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Hamiltonian dynamics of Boolean networks

Arturo Zapata-Cortés, Julio Aracena

TL;DR

This work investigates how Hamiltonian dynamics, defined by trajectories that visit all $2^n$ configurations, shape the interaction graph in Boolean networks and its subnetworks. It develops a rigorous framework linking global Hamiltonian behavior to local in degree and connectivity properties, proving that certain Hamiltonian regimes force connected or strongly connected interaction graphs and derive parity-based in-degree constraints. A key contribution is the construction of unate Boolean networks that realize Hamiltonian dynamics, including a self-dual Hamiltonian cycle family with complete interaction graphs, and a general method to realize any Hamiltonian digraph via 2-Hamiltonian concepts. These results provide theoretical tools for modeling complex dynamic interactions and extend the reach of unate networks to bijective and quasi-bijective dynamics, with potential generalization to $q\ge2$ alphabets.

Abstract

This article examines the impact of Hamiltonian dynamics on the interaction graph of Boolean networks. Three types of dynamics are considered: maximum height, Hamiltonian cycle, and an intermediate dynamic between these two. The study addresses how these dynamics influence the connectivity of the graph and the existence of variables that depend on all other variables in the system. Additionally, a family of unate Boolean networks capable of describing these three Hamiltonian behaviors is introduced, highlighting their specific properties and limitations. The results provide theoretical tools for modeling complex systems and contribute to the understanding of dynamic interactions in Boolean networks.

Hamiltonian dynamics of Boolean networks

TL;DR

This work investigates how Hamiltonian dynamics, defined by trajectories that visit all configurations, shape the interaction graph in Boolean networks and its subnetworks. It develops a rigorous framework linking global Hamiltonian behavior to local in degree and connectivity properties, proving that certain Hamiltonian regimes force connected or strongly connected interaction graphs and derive parity-based in-degree constraints. A key contribution is the construction of unate Boolean networks that realize Hamiltonian dynamics, including a self-dual Hamiltonian cycle family with complete interaction graphs, and a general method to realize any Hamiltonian digraph via 2-Hamiltonian concepts. These results provide theoretical tools for modeling complex dynamic interactions and extend the reach of unate networks to bijective and quasi-bijective dynamics, with potential generalization to alphabets.

Abstract

This article examines the impact of Hamiltonian dynamics on the interaction graph of Boolean networks. Three types of dynamics are considered: maximum height, Hamiltonian cycle, and an intermediate dynamic between these two. The study addresses how these dynamics influence the connectivity of the graph and the existence of variables that depend on all other variables in the system. Additionally, a family of unate Boolean networks capable of describing these three Hamiltonian behaviors is introduced, highlighting their specific properties and limitations. The results provide theoretical tools for modeling complex systems and contribute to the understanding of dynamic interactions in Boolean networks.
Paper Structure (8 sections, 22 theorems, 43 equations, 9 figures, 2 tables)

This paper contains 8 sections, 22 theorems, 43 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. In-degree and connectivity of the interaction graph
  4. Hamiltonian unate Boolean networks
  5. Self-dual Boolean networks
  6. Family of Hamiltonian cycle self-dual unate Boolean networks
  7. General case of Hamiltonian Boolean networks
  8. Conclusions

Key Result

Lemma 3.1

Let $f$ be a Boolean network with $n \in \mathbb{N}$ variables and $j \in [n]$ such that $|T(f_j)|$ is odd. Then, the in-degree of vertex $j$ in the interaction graph is $n$.

Figures (9)

  • Figure 1: Boolean network $f$ from \ref{['eje0003']}.
  • Figure 2: Boolean network $f$ from \ref{['eje0004']}.
  • Figure 3: Boolean network $f$ from \ref{['eje0005']}.
  • Figure 4: Boolean network $f$ from \ref{['eje0006']}.
  • Figure 5: Quasi-Hamiltonian dynamics.
  • ...and 4 more figures

Theorems & Definitions (61)

  • Example 2.1
  • Definition 2.1
  • Definition 2.2
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.3: crama2011boolean
  • Lemma 3.1: 10.1093/pnasnexus/pgac017, mitesis
  • proof
  • Proposition 3.1
  • ...and 51 more