{"id":"78d17ca2-7a5e-4531-b5d0-72688707121d","arxiv_id":"2501.04157","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A Kernighan-Lin plus information-entropy heuristic automatically partitions Boolean reaction networks for low-rank master equation integration, outperforming manual and cut-minimal partitions in tests on mTOR, pancreatic cancer, and apoptosis models.","lead":"This paper presents an automatic way to split a biochemical reaction network into parts before running a low-rank simulation of its master equation. The method combines the Kernighan-Lin graph heuristic with an information-entropy ranking and beats hand-made splits on three cancer-related models.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The entropy score in Eqs. (6)-(8) averages over a uniform distribution, but the DLR error depends on the actual time-dependent distribution; if rankings diverge, the automatic selection could be suboptimal.","rationale":"The paper's novelty and central claim rest on the entropy heuristic: selecting the lowest-H partition from a Kernighan-Lin pool is asserted to yield DLR accuracy superior to human and min-cut choices. The numerical evidence is genuine and the preprocessing cost is small, but the heuristic itself is the least secured link. Equation (8) defines p_i(x_i) as a uniform average over the complementary partition, and Eq. (6) averages over x_i uniformly. There is no a priori reason this matches the measure induced by the master equation solution; Boolean networks often have highly concentrated stationary distributions. The only exact-reference experiment (mTOR) shows the correlation is not perfect, and the larger experiments use high-rank DLR references rather than ground truth. The reader's weakest assumption correctly identifies this uniform-measure gap. My proposed test directly compares uniform-entropy rankings against empirical-entropy rankings and DLR errors for a case with ground truth, so it would settle whether the gap actually hurts the selection. Until then, the conditional acceptance is appropriate; the concern is addressable and does not invalidate the contribution.","tokens_in":24020,"tokens_out":10850,"duration_ms":110664,"concrete_test":"On the mTOR pathway (d=22, exact reference available), compute for each of the 32 KL-pool partitions the DLR error at r=8 and r=16. Also compute two entropy scores: H_uniform from Eqs. (6)-(8), and H_emp obtained by replacing the uniform averages with the marginals of the converged reference solution P_ref(t=300,x) (or time-averaged over [0,300]) in Eqs. (6)-(8). Rank partitions by each score and by error. If Spearman's rho for H_emp is substantially larger than for H_uniform, and the H_emp-selected partition has lower error than the H_uniform-selected partition, the uniform assumption is the limiting step. If the two rankings agree, the concern does not land.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central route to the paper's claim is that the total entropy H (Eq. 9) ranks partitions by how much information is lost when cutting reaction pathways. But H is computed from rule-specific entropies h_i in which both the partition state x_i and the complementary state x_i^c are averaged uniformly (Eqs. 6 and 8). The DLR approximation error, however, is governed by the actual probability distribution P(t,x) solving the master equation. A Boolean rule can have large uniform conditional entropy h_i yet rarely be active under P, so severing that pathway may barely increase the required rank; conversely a rule with small uniform entropy could act on high-probability states and be costly to cut. The numerical experiments do show a positive correlation between H and error, but only on three models and with a non-uniform initial condition in only one of them (apoptosis). The mTOR example, the only one with an exact reference, even shows that the lowest-entropy partition is not the most accurate at every rank (Fig. 6). The paper does not check whether the uniform ranking matches the ranking induced by the actual distribution, leaving the core proxy unvalidated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The authors propose a preprocessing heuristic for choosing the bipartition (and hierarchical tree partition) of a Boolean reaction network for dynamical low-rank (DLR) approximation of the master equation. Candidate partitions are generated by repeated Kernighan-Lin runs, and each candidate is scored by a rule-specific Shannon entropy H (Eqs. (6)-(9)) that measures information lost when a reaction pathway crosses the partition boundary; the lowest-H partition is selected. The scheme is tested on three biological models (mTOR with 22 species, pancreatic cancer with 34 species, and apoptosis with 41 species) for both one-level and hierarchical decompositions, and the authors report that the entropy-guided choice outperforms both manual literature partitions and pure cut-minimizing partitions.","tokens_in":24264,"tokens_out":5506,"duration_ms":53929,"significance":"If the entropy proxy is reliable, the method is a cheap and useful preprocessing step for an existing DLR simulation pipeline; the hierarchical extension appears new, and the reported preprocessing overhead is negligible (Table 3). The paper is clearly written, and the numerical experiments span a reasonable range of model sizes. However, the central proxy (uniform-distribution entropy) is not directly validated against the actual time-dependent distribution that governs the DLR rank, and the two largest benchmarks use DLR reference solutions rather than exact or independent stochastic references. The claim of superiority to human-expert partitions currently rests on a single direct comparison (one-level pancreatic cancer). While the paper does not ship code, the algorithm is straightforward enough to reproduce from the description.","major_comments":[{"comment":"The rule-specific entropy h_i averages the binary entropy over a uniform distribution on the complementary partition states in Eq. (8). Because the DLR approximation error is governed by the actual master-equation distribution P(t,x), the uniform proxy may rank partitions incorrectly: a rule with large h_i may act only on states of negligible probability, and vice versa. The paper does not test whether the H-ordering matches the ordering induced by P(t,x). Figure 6 itself shows that the lowest-entropy partition (M11) is not the most accurate at any rank, and at r=8 two higher-entropy partitions (M4, PURPLE) beat it; the text acknowledges this in Section 5.1.1 ('a smaller entropy does not automatically yield better accuracy'). To support the central claim, please provide a direct comparison on a small exact model: for example, compute marginal probabilities from the exact solution (or from SSA samples) and show that the entropy ranking agrees with a distribution-aware importance ranking (e.g., the rank needed for a prescribed accuracy or a KL-divergence measure).","section":"Section 4.2, Eqs. (6)-(8)"},{"comment":"The reference solutions for the pancreatic cancer and apoptosis models are DLR approximations at rank 50 and 60, not solutions of the full master equation. The reported differences (3.34e-3 and 4.77e-3) between the r=50 and r=60 solutions only establish self-consistency of the DLR solver; a shared bias in the low-rank manifold could still affect the ranking of partitionings. Since the abstract claims superiority to manual and cut-minimal partitionings for these systems, at least one of the larger models should be validated against an independent reference (e.g., SSA estimates of the first moments, or the exact solution of a reduced model with similar topology).","section":"Sections 5.1.2 and 5.2.2, reference solutions"},{"comment":"The hierarchical entropy computation is underspecified. The sentence 'when a reaction pathway was approximated on the parent node, then it must not be considered for the computation of the entropies in the child nodes' needs a precise definition of how the Boolean rule's dependence is modified when computing Eq. (8) at child nodes (e.g., set to a constant, or excluded from the sum over species). The current Algorithm 1 does not show this step, yet the numerical values in Tables 6 and 7, and therefore the selection of 'best', depend on it.","section":"Section 4.3, paragraph preceding Algorithm 1"},{"comment":"The claim that the resulting accuracy is 'superior to both partitionings chosen by human experts and those found by simply minimizing the number of reaction pathways' is stronger than the evidence presented. The only direct comparison with a human-expert partition is the one-level pancreatic cancer case (Section 5.1.2, where 'best' beats 'literature'); in the mTOR example, manual partitions (BLUE, PURPLE, GREEN) are used to illustrate the entropy correlation, but the algorithm's single selected partition is not explicitly benchmarked against them, and Figure 6 shows cases where it would not be the most accurate. Please qualify the claim to what is demonstrated, or add the missing direct comparisons.","section":"Abstract and Section 6"}],"minor_comments":[{"comment":"The conclusion reads 'we proposed an heuristic algorithm'; 'an heuristic' should be 'a heuristic'.","section":"Section 6"},{"comment":"The notation Ω∁_i for the complementary partition is not defined in the text; please introduce it explicitly or use a simpler notation such as Ω_i^c.","section":"Section 4.2, Eq. (8)"},{"comment":"The caption states 'A linear fit is shown for solutions with the same rank', but no fit parameters or R² values are reported; please include them or describe the fit as a qualitative guide only.","section":"Figure 6"},{"comment":"The caption contains a duplicated word: 'for for three hierarchical partitionings' should be 'for three hierarchical partitionings'.","section":"Figure 12 caption"},{"comment":"The text notes that the best partition depends on the time horizon (around t=150 vs. steady state); please clarify which error measure (maximum over time, value at t=300, or steady state) is used for the entropy correlation in Figure 6, since the ranking is sensitive to this choice.","section":"Section 5.1.1, Figure 5 discussion"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the scope of math.NA and presents a useful heuristic with a clear algorithmic description. The main risk is that the entropy proxy is taken as validated when the evidence shows only a weak correlation (and one counterexample even in the small model). I would encourage the editor to request the additional validation described in the major comments, especially an independent reference for at least one of the larger models and a distribution-aware comparison on the mTOR model. The authors may also consider releasing the partitioning code (it is cheap) to increase reproducibility."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: worth refereeing. The paper replaces hand-picked partitions, the only option so far for DLR simulation of Boolean reaction networks, with an automatic two-step heuristic: generate a pool of Kernighan-Lin partitions, then pick the lowest information entropy. On three test networks, the selected partition beats both the expert-chosen partition from the earlier DLR literature and the pure cut-minimal partition, often by a lot in terms of required rank. The entropy is computed from the Boolean rules alone, costs seconds, and the implementation is described clearly enough to reproduce. The hierarchical tree-tensor extension is new for this application, and the memory numbers are dramatic (137 GB down to 64 kB at rank 5 in the pancreatic cancer case).\n\nThe main soft spot is also the conceptual heart: the entropy in Eqs. (6)-(8) averages over a uniform distribution on the complementary states, while the true DLR error is governed by the time-dependent master equation distribution. A rule can look expensive to cut under a uniform average yet be rarely active under the actual dynamics, and the paper never checks whether the heuristic ranking matches a ranking derived from the simulated distribution. The authors do show the trend is real on three models, and they honestly note (Fig. 6, p. 13) that lower entropy does not automatically give lower error at every rank. So that concern lands, but it lands on a limitation the paper itself admits: this is a heuristic, not an optimality proof. I would have liked to see the ranking validated on the mTOR example, where an exact reference exists, against a distribution-aware importance measure.\n\nFor the two larger models, the reference is itself a DLR run at rank 60 rather than the true master equation solution. That is acceptable for benchmarking, but worth flagging. Second, no code or data is shipped and the Boolean rules for the three networks are only cited, not included; that makes the results hard to reproduce. For a methods paper this is a real but fixable deficiency. Minor: only Kernighan-Lin is tested as the pool generator.\n\nWho gets value: anyone working on DLR for chemical master equations or tensor-tensor simulation of biological networks. The central empirical claim holds up; the paper does not overclaim optimality. I would send it to referees. The suggested revisions - validate the entropy proxy and ship the rules - are doable and would make it stronger.","headline":"Worth refereeing: a genuinely useful automatic partitioning heuristic for DLR Boolean-network simulation, with honest numerical evidence, but the entropy criterion's uniform-state assumption is a real soft spot.","tokens_in":24709,"tokens_out":3533,"would_cite":true,"duration_ms":35003,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65F55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a good partition for dynamical low-rank simulation of a Boolean reaction network is one whose severed reaction pathways carry little information, and that this partition can be chosen automatically by picking the…","keywords":["Boolean reaction networks","dynamical low-rank approximation","chemical master equation","graph partitioning","Kernighan-Lin algorithm","information entropy","tree tensor networks"],"falsifier":"Take the pancreatic cancer model, generate a large Kernighan-Lin pool, and run DLR for the ten lowest- and ten highest-entropy partitions; if any higher-entropy partition requires a lower rank than a lower-entropy partition to reach the same error in the first moments, the uniform-counting assumption behind Eq. (8) is violated in that regime. A more direct test would compare the ranking predicted by uniform-counting entropy with a ranking computed using the actual time-marginal distribution of the complementary partition at the simulation end time.","tokens_in":23834,"feed_emoji":"🧬","tokens_out":8899,"duration_ms":73858,"temperature":0.7,"pith_summary":"The paper argues that the accuracy of dynamical low-rank (DLR) simulations of stochastic Boolean reaction networks depends strongly on how the species are partitioned, and that this choice can be automated. It proposes a two-step preprocessing heuristic: generate a pool of balanced candidate partitions with the Kernighan-Lin graph algorithm, then select the partition with the lowest Shannon information entropy, a rule-by-rule measure of how much information is lost when a reaction pathway crosses a partition boundary. The pay-off, measured on three biochemical networks, is that the entropy-selected partition reaches a given accuracy with a smaller rank than human-expert partitions or partitions that merely minimize the number of cut pathways. If the claim holds, automatic partitioning removes a manual bottleneck and makes low-rank simulation practical for exploring many candidate network models.","feed_headline":"Low-entropy cuts beat expert partitioning in network simulations","feed_subtitle":"A cheap entropy score picks partitions that beat manual cuts at lower DLR rank.","key_machinery":"The load-bearing object is the rule-specific information entropy $h_i$ defined in Eqs. (6)-(8). For each Boolean rule $B_i$ that updates species $S_i$, and for each value $x_i$ of the species on which the rule directly depends, the scheme computes the fraction $p_i(x_i)$ of states in the complementary partition for which the rule would fire, and then takes the binary Shannon entropy of that fraction: $h_i$ is the average over all $x_i$. The total entropy $H$ is the sum of the $h_i$ over all species. The Kernighan-Lin algorithm supplies the candidate pool of balanced partitions with few reaction-pathway cuts, and $H$ is then used to pick the partition whose approximated pathways lose the least information; pathways that are logically redundant (e.g., an absorption-law rule) cost zero entropy, while a complete logical dependence costs one.","core_discovery":"On the paper's own terms, the central discovery is that cut count is the wrong objective for partitioning Boolean reaction networks in DLR methods. A partition should instead sever reaction pathways that carry little information, and the rule-specific Shannon entropy of Eqs. (6)-(8) quantifies exactly that: a rule whose output is completely determined by the state of the other partition contributes entropy one, a logically redundant rule contributes zero, and partial dependencies contribute intermediate values. The paper shows numerically that choosing the minimal-entropy partition from a Kernighan-Lin pool yields a DLR approximation that is more accurate at fixed rank---and equivalently needs a lower rank for a fixed error---than both the manual partitions previously used in the literature and the minimal-cut partition, in single-level and hierarchical tree tensor network settings.","pith_inferences":["An untested implication is that the uniform-counting assumption in Eq. (8) could be relaxed cheaply: using a short stochastic simulation or a steady-state estimate of the marginal distribution over the complementary partition would likely sharpen the entropy ranking for regimes where the distribution is far from uniform.","The entropy score could be folded back into the graph weights before running Kernighan-Lin, as the authors note in their outlook; that would let the heuristic directly minimize information loss instead of only generating candidates that are then re-ranked.","A practical extension would be to weight the per-rule entropies by rule probabilities for probabilistic Boolean networks, which is straightforward because the entropy is already defined per rule; the paper mentions this but does not test it.","The rank-versus-entropy monotonicity observed in the experiments suggests a cheap model-selection tool: in parameter inference, where many network variants must be simulated, ranking partitions by $H$ may predict which models are simulable at low rank before any DLR run is attempted."],"forward_implications":["For a fixed rank, the automatically selected partition yields a lower error, and for a fixed error it permits a smaller rank, cutting memory by roughly the ratio of the ranks---about fourfold when moving from $r=20$ to $r=5$ in the pancreatic cancer example.","The preprocessing cost is negligible compared with a single DLR time integration: generating and scoring a pool of thousands of partitions for the 41-species apoptosis network takes tens of seconds on a laptop.","The same selection scheme carries over to hierarchical tree tensor network decompositions, so the method scales to networks whose single partitions would still be too large to store exactly.","Because the entropy is computed directly from truth tables, the scheme needs no expert knowledge of the network and can be rerun automatically whenever a network topology changes, which matters for model inference pipelines.","The numerical experiments make the accuracy difference visible: among one-level partitions of the mTOR network, the worst partition is about two orders of magnitude less accurate than the best at fixed rank."],"supporting_citations":[{"why":"Supplies the Kernighan-Lin graph-partitioning heuristic that generates the candidate pool of balanced partitions.","marker":"[42]"},{"why":"Supplies the Shannon entropy used to quantify information loss of severed reaction pathways.","marker":"[56]"},{"why":"Introduced the DLR approach for Boolean reaction networks and the manual partitions the paper compares against.","marker":"[54]"},{"why":"Provides the hierarchical DLR integrator for tree tensor networks used in the two-level numerical tests.","marker":"[24]"},{"why":"Provides the low-rank integrator for the chemical master equation that underlies the one-level DLR simulations.","marker":"[25]"},{"why":"Establishes time integration of tree tensor networks that the hierarchical discretization relies on.","marker":"[11]"},{"why":"Supplies the 22-species mTOR pathway Boolean network used as the first test case with an exact reference solution.","marker":"[3]"},{"why":"Supplies the 34-species pancreatic cancer signaling network used for one-level and hierarchical tests.","marker":"[33]"},{"why":"Supplies the 41-species apoptosis network used as the largest test case.","marker":"[50]"},{"why":"Provides the implementation of the Kernighan-Lin algorithm used to build the partitioning pools in the experiments.","marker":"[35]"}],"fun_headline_variants":["Entropy-based partitioning outperforms expert cuts in DLR","Automatic info-theoretic partitioning beats manual cuts","Entropy-guided Kernighan-Lin partitions improve DLR accuracy","Shannon entropy picks better Boolean network partitions for DLR","Automatic entropy partitioning outperforms human expert choices"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entropy score in Eqs. (6)-(8) assumes that, for each fixed value of the species a rule directly depends on, all states of the complementary partition are equally likely; if the true probability distribution over those states is far from uniform, the entropy ranking may not reflect the rank the DLR approximation actually needs.","fun_headline_variants_meta":{"raw":{"variants":["Entropy-based partitioning outperforms expert cuts in DLR","Automatic info-theoretic partitioning beats manual cuts","Entropy-guided Kernighan-Lin partitions improve DLR accuracy","Shannon entropy picks better Boolean network partitions for DLR","Automatic entropy partitioning outperforms human expert choices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001053,"raw_usage":{"total_tokens":4377,"prompt_tokens":853,"completion_tokens":3524,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":469,"completion_tokens_details":{"reasoning_tokens":3445}},"tokens_in":469,"tokens_out":3524,"duration_ms":27658,"temperature":1.0,"reasoning_tokens":3445,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:39:57.187405+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the pancreatic cancer model, generate a large Kernighan-Lin pool, and run DLR for the ten lowest- and ten highest-entropy partitions; if any higher-entropy partition requires a lower rank than a lower-entropy partition to reach the same error in the first moments, the uniform-counting assumption behind Eq. (8) is violated in that regime. A more direct test would compare the ranking predicted by uniform-counting entropy with a ranking computed using the actual time-marginal distribution of the complementary partition at the simulation end time.","supporting_citations":[{"cited_title":"Boolean network-based analysis of the apoptosis network: Irreversible apoptosis and stable surviving","cited_arxiv_id":null,"evidence_quote":"Supplies the 41-species apoptosis network used as the largest test case."},{"cited_title":"Symbolic Model Checking of Signaling Pathways in Pancreatic Cancer","cited_arxiv_id":null,"evidence_quote":"Supplies the 34-species pancreatic cancer signaling network used for one-level and hierarchical tests."},{"cited_title":"Time integration of tree tensor networks","cited_arxiv_id":null,"evidence_quote":"Establishes time integration of tree tensor networks that the hierarchical discretization relies on."},{"cited_title":"An extended gene protein/products boolean network model including post- transcriptional regulation","cited_arxiv_id":null,"evidence_quote":"Supplies the 22-species mTOR pathway Boolean network used as the first test case with an exact reference solution."}],"review_version":1}