An Extensive Study of Two-Node McCulloch-Pitts Networks

TL;DR

This paper comprehensively analyzes the dynamics of all 39 two-node McCulloch-Pitts neural network models with various configurations, exploring their behaviors, stability, and robustness under different conditions.

Contribution

It provides a complete classification of two-node McCulloch-Pitts networks, including new insights into their dynamics, stability, and the effects of model variations and parameter changes.

Findings

39 distinct network models analyzed

Dynamics vary with Boolean variable encoding

Certain models exhibit high robustness and stability

Abstract

Networks with two nodes are previously grouped into either two classes (mutually interactive, master-slave) or five classes (mutualism, competition, predator-prey, commensalism, amensalism). By allowing self-loops, the number of signed regulatory graphs increases to 39. We provide a complete summary of dynamical behaviors of the 39 two-node McCulloch-Pitts models when the link weights are constrained to three values [$-1$,0,$+1$] and Boolean node variables. Depending on whether the Boolean values are [$-1,1$] (bipolar) or [0,1] (binary), we show that the dynamics could also be different with the same signed regulatory graphs. We demonstrate that slight variations in the McCulloch-Pitts model (called variants) may lead to fundamentally different dynamics. We study the full model space and three kinds of robustness or stability: a) of a rule against parameter change on its overall dynamics, b) for a given state against parameter change on its final state, and c) against an initial state change on its final state. All these stability properties are loosely related to a model's limiting dynamics, with the fixed-point rules to be more stable in the first two types of robustness, but less stable in the third robustness type. These analyses pave the way towards a better understanding of a minimum complex system.