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arxiv: 2505.14923 · v2 · submitted 2025-05-20 · 💻 cs.DM

Robustness of Boolean networks to update modes: an application to hereditary angioedema

Jacques Demongeot , Eric Goles , Houssem ben Khalfallah , Marco Montalva-Medel , Sylvain Sen\'e This is my paper

Pith reviewed 2026-05-22 13:26 UTC · model grok-4.3

classification 💻 cs.DM
keywords Boolean networksupdate modeshereditary angioedemagene regulatory networksblock-parallel updatesrobustnessinteraction graphs
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The pith

The hereditary angioedema interaction graph contains subgraphs that activate in an alternating block-parallel mode with a constantly updated core and two alternating subsets, exhibiting robustness or instability relative to classical update

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

The paper examines how different update schedules for gene states affect behavior in a Boolean network model tied to hereditary angioedema. It isolates subgraphs that follow an alternating block-parallel pattern, with one core set updated at every step and two complementary groups taking turns. A sympathetic reader would care because these patterns may clarify how regulatory networks sustain or lose coordinated activity in a disease setting when the timing of gene updates changes. The analysis compares the structural properties of these subgraphs against standard periodic modes such as fully parallel or sequential updates.

Core claim

Subgraphs within the hereditary angioedema gene interaction graph activate through an alternating block-parallel mode featuring one core that is constantly updated and two complementary subsets that alternate their updating, and these subgraphs display structural robustness or instability when compared to classical periodic update modes.

What carries the argument

The alternating block-parallel update mode with one core constantly updated and two complementary subsets alternating, applied to subgraphs of the hereditary angioedema interaction graph.

If this is right

  • The identified subgraphs maintain consistent activation under the alternating block-parallel schedule.
  • Robust subgraphs indicate network features that remain stable across changes in update timing.
  • Unstable subgraphs mark locations where small shifts in update order can disrupt coordinated gene activity.
  • Direct comparison with classical periodic modes isolates which structural traits are independent of the update schedule.

Where Pith is reading between the lines

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

  • Similar alternating-update subgraphs may exist in other familial-disease networks and could be searched for by the same structural criteria.
  • Varying the assignment of genes to core versus alternating groups in simulation would test how sensitive the robustness result is to the initial partitioning.
  • If the pattern generalizes, it supplies a concrete way to rank candidate intervention points by their update-mode stability.

Load-bearing premise

The particular interaction graph chosen and the division of genes into the constantly updated core and the two alternating subsets correctly reflect the actual regulatory logic in hereditary angioedema.

What would settle it

A simulation of the described subgraphs under the alternating block-parallel mode that produces activation patterns inconsistent with observed disease biology or that loses its reported robustness after modest rewiring of the graph edges would falsify the central claim.

Figures

Figures reproduced from arXiv: 2505.14923 by Eric Goles, Houssem ben Khalfallah, Jacques Demongeot, Marco Montalva-Medel, Sylvain Sen\'e.

Figure 1
Figure 1. Figure 1: Boolean automata network f of Example 1 on page 7; (a) its definition by means of functions; (b) its associated interaction graph. modes of a network of size n is denoted by BSn. The update of f under µ ∈ BSn is given by fµ : B n → B n as follows: fµ(x) = fWm ◦ · · · ◦ fW2 ◦ fW1 (x), where for all i ∈ M, for all k ∈ N, fWi (x)k = fk(x), if k ∈ Wi , and fWi (x)k = xk otherwise. 2.1.4 Block-parallel update m… view at source ↗
Figure 2
Figure 2. Figure 2: Different periodic dynamics of the Boolean automata network of Example 1 on [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) Boolean automata network f of Example 2; (b) its associated interaction graph G = (V, E), where the set of vertices is V = {1, 2, 3} and the set of edges is E = {(1, 2),(2, 1),(2, 3)}. Theorem 1. A labeled digraph is an update digraph if and only if changing the direction of only its arcs with <-labels results in a new digraph (possibly a multidigraph) that does not contain any cycle with a <-edge. Add… view at source ↗
Figure 4
Figure 4. Figure 4: All the update digraphs associated to the digraph [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The six distinct possible block-sequential dynamics of the Boolean automata network of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) An update digraph (G, lab); (b) its related quotient (G, lab)[≥] according to the equivalence relation “is accessible through ≥-edges” (cf. Algorithm); (c) (G, lab)[≥] rev obtained from (b) by reversing all the <-edges. from G′ such the latter is now composed of only vertex 3 and no edge. Then, we go back to step 4). At this stage, since G′ has no edge, there is no path and P< = ∅, which leads to step … view at source ↗
Figure 7
Figure 7. Figure 7: (a) Genetic regulatory network integrating different genes and proteins involved in [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) The Boolean automata network associated with the simplified version of the genetic [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: All the possible block-sequential attractors of Boolean automata network Ψ defined [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Six and/or Boolean automata networks with their associated interactions which conserve both the type of local functions of Ψ and the underlying structure of that of Ψ. – The parallel dynamics of α and δ includes two fixed points and there exist other dynamics with no limit cycles; – The parallel dynamics of β and ζ has no fixed point but has a limit cycle of length 5; – The parallel dynamics of γ and η in… view at source ↗
Figure 11
Figure 11. Figure 11: Dynamics of BAN6 f presented in [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
read the original abstract

Many familial diseases are caused by genetic accidents, which affect both the genome and its epigenetic environment, expressed as an interaction graph between the genes as that involved in one familial disease we shall study, the hereditary angioedema. The update of the gene states at the vertices of this graph (1 if a gene is activated, 0 if it is inhibited) can be done in multiple ways, well studied over the last two decades: Parallel, sequential, block-sequential, block-parallel, random, etc. We will study a particular graph, related to the familial disease proposed as an example, which has subgraphs which activate in an intricate manner (\emph{i.e.}, in an alternating block-parallel mode, with one core constantly updated and two complementary subsets of genes alternating their updating), of which we will study the structural aspects, robust or unstable, in relation to some classical periodic update modes.

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 / 2 minor

Summary. The manuscript studies the robustness of Boolean networks to update modes by analyzing a specific interaction graph associated with hereditary angioedema. It identifies subgraphs that activate in an alternating block-parallel mode (one core constantly updated, two complementary subsets alternating) and compares their structural robustness or instability to classical periodic modes such as parallel and sequential updates.

Significance. If the central claims hold, the work would contribute to the discrete-mathematics literature on Boolean-network dynamics by providing a concrete biological application that links update-mode sensitivity to disease-related regulatory graphs. The focus on structural properties rather than simulation alone offers a potentially falsifiable framework for assessing robustness across update schedules.

major comments (2)
  1. [Abstract and §1] Abstract and §1: the central claim that the hereditary-angioedema graph contains subgraphs exhibiting structural robustness (or instability) under the described alternating block-parallel mode rests on an unvalidated choice of graph and an unmotivated partitioning of vertices into a constant core plus two alternating blocks; no derivation from primary biological data or cross-validation against known regulatory interactions is supplied, so the disease-specific interpretation does not follow from the mathematical properties alone.
  2. [§3] §3 (or the section presenting the update-mode definitions): the alternating block-parallel schedule is introduced as 'intricate' without an explicit formal definition (e.g., no equation specifying the periodic schedule, the size of the core, or the complementarity condition on the alternating subsets), preventing verification that the reported robustness properties are independent of the particular partitioning chosen.
minor comments (2)
  1. [Figures] Figure captions and notation: ensure that every gene label appearing in the interaction graph is defined in a table or legend so that the core-versus-alternating assignment can be inspected without external lookup.
  2. [References] References: add citations to prior work on block-parallel update modes in Boolean networks (e.g., the original definitions of block-sequential and block-parallel schedules) to situate the alternating variant.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and indicate the revisions made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: the central claim that the hereditary-angioedema graph contains subgraphs exhibiting structural robustness (or instability) under the described alternating block-parallel mode rests on an unvalidated choice of graph and an unmotivated partitioning of vertices into a constant core plus two alternating blocks; no derivation from primary biological data or cross-validation against known regulatory interactions is supplied, so the disease-specific interpretation does not follow from the mathematical properties alone.

    Authors: The hereditary angioedema interaction graph is presented as a concrete illustrative example drawn from the literature on the disease to motivate the mathematical study of update-mode robustness, rather than as a model derived directly from primary experimental data within this work. The partitioning into a constant core and two complementary alternating blocks is chosen based on the observed structure of the graph that supports the alternating block-parallel dynamics. We have revised the abstract and §1 to clarify the illustrative nature of the biological application and to emphasize that the robustness results are structural and mathematical. We agree that additional cross-validation against primary data would strengthen the disease link but consider it beyond the discrete-mathematics scope of the paper. revision: partial

  2. Referee: [§3] §3 (or the section presenting the update-mode definitions): the alternating block-parallel schedule is introduced as 'intricate' without an explicit formal definition (e.g., no equation specifying the periodic schedule, the size of the core, or the complementarity condition on the alternating subsets), preventing verification that the reported robustness properties are independent of the particular partitioning chosen.

    Authors: We agree that an explicit formal definition is required for clarity and reproducibility. In the revised manuscript we have added a precise definition in §3, including a mathematical equation for the periodic schedule, the fixed core vertex set, and the two complementary alternating subsets together with their update rules. This formalization makes the robustness properties verifiable independently of any specific partitioning. revision: yes

Circularity Check

0 steps flagged

No significant circularity; analysis applies standard update-mode methods to a chosen input graph

full rationale

The paper takes as given a particular interaction graph related to hereditary angioedema and analyzes the activation patterns and robustness of its subgraphs under alternating block-parallel and classical periodic update modes. No equations or claims reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the graph and its partitioning are explicit modeling inputs, and the reported structural properties follow from direct examination of that graph rather than from any circular renaming or imported uniqueness theorem. The derivation chain is therefore self-contained against external Boolean-network benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the published interaction graph for hereditary angioedema is both complete and correctly signed, and that the alternating block-parallel partitioning chosen by the authors reflects a biologically meaningful division of labor among genes. No independent evidence for either is supplied in the abstract.

axioms (2)
  • domain assumption The gene interaction graph for hereditary angioedema is known and fixed.
    Invoked when the authors state they will study 'a particular graph, related to the familial disease'.
  • ad hoc to paper Alternating block-parallel update with a constant core and two complementary alternating subsets is a relevant mode for this network.
    The abstract introduces this specific schedule as the intricate activation pattern to be analyzed.

pith-pipeline@v0.9.0 · 5699 in / 1480 out tokens · 25380 ms · 2026-05-22T13:26:20.508111+00:00 · methodology

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Reference graph

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