between ECA and BNN.
• Figure 2: Example of 1 dimensional CA with rule 90 with TT, starting from a central cell on, executing 7 time-steps.
• Figure 3: Common 2-dimensional neighbourhood schemes.
• Figure 4: Single time-step of a 2-dimensional CA with Conway's Game of Life rules. It features an oscillating blinker, a stable block, and a spaceship glider.
• Figure 5: Example of an RBN with 7 Nodes and 3 neighbours, with a transition Table in two forms and a short execution example.