An Extensive Study of Two-Node McCulloch-Pitts Networks

TL;DR

This work systematically catalogs all two-node McCulloch-Pitts networks with weights in $\{-1,0,1\}$ and binary or bipolar node states, expanding the regulatory landscape from five ecological interaction classes to 39 signed graphs with self-loops. By analyzing the full model space through a spectrum-based state-transition framework, the authors classify attractors as fixed points or cycles and quantify robustness to parameter changes, initial conditions, and rule mutations across six variants, including a new V7 that mirrors V4. They demonstrate that the same signed regulatory graph can yield different dynamics under different variants and update schemes, challenging the sufficiency of structural graphs for predicting behavior and highlighting strong links between canalization, update rules, and stability. The findings reveal minimal yet rich dynamical complexity in a small network, offering insights into the design and interpretation of regulator networks in biology and beyond.

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.

Figures (7)

• Figure 1: (A) The two basic types for two-node interaction: master-slave(unidirectional) and mutual-interaction (bidirectional). (B) The five basic types for two-node interaction when positive (red) or negative (black) signs are added to the arrows: commensalism (unidirectional positive), amensalism (unidirectional negative), mutualism (two positives links), predator-prey (one positive and one negative links), and competition (two negatives links).
• Figure 2: The five basic types in Fig.\ref{['fig1']} are expanded to 39 models (34 here and 5 in Fig.\ref{['fig1']}) by including self links. Positive self links are called autocatalysis (autocatalytic), and negative self links self-regulation (self-regulating).
• Figure 3: Illustration that given the same regulatory graph, the limiting dynamics can depend on the variants. The rule R8 is used, with $a=b=-1$, $c=1$, $d=0$. The $x$ axis represents the node-1, and $y$ for node-2. grey/pink line represents the threshold line for $x$/$y$ of Eq.(\ref{['eq-mpn']}). The dashed lines are the state transition (a state without a dashed line means that it has a transition to itself). (A) bipolar as it is (V1), 4-cycle; (B) bipolar positive (V2), 3-cycle; (C) bipolar negative (V3), 3-cycle; (D) binary as it is (V4), two fixed-points; (E) binary positive (V5), one fixed point; and (F) binary negative (V6), one fixed-point.
• Figure 4: Illustration of the construction of global interaction graph by the MPN model R7, for three different variants. (A) The original regulatory graph where the sign of an arrow is determined by the parameters $(a,b,c,d)$. (B) The reconstructed global interaction graph for V1 (bipolar, as-it-is). The four network states marked at the four corners of a square represent four mappings: $(1,-1)$ at lower-left for $(-1,-1) \rightarrow (1,-1)$, $(1,1)$ at lower-right for $(1,-1) \rightarrow (1,1)$, $(-1,-1)$ at upper-left for $(-1,1) \rightarrow (-1,-1)$, and $(-1,1)$ at upper-right for $(1,1) \rightarrow (-1,1)$. Moving from left to right, the first bits are unchanged and the second bits increase. It implies $x$ has positive impact on $y$. Moving from bottom to top, the second bits are unchanged whereas the first bits decrease. It implies $y$ has a negative impact on $x$. (C) similar to (B) for variant-2 (bipolar positive). (D) similar to (B) for variant-6 (binary negative).
• Figure 5: UMAP of unfolded rule space of 81 two-node MPN models. The two axes in UMAP do not have a direct interpretable meaning. When two rules, represented by two points, are linked, the Hamming distance between the two is one. (A) nodes are colored by pink (V1 F$_4$ rules), red (V1 F$_2$ rules), orange (V1 M rules), blue (V1 2-cycle rules), and black (V1 4-cycle rules). The $N=1$ or $N=0$ rules are shown as green dots and link (one Hamming distance) to them are in dashed lines. (B) nodes are colored by the robustness (percentage of point mutation in rule that does not change the limiting state) in V4, with darker colors for more robust rules. In legend of (B), %U denotes the percentage of unchanged limiting dynamics.