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arxiv: 2501.03175 · v3 · pith:GZDGNJT2new · submitted 2025-01-06 · 💻 cs.DM

Hamiltonian dynamics of Boolean networks

Arturo Zapata-Cort\'es , Julio Aracena This is my paper

Pith reviewed 2026-05-23 06:00 UTC · model grok-4.3

classification 💻 cs.DM
keywords Boolean networksHamiltonian dynamicsinteraction graphunate functionsnetwork connectivitystate spacecomplex systems modeling
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The pith

Hamiltonian dynamics in Boolean networks force connectivity in the interaction graph and require variables depending on all others.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines three Hamiltonian dynamics—maximum height, cycle, and an intermediate type—on the state space of Boolean networks. It establishes that these dynamics impose connectivity requirements on the interaction graph and necessitate the existence of variables that depend on every other variable. The authors introduce a family of unate Boolean networks that realize these dynamics while satisfying the resulting structural conditions. This work supplies explicit constructions and limitations for networks that exhibit the behaviors. A reader would care because the results tie specific dynamical trajectories directly to graph-level properties used in modeling regulatory and logical systems.

Core claim

The three types of Hamiltonian dynamics influence the connectivity of the interaction graph and the existence of variables that depend on all other variables; a family of unate Boolean networks is introduced that can describe these behaviors, with their specific properties and limitations.

What carries the argument

The three Hamiltonian dynamics (maximum height, Hamiltonian cycle, intermediate) realized on the state space by unate Boolean functions.

If this is right

  • The interaction graph of any such network must be connected.
  • At least one variable must depend on every other variable in the network.
  • Unate functions suffice to realize all three dynamics.
  • The constructions come with explicit limitations on the admissible networks.
  • The results supply theoretical tools for modeling systems whose dynamics are Hamiltonian.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The constructions may allow systematic generation of Boolean networks with prescribed global dynamical properties.
  • Similar structural constraints could appear when Hamiltonian dynamics are imposed on other discrete dynamical systems.
  • Testing whether real gene-regulatory networks exhibit these Hamiltonian behaviors would require checking the dependence condition directly on their update functions.

Load-bearing premise

The three Hamiltonian dynamics are well-defined on the state space of Boolean networks and can be realized by unate functions without additional constraints on the interaction graph structure.

What would settle it

A concrete Boolean network that follows one of the three Hamiltonian dynamics yet has a disconnected interaction graph or lacks any variable depending on all others.

Figures

Figures reproduced from arXiv: 2501.03175 by Arturo Zapata-Cort\'es, Julio Aracena.

Figure 1
Figure 1. Figure 1: Boolean network f from Example 2.1. Definition 2.1. For x, y ∈ {0, 1} n , we define x ≤ y if, for every component, xi ≤ yi holds. Given f, a Boolean network with n ∈ N variables, the local activation function fj is said to be: 3 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Boolean network f from Example 2.2. Example 2.3. Given the Boolean network f = (f1, f2, f3) defined in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Boolean network f from Example 2.3. Example 2.4. Given f = (f1, f2, f3), described by the local activation functions and the interaction graph shown in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Boolean network f from Example 2.4. Our objective is on the properties of a BN with Hamiltonian dynamic. Next, we define a partition of the configuration space to identify patterns in the variable behavior that will be useful in the following results. Definition 2.3 ([10]). Given f, a Boolean network with n variables, and j ∈ [n], the set T(fj ) ⊆ {0, 1} n is defined as the set of true points, and F(fj ) ⊆… view at source ↗
Figure 5
Figure 5. Figure 5: Quasi-Hamiltonian dynamics. Lemma 3.2 strengthens the connection between the dynamics digraph and the resulting in￾teraction graph, highlighting how the global properties of a network influence the characteristics of its induced subnetworks. Next, we explore how these properties affect the connectivity of the interaction graph. Proposition 3.2. If GΓ is Hamiltonian, then for any f ∈ F(GΓ) is guaranteed G(f… view at source ↗
Figure 6
Figure 6. Figure 6: Interaction graph with signs for f [2](x1, x2) = (x2, x1) and its dynamics. To extend this construction to the case n + 1, we use f [n] a Hamiltonian cycle BN with n variables. Definition 4.3. Let f [1] = (x1) be a Boolean network with a single variable. Recursively, networks h [n+1], f[n+1] : {0, 1} n+1 → {0, 1} n+1 are constructed from f [n] , n ∈ N. Specifically, if z [n+1] =: z ∈ {0, 1} n+1 such that f… view at source ↗
Figure 7
Figure 7. Figure 7: shows f [2](x1, x2) = (x2, x1) and the associated auxiliary network h [3](x1, x2, x3) = (x2, x1, x3). By modifying the auxiliary network, swapping the preimages of ⃗0 ∈ {0, 1} n+1 and ⃗1 ∈ {0, 1} n+1, a new BN f [n+1] is constructed, which is a Hamiltonian cycle, unate, and self-dual in [n + 1] [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Dynamics of the network f [3] constructed from the auxiliary network h. self-dual in [2] and [3], respectively, according to the local activation functions: f [2] 1 (x) = x2, f [2] 2 (x) = x1, f [3] 1 (x) = (x1 ∧ x2) ∨ (x2 ∧ x3) ∨ (x1 ∧ x3), f [3] 2 (x) = (x1 ∧ x2) ∨ (x1 ∧ x3) ∨ (x2 ∧ x3), f [3] 3 (x) = (x1 ∧ x3) ∨ (x2 ∧ x3) ∨ (x1 ∧ x2). Induction for n ≥ 3: Assume that f [n] is a Hamiltonian cycle and sel… view at source ↗
Figure 9
Figure 9. Figure 9: Example of a 2-Hamiltonian digraph. 2-Hamiltonian digraphs illustrate the ability to modify two images of the auxiliary net￾work h [n] while maintaining the property of being a unate Boolean network. Examples of 2- Hamiltonian digraphs include Hamiltonian digraphs, Γ(h [n] ), or the one described in [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper 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.

Significance. If the constructions and claims hold with rigorous proofs, the results could provide theoretical tools for modeling complex systems and contribute to understanding dynamic interactions in Boolean networks, particularly by linking Hamiltonian properties to graph connectivity and full-dependence variables via unate functions. The introduction of a specific unate family is a potential strength if it avoids extra graph constraints.

major comments (2)
  1. [Abstract] Abstract and introduction: the central claim that the three Hamiltonian dynamics (maximum height, cycle, intermediate) can be realized by a family of unate Boolean networks without additional constraints on the interaction graph is load-bearing; the manuscript must explicitly show (via construction or theorem) that the update functions remain unate while achieving the claimed effects on connectivity and full-dependence variables, or clarify any required graph assumptions.
  2. [Abstract] No derivations, proofs, or explicit constructions are referenced in the abstract; if the full text lacks detailed proofs that the dynamics are well-defined on the 2^n state space and achievable via unate functions, the results cannot be verified.
minor comments (1)
  1. [Abstract] The abstract refers to 'specific properties and limitations' of the unate family without naming them; these should be stated explicitly early in the paper.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and valuable comments on our manuscript. We address each major comment below and will revise the abstract to reference the key constructions and proofs more explicitly.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introduction: the central claim that the three Hamiltonian dynamics (maximum height, cycle, intermediate) can be realized by a family of unate Boolean networks without additional constraints on the interaction graph is load-bearing; the manuscript must explicitly show (via construction or theorem) that the update functions remain unate while achieving the claimed effects on connectivity and full-dependence variables, or clarify any required graph assumptions.

    Authors: We agree this claim is central to the paper. The full manuscript provides explicit constructions of the unate update functions for each of the three dynamics (maximum-height, Hamiltonian-cycle, and intermediate), along with theorems proving that these functions remain unate and produce the stated effects on graph connectivity and full-dependence variables without imposing extra constraints on the interaction graph. We will revise the abstract to cite the relevant constructions and theorems. revision: yes

  2. Referee: [Abstract] No derivations, proofs, or explicit constructions are referenced in the abstract; if the full text lacks detailed proofs that the dynamics are well-defined on the 2^n state space and achievable via unate functions, the results cannot be verified.

    Authors: The abstract serves as a concise summary and does not reference specific proofs. However, the full text contains detailed proofs that the three Hamiltonian dynamics are well-defined over the 2^n state space and are realized by unate functions. We will update the abstract to reference these proofs and constructions for improved verifiability. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained in standard graph and Boolean function properties

full rationale

The abstract and available text introduce Hamiltonian dynamics (maximum height, cycle, intermediate) on Boolean network state spaces and a family of unate networks, claiming effects on interaction graph connectivity and full-dependence variables. No equations, parameter fits, self-citations, or ansatzes are quoted that reduce any prediction or uniqueness claim to the inputs by construction. The central statements rely on standard definitions of unate functions and graph connectivity without load-bearing self-referential steps, so the work is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on abstract; relies on standard definitions from Boolean network theory and graph theory with no explicit free parameters, ad-hoc axioms, or invented entities visible.

axioms (2)
  • standard math Boolean networks have finite state spaces and well-defined interaction graphs.
    Implicit in any study of Boolean network dynamics; invoked throughout the described analysis.
  • domain assumption Unate Boolean functions can realize the three Hamiltonian update patterns.
    Central to the introduced family; stated as a capability of the networks in the abstract.

pith-pipeline@v0.9.0 · 5608 in / 1277 out tokens · 26565 ms · 2026-05-23T06:00:56.766531+00:00 · methodology

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