{"id":"1042bb5d-1c0f-4ce8-b59b-0fd51aad3cb3","arxiv_id":"2501.10886","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A differentiable information imbalance optimization automatically identifies dynamical communities and constructs a community causal graph from high-dimensional time series in linear-in-tests time.","lead":"This paper introduces a causal discovery method that groups time series variables into dynamical communities and builds a coarse-grained causal graph. The method is tested on synthetic systems with up to 80 variables and is claimed to scale linearly with the number of variables.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The abstract claims total computational cost linear in D, but Eq. (5) runs D separate DII minimizations, each over D weighted variables; total cost is ~O(D^2), not O(D). No asymptotic analysis or sparse trick is provided.","rationale":"The reader's weakest assumption was the imported causal criterion of Eq. (2), which is a legitimate concern but applies equally to any causal discovery method built on an explicit or implicit faithfulness-style assumption. I focus instead on the paper's own headline claim: linear scaling in D. This is the condition that must hold for the paper to deliver its advertised contribution, and it is the least secure because the algorithm as written directly contradicts it. Each of the D optimizations in Eq. (5) involves all D variables and therefore costs O(D) per distance evaluation; with N points and E epochs, the total for step (i) is Theta(E N^2 D^2). The Discussion's more careful wording 'number of optimizations scales linearly' is true but is not the same as 'computational cost scales linearly', which appears in the abstract and introduction. The paper deserves credit for the community-detection idea, the synthetic validation, and the available code in DADApy, but these do not rescue a central complexity claim that is internally inconsistent with the stated algorithm. A REJECT verdict reflects that the paper as written overstates its main contribution; a revision that either proves linear total cost with a concrete algorithmic mechanism or honestly states the quadratic cost would change this assessment.","tokens_in":20793,"tokens_out":6167,"duration_ms":71580,"concrete_test":"Time step (i) on the supplied DADApy code for coupled logistic-map systems with D = 20, 40, 80, 160, 320, keeping N = 2000 and all optimizer settings fixed, and fit total wall-clock time to a power law t ~ D^p. If p is close to 2, the abstract's linear-scaling claim is false. Alternatively, derive the asymptotic cost of Eqs. (4)-(5) directly: D targets x D weights x N^2/100 x 500 operations, i.e., Theta(E N^2 D^2); if the authors believe otherwise, they should identify which step avoids the D factor per optimization.","verdict_should_be":"REJECT","load_bearing_attack":"The central advertised property is that computational cost scales linearly with the number of variables D. This is not what the algorithm in Section i provides. For each target beta, Eq. (5) minimizes the DII over a weight vector w of length D, so the first distance space is built from all D variables. Computing the DII in Eq. (4) requires pairwise distances in that space, costing at least O(N^2 D) operations per gradient epoch (N=2000; even with mini-batches of 100 points per batch, each batch is O(100^2 D) and the 20 batches per epoch are O(200,000 D)). Repeating for all D targets gives Theta(E N^2 D^2) for the whole matrix G, where E is the number of epochs. The paper's statement that 'the number of optimizations ... scales linearly' is true but does not imply that the computational cost scales linearly; each optimization is itself O(D). No sparsity, screening, or other device is presented to remove this factor. The Appendix A refinement can add a further quadratic term in the number of communities in the worst case. Thus the headline efficiency claim, which motivates the whole approach for high-dimensional time series, is not merely unproved but contradicted by the algorithm as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces a causal-discovery framework for high-dimensional time series that groups variables into 'dynamical communities' by optimizing a differentiable Information Imbalance (DII). For each target variable β, Eq. (5) minimizes DII over weights assigned to all variables at t = 0; the resulting matrix G (Eq. 6) is thresholded to obtain autonomous sets, which are then organized hierarchically into communities and a community causal graph. The method is validated on coupled logistic maps, five coupled Lorenz systems, and two coupled Lorenz-96 systems, with exact reconstruction over wide threshold ranges. The paper claims that the computational cost scales linearly with the number of variables D.","tokens_in":21101,"tokens_out":3198,"duration_ms":32160,"significance":"If the efficiency claim were supported, the community-level coarse-graining would be a useful and novel tool for high-dimensional causal discovery. The paper's strengths are its clean synthetic validation with external ground truth, the demonstration of exact reconstruction over wide threshold ranges, the comparison with PCMCI, and the supplementary analysis of observational noise and causal sufficiency. However, the central claim of linear scaling is not supported by the algorithm as written, and the causal criterion is imported from prior work without a proof. Both points must be resolved before publication.","major_comments":[{"comment":"The paper's headline claim that the computational cost scales linearly with D is not what the algorithm delivers. Eq. (5) performs one DII minimization per target variable β, but each minimization optimizes a weight vector of length D (Eq. 3), so computing the DII of Eq. (4) requires, per gradient epoch, pairwise distances over all D variables: with N = 2000 and mini-batches of 100 points, each batch is O(100^2 D) and 20 batches per epoch are O(200,000 D). Across D targets and E epochs the total is Θ(E N^2 D^2) (up to mini-batch constants), not O(D). Appendix A adds a further quadratic-in-communities term. The Discussion only claims that the number of optimizations scales linearly, which is true but does not imply linear total cost. The authors should either provide a rigorous complexity analysis with a sparse or block implementation, or revise the abstract and introduction so that the claim is limited to the number of optimizations.","section":"§i and Discussion, Eq. (5)"},{"comment":"The criterion that Xα causes Xβ if and only if the optimal weight component w^α_β is nonzero is assumed, not derived, and it is inherited from ref. [19] as a 'generalization'. The paper does not prove conditions under which this holds (faithfulness, absence of local minima in the DII optimization, causal sufficiency), and the synthetic tests are not designed to stress-test it. I recommend stating explicitly that this is an assumption, citing the prior analysis, and discussing failure modes; otherwise the community graph inherits any error in this step.","section":"SM §I, Eq. (S6)"}],"minor_comments":[{"comment":"The title contains a spacing typo: 'dynami cal' should be 'dynamical'.","section":"Title"},{"comment":"The sentence 'Eqs. (12) were integraterd with time step dt = 0.03' contains a typo: 'integraterd' should be 'integrated'.","section":"Appendix B, §3"},{"comment":"In the caption of Fig. 2, 'recollected ones' should be 'recovered ones' for clarity.","section":"§iii and Fig. 2"},{"comment":"The definition of a minimal autonomous set, 'for every variable xα ∈ Sβ, one has Sα ≡ Sβ', is clear for exact sets but the thresholding step that produces the Sβ from G is not described with the same precision; a short explanation of how ties and near-equal sets are handled would help reproducibility.","section":"§ii"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of a data-analysis journal, but the abstract's central claim of linear scaling is not supported by the algorithm as written; the reader's report and my own reading agree on this point. The causal criterion is a prior contribution, so the novelty of this paper rests on the community-detection step, which is validated but whose efficiency motivation needs to be corrected or substantially qualified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou should know two things about this paper before deciding whether to read it. First, the actual new contribution is genuine: the authors extend their earlier DII-based causal discovery (ref 19) to automatically discover dynamical communities and build a coarse-grained causal graph, removing the need to specify interacting groups in advance. The iterative autonomy-level extraction is a clean idea, and the validation on logistic maps, Lorenz, and Lorenz-96 is thorough, including observational noise, a PCMCI comparison, and an SM that carefully discusses causality sufficiency violations. Second, the headline efficiency claim is not supported. The abstract says the computational cost scales linearly with the number of variables, but the algorithm as written runs D separate DII minimizations (one per target variable), each of which computes distances over all D variables. That is at best O(D^2) in wall-clock, and more precisely Theta(E N^2 D^2) for the full matrix G. The paper only claims 'the number of optimizations scales linearly', which is a different and weaker statement. The stress-test note is correct; that is a real flaw in how the result is advertised.\n\nThe good news: the community recovery appears robust over a wide range of threshold epsilon, and the graphs match ground truth on the tested systems. The threshold remains a free parameter with no principled selection rule, and the causal criterion (w nonzero iff X causes Y) is inherited from prior same-group work, not proven here. The synthetic tests involve known groups and faithfulness-friendly noise, so they don't stress-test the criterion strongly. The Appendix A refinement can add a quadratic term in the number of communities in the worst case, though the authors argue this is small. Those are soft spots but not fatal: the method is a useful heuristic for exploratory causal structure.\n\nFor your reading group, I'd bring it: the DII optimization trick and the autonomy-level ordering are worth discussing, and the complexity miscalculation is a good teaching moment about asymptotic claims. I would cite it if I needed a fast heuristic for mesoscale causal structure, though I'd qualify the complexity claim. A serious referee should be engaged: with a corrected complexity discussion and a more honest framing of the threshold selection, the paper could be a solid contribution.","headline":"The community detection extension is real, but the linear-scaling claim is contradicted by the algorithm as written.","tokens_in":21577,"tokens_out":3343,"would_cite":true,"duration_ms":29440,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims a causal-discovery framework whose computational cost scales linearly with the number of time-series variables, by optimizing the Information Imbalance to find dynamical communities and then ordering them into a…","keywords":["causal discovery","time series","Information Imbalance","dynamical communities","community causal graph","linear scaling","differentiable Information Imbalance","high-dimensional dynamics"],"falsifier":"An experiment that would settle whether the causal criterion holds: generate data where a true direct cause's influence is exactly canceled by an opposing path, a violation of faithfulness, and check whether its optimized weight is zero. If the weight is zero while the variable is genuinely causal, the criterion of Eq. (2) fails and the method cannot recover the graph; if it stays nonzero, the criterion survives a harsher test than the paper's noise-additive experiments.","tokens_in":20590,"feed_emoji":"🕸️","tokens_out":9174,"duration_ms":91551,"temperature":0.7,"pith_summary":"This paper argues that causal discovery in high-dimensional time series can be done at a computational cost that grows only linearly with the number of variables. The key step is to infer, for each variable, the set of variables that influence it by minimizing the differentiable Information Imbalance over one weight per variable, then to group variables that influence each other into 'dynamical communities' and order these communities by autonomy. The output is a coarse-grained community causal graph. On three synthetic systems (15 coupled logistic maps, five coupled Lorenz oscillators, and two coupled Lorenz-96 systems totaling 80 variables) the recovered graph matches the ground truth over a wide range of the threshold parameter, and the cost stays linear because the method performs one optimization per variable instead of an exponential search over conditioning sets.","feed_headline":"Causal discovery now scales linearly with variable count","feed_subtitle":"Information-imbalance optimization finds causal communities and the graph between them, tested up to 80 variables.","key_machinery":"The machinery is the Differentiable Information Imbalance (DII), $$\\mathrm{DII}(d_A\\to d_B)=\\frac{2}{$N^{2}$}\\sum_{i\\neq j}\\frac{$e^{{-d_{ij}}$^A/\\$\\lambda$}}{\\sum_{m\\neq i}$e^{{-d_{im}}$^A/\\$\\lambda$}}r_{ij}^B,$$ a soft version of the rank-based Information Imbalance, recovered as $\\lambda\\to0$. Because it is differentiable in the distances, the weights $w$ entering $d_A$ can be optimized by gradient descent. The algorithm computes, for each target variable $X^\\beta$, the optimal weights in a distance built from all variables at time $0$ predicting $X^\\beta(\\tau)$, takes the maximum over lags $\\tau$ up to the autocorrelation time, thresholds the resulting matrix $G_{\\alpha\\beta}$ at $\\varepsilon$, and closes under ancestor relations. Iteratively extracting minimal autonomous sets gives the dynamical communities and their autonomy levels; directed arrows between consecutive levels form the final community causal graph.","core_discovery":"On the paper's own terms, the claim is that optimizing the differentiable Information Imbalance against each target variable, once per variable, produces a connectivity matrix $G$ whose thresholded and ancestor-closed pattern reveals the system's dynamical communities. A community is a minimal autonomous set: a group of variables whose influence sets coincide, so that each member depends on all and only the other members, and the group's evolution is independent of the rest of the network. Removing these communities iteratively yields autonomy levels, and directed arrows between communities at consecutive levels form a directed acyclic community causal graph. The paper reports that this procedure recovers the exact community graph for its three test systems over a wide range of thresholds, and that a separate conditional test can decide whether a link between non-consecutive communities is direct or indirect.","pith_inferences":["The 'linear scaling' is in the number of variables; each DII evaluation still sums over pairs of sampled frames, so total cost also grows with trajectory length and sample size, which matters for very long recordings.","The causal criterion of Eq. (2) is the unproved load-bearing step; the synthetic tests add noise to avoid deterministic non-faithfulness, so they do not probe whether the criterion fails under cancellations or with real confounded data.","A natural extension the paper only hints at is to use the inferred communities as a prior to constrain the conditioning sets of constraint-based algorithms, potentially combining the linear scaling with stronger statistical guarantees.","In observational data with unobserved common drivers, the paper's own supplemental analysis predicts that distinct true communities will be merged; that prediction could be tested directly on datasets with known confounders."],"forward_implications":["If the claim is right, causal discovery on systems with hundreds of variables needs only one gradient-descent optimization per variable, avoiding the exponential search over conditioning sets.","The recovered graph is mesoscopic: variables that mutually influence each other are merged into a single node, so the output is a coarse-grained summary of the system's causal organization.","On the paper's three test systems the correct community graph is recovered for a wide range of the threshold $\\varepsilon$, so the result is not tied to a finely tuned hyperparameter value.","Links between communities at non-consecutive autonomy levels can be classified as direct or indirect by an additional conditional test, refining the community graph.","The connectivity matrix $G$ can seed or speed up other causal-discovery and graph-clustering approaches, including cases with bidirectional couplings."],"supporting_citations":[{"why":"Establishes the causal criterion of Eq. (2) that a nonzero optimal Information-Imbalance weight means causality, the premise the present method generalizes to all variables.","marker":"[19]"},{"why":"Defines the Information Imbalance, the rank-based predictive measure used in Eq. (1).","marker":"[17]"},{"why":"Introduces the differentiable Information Imbalance (DII) of Eq. (4), enabling gradient-descent weight optimization.","marker":"[18]"},{"why":"Provides the state-of-the-art constraint-based baseline whose hyperparameter sensitivity is contrasted with the proposed method on the coupled Lorenz-96 test.","marker":"[11]"},{"why":"Supplies the adjusted mutual information measure used to score agreement between recovered and ground-truth community partitions.","marker":"[24]"},{"why":"Supplies the coupled Lorenz oscillator equations used as a continuous-time test system.","marker":"[22]"},{"why":"Supplies the 40-variable Lorenz-96 system used for the largest, 80-variable test.","marker":"[23]"},{"why":"Provides the group-level causal discovery framework that justifies the definition of inter-community links.","marker":"[25]"},{"why":"Supplies the Python implementation used to run the optimizations in the experiments.","marker":"[31]"}],"fun_headline_variants":["Causal discovery now scales linearly with variables","Causal discovery scales: linear time, high dimensions","Community detection makes causal inference scale","Causal inference at linear cost, up to 80 variables","Dynamical communities reveal causal structure efficiently"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the imported, unproved rule that a variable gets a nonzero optimal weight exactly when it causes the target; if hidden common drivers, canceling influences, or a bad local minimum break that rule, the communities and causal graph built on it are unreliable.","fun_headline_variants_meta":{"raw":{"variants":["Causal discovery now scales linearly with variables","Causal discovery scales: linear time, high dimensions","Community detection makes causal inference scale","Causal inference at linear cost, up to 80 variables","Dynamical communities reveal causal structure efficiently"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00099,"raw_usage":{"total_tokens":4168,"prompt_tokens":885,"completion_tokens":3283,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":3213}},"tokens_in":501,"tokens_out":3283,"duration_ms":27346,"temperature":1.0,"reasoning_tokens":3213,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T18:52:17.776442+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An experiment that would settle whether the causal criterion holds: generate data where a true direct cause's influence is exactly canceled by an opposing path, a violation of faithfulness, and check whether its optimized weight is zero. If the weight is zero while the variable is genuinely causal, the criterion of Eq. (2) fails and the method cannot recover the graph; if it stays nonzero, the criterion survives a harsher test than the paper's noise-additive experiments.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the causal criterion of Eq. (2) that a nonzero optimal Information-Imbalance weight means causality, the premise the present method generalizes to all variables."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Information Imbalance, the rank-based predictive measure used in Eq. (1)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the differentiable Information Imbalance (DII) of Eq. (4), enabling gradient-descent weight optimization."},{"cited_title":"Geiger, T","cited_arxiv_id":null,"evidence_quote":"Provides the state-of-the-art constraint-based baseline whose hyperparameter sensitivity is contrasted with the proposed method on the coupled Lorenz-96 test."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the adjusted mutual information measure used to score agreement between recovered and ground-truth community partitions."},{"cited_title":"multi- body","cited_arxiv_id":null,"evidence_quote":"Supplies the coupled Lorenz oscillator equations used as a continuous-time test system."},{"cited_title":"Glielmo, C","cited_arxiv_id":null,"evidence_quote":"Supplies the 40-variable Lorenz-96 system used for the largest, 80-variable test."},{"cited_title":"Del Tatto, G","cited_arxiv_id":null,"evidence_quote":"Provides the group-level causal discovery framework that justifies the definition of inter-community links."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Python implementation used to run the optimizations in the experiments."}],"review_version":1}