{"id":"3e38e286-1bd1-45e9-93b4-d000a04e740d","arxiv_id":"1908.07565","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Spectral clustering based on the random-walk transition matrix identifies high-effective-information macro-scales of preferential attachment networks faster than greedy or gradient-descent search.","lead":"This paper tests three computer methods for finding the best simplified scale of a network, where groups of nodes become single macro-nodes. A spectral clustering method finds these informative scales faster and as well or better than greedy and gradient-descent approaches.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unproved Markov-blanket pruning rule in §2.2.2 could exclude the true EI-maximizing partition, so exhaustive small-network tests are needed before claiming the spectral method finds the right scale.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: the unproved Markov-blanket restriction in §2.2.2. This is the right focus because the spectral algorithm's search space is explicitly pruned by that rule, and the greedy algorithm inherits the same restriction, so both could fail in the same way. The concern is not merely theoretical: the paper provides no derivation, citation, or brute-force evidence for the rule, and the kernel-degeneracy discussion addresses a different, global property. The absence of an exact ground truth in the experiments means the performance comparison only shows that spectral beats two heuristics, not that it finds the optimum. An exact enumeration on small networks would directly test the rule and settle whether the conditional verdict should be upgraded or the claim softened. I therefore agree with the reader and recommend keeping the CONDITIONAL verdict unchanged, pending this concrete test.","tokens_in":10230,"tokens_out":5205,"duration_ms":258427,"concrete_test":"Enumerate all set partitions of networks with n=5–8 micro-nodes (e.g., 100 random directed preferential-attachment networks per n, plus random directed graphs with varied α) and compute EI for every partition using the same µ|π macro-node construction as the paper. Record whether the EI-maximizing partition ever contains a pair of nodes that are not in each other's Markov blankets, and compare the EI of the spectral and greedy outputs to the exact maximum. If any violation occurs, the pruning rule in §2.2.2 is false and the headline claim requires revision; if no violations occur across a broad family, the rule gains empirical support and the conditional verdict can be upgraded.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central practical claim is that the spectral method finds informative and complex higher scales and outperforms alternatives. That claim depends on the pruning rule stated in §2.2.2: if two nodes are not in each other's Markov blankets, then grouping them cannot increase EI. This rule is asserted without proof or citation, and it is used both to set pairwise distances to infinity in the spectral distance matrix and to restrict the greedy search inherited from [2]. If the rule is false, both the spectral and greedy algorithms can miss the true EI-maximizing macroscale, so the reported causal emergence would only be the best within a restricted class of groupings, not the 'right scale' the paper claims to find. The kernel-degeneracy derivation in §2.2.2 does not rescue the rule: it establishes a global relation between kernel dimension and degeneracy, but says nothing about whether merging two specific non-blanket nodes can increase EI. The empirical comparison in Fig. 1C/D is against greedy and gradient-descent heuristics, not against an exact optimum, so superior performance among heuristics is compatible with all three missing the true optimum. A direct exact-search test on small networks is the decisive check.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper addresses the problem of finding informative higher-scale (macro) representations of networks under the effective information (EI) criterion of causal emergence. The authors compare three search heuristics: a greedy algorithm from prior work, gradient descent on a relaxed soft-assignment matrix, and a spectral clustering method that uses eigenvectors of the transition matrix, cosine similarity, and an OPTICS clustering step with a Markov-blanket restriction on permissible merges. On preferential attachment networks, the spectral method is reported to be orders of magnitude faster and to find EI equal to or greater than the other heuristics. The paper also reports variations of network properties (entropy rate, efficiency, centrality, assortativity, kernel dimension) between micro and macro scales. The spectral algorithm has been added to the publicly available einet Python package.","tokens_in":10485,"tokens_out":8879,"duration_ms":83907,"significance":"If the central claims are correct, the paper makes a useful methodological contribution: it identifies a scalable heuristic for network coarse-graining under an information-theoretic objective. The reported runtime advantage of the spectral method is substantial, and the inclusion of the algorithm in a public repository is a strength. The significance, however, is conditional on the validity of the EI objective and on the completeness of the search. The paper relies on an unproved assertion that only Markov-blanket pairs can be merged to increase EI, and the empirical evaluation is only against other heuristics. If that assertion is false, the claimed 'right scale' is not the true EI-maximizing scale. The paper also does not currently provide the hyperparameter details needed for exact reproduction of the comparisons.","major_comments":[{"comment":"The assertion that 'If a pair of nodes are not in each other's Markov blankets, then grouping them together cannot increase EI' is load-bearing for both the spectral algorithm (it sets pairwise distances to infinity) and the greedy algorithm inherited from [2]. The statement is given without proof, reference, or empirical support. The subsequent kernel-dimension derivation is a global statement about the transition matrix and does not imply anything about whether merging two specific non-blanket nodes can increase EI. Because Figure 1 compares only against other heuristics, a scenario in which all tested methods miss the true optimum would be invisible. The authors should add a brute-force exhaustive search over all partitions for small networks (n <= 10) to test this rule directly: if any EI-maximizing partition merges nodes that are not in each other's Markov blankets, the rule is false and the reported causal emergence values are systematically biased. This is the decisive missing experiment.","section":"Section 2.2.2"},{"comment":"The comparison in Section 3.1 establishes only that the spectral heuristic finds EI at least as high as the greedy and gradient-descent heuristics on the tested instances. In the absence of a brute-force EImax baseline, the abstract's claim that the spectral method finds 'the right scale' is unsupported; it may find the best restricted scale, not the best scale. This matters because gradient descent is not restricted to Markov-blanket merges, so if the pruning rule of Section 2.2.2 is false, gradient descent is in principle capable of finding better EI than the spectral method, and the reported ordering of algorithms could reverse. I request an exact enumeration on small networks (e.g., all partitions for n = 5 to 10) reporting the ratio of each heuristic's EI to the true EImax, together with the fraction of instances on which each heuristic attains the global optimum.","section":"Section 3.1, Fig. 1C/D"},{"comment":"The claims that macroscales have higher global efficiency, betweenness centrality, and eigenvector centrality are based on comparisons between networks of different sizes. Global efficiency is defined as the inverse of the mean shortest path length, which trivially tends to increase as the number of nodes decreases; average betweenness and eigenvector centrality also scale with n. Without a null model (e.g., random coarse-grainings with the same number of macro-nodes) or size-normalized indices, the observed differences in Fig. 2E-G may be artifacts of dimension reduction rather than evidence that macroscales are more efficient or more central. The authors should re-analyze these properties with appropriate normalization or explicitly state the size-dependence caveat.","section":"Section 3.2, Fig. 2E-G"},{"comment":"The spectral algorithm depends on the OPTICS distance threshold epsilon (over which a range is scanned) and on OPTICS parameters such as min_samples; the gradient-descent method depends on learning rate, momentum, maximum iterations, and initialization. None of these settings are reported in the text or figure captions. Since Figure 1 compares runtimes and achieved EI, and since the spectral method's epsilon scanning may dominate its runtime, the absence of these details makes the comparison irreproducible and leaves open the possibility that the algorithms were not compared on equal footing. Please report all hyperparameters, the epsilon grid, and the selection rule for the reported results.","section":"Sections 2.2.2 and 2.2.3"}],"minor_comments":[{"comment":"The notation <Wout_i> and <H(Wout_i)> is not defined precisely; the average is presumably uniform over nodes, but this should be stated explicitly.","section":"Section 2.1, Eq. (1)"},{"comment":"The complexity statement that 'computing the OPTICS reachability graph is O(n log n)' is not accurate for a precomputed dense distance matrix, where OPTICS requires O(n^2) distance comparisons. Please revise the complexity claim.","section":"Section 2.2.2"},{"comment":"If the transition matrix is not symmetric, the eigendecomposition can have complex eigenvalues and eigenvectors, and the cosine similarity of complex vectors is not defined in the text. Please state the assumption (e.g., undirected graphs, for which the transition matrix is similar to a symmetric matrix) or specify the complex similarity measure.","section":"Section 2.2.2"},{"comment":"The description of the greedy algorithm contains a notational inconsistency: 'a list of neighboring nodes is constructed, {vj} in Bi' should read 'v_j in B_i'; the text also uses 'it's' instead of 'its' in several places.","section":"Section 2.2.1"},{"comment":"The number of macro-nodes k in the gradient descent relaxation is not specified; the paper should state how k is chosen or optimized.","section":"Section 2.2.3"},{"comment":"The text says 'the kernel dimension of the micro-nodes increasing rapidly, while the macro-nodes decrease,' but it is not clear whether the figure shows kernel dimension for the micro-network, the macro-network, or both. Please clarify the legend.","section":"Section 3.2, Fig. 2L"},{"comment":"The claim that 'the number of possible dimension reductions exponentially increases with the size of the system' is imprecise; the number of partitions grows super-exponentially (according to the Bell numbers), not merely exponentially.","section":"Section 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is a candidate for publication if the authors can supply the small-network exact-baseline experiments and address the normalization issues in the network-property comparisons. The Markov-blanket pruning rule is the central correctness risk; it is asserted rather than derived, and the existing evaluation cannot detect whether it excludes the true optimum. I would not recommend rejection, as the concern is testable and fixable within the manuscript's scope. The paper's framing as finding 'the right scale' overclaims relative to what is demonstrated; a revised version that honestly reports 'best among heuristics under a restricted search' and then validates the restriction would be much stronger."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The spectral method in this paper is a genuine advance in the narrow but useful problem of identifying coarse-grainings that increase effective information (EI). Credit where due: the eigenvalue-weighted embedding, the Markov-blanket distance constraint, and the OPTICS epsilon sweep are a new combination; the runtime numbers in Figure 1 are credible; and the authors ship code in the einet package. The claims that macroscales can lower entropy rate, raise efficiency, and change centrality are observations, not theorems, but they are clear and reproducible.\n\nThe soft spots are real, and the stress-test note identifies the main one correctly. The worst is the unproved assertion in Section 2.2.2 that grouping two nodes not in each other's Markov blankets cannot increase EI. Both the spectral distance matrix (setting distance to infinity) and the inherited greedy search rely on it. If the rule is false, both methods can miss the true EI-maximizing partition. The kernel-degeneracy discussion does not save it: it establishes a global relation between kernel dimension and degeneracy, but says nothing about whether a specific merge can increase EI. The empirical comparison in Figure 1C/D is against two other heuristics, not against an exact optimum, so \"outperforms\" means \"finds at least as high EI as greedy and gradient descent\"—not \"finds the right scale.\" A brute-force search over all partitions on networks small enough (10-15 nodes) would settle whether the blanket rule actually excludes better partitions.\n\nA lesser issue: the free parameters in OPTICS (epsilon, min_samples) and gradient descent are tuned without a sensitivity analysis. The runtime advantages are robust enough that this is minor, but reporting the chosen values and a few neighbors would be cleaner.\n\nThe paper is not circular. The success metric is EI from the authors' own prior work, but the algorithmic comparison is empirical: all three methods optimize the same metric, and the question is which search finds more of it. That is legitimate, though independent replication of the EI framework itself would be welcome.\n\nWho gets value: network scientists who want a practical tool for finding informative higher scales of networks up to thousands of nodes. It does not prove a deep theory of scale, and it does not claim to. I would send it to peer review with a request for small-network exact tests and either a proof or a relaxed version of the Markov blanket constraint. The authors should also soften \"finding the right scale\" to \"finding a high-EI scale among those considered.\"\n\nIt deserves a serious referee, and the fixes are addressable.","headline":"A useful engineering contribution to causal-emergence search, with an unproven pruning rule and no exact baseline; deserves a serious referee but needs revision.","tokens_in":10980,"tokens_out":1782,"would_cite":true,"duration_ms":18497,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C82","68R10","94A17"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a modified spectral-clustering algorithm identifies causally emergent macro-scales in networks faster and as well as or better than greedy and gradient-descent methods, and that these macro-scales improve…","keywords":["causal emergence","effective information","spectral clustering","macroscale networks","preferential attachment","Markov blankets","kernel dimension","entropy rate"],"falsifier":"Run an exhaustive search over all partitions of a small network (12–15 nodes), compute effective information for every partition, and compare the maximum against what the spectral algorithm finds under the Markov-blanket restriction; a single network where a non-blanket merge yields higher EI than any blanket-restricted grouping would falsify the restriction and show the algorithm's search is incomplete.","tokens_in":9994,"feed_emoji":"🕸️","tokens_out":8432,"duration_ms":80202,"temperature":0.7,"pith_summary":"This paper tries to establish that the right scale of a network—the coarse-grained representation that maximizes effective information—can be found efficiently with a spectral-clustering algorithm, and that this algorithm beats both a greedy search and gradient descent on speed and matches or exceeds them on quality. Effective information measures how much certainty a random walker has about its next step on the network, and a higher scale can be genuinely more informative than the microscale. The paper shows on preferential attachment networks that causally emergent macroscales are not a curiosity: they arise for superlinear attachment, and they carry lower entropy rate, higher determinism, lower degeneracy, higher global efficiency, and higher centrality compared with the microscale. The paper additionally links degeneracy to the kernel dimension of the adjacency matrix, which explains why spectral analysis is the right tool.","feed_headline":"Spectral clustering finds network macroscales fastest","feed_subtitle":"Coarse-grained networks can carry more information and run more efficiently than the original microscale.","key_machinery":"The load-bearing object is the spectral embedding of the transition matrix $W^{\\mathrm{out}}$ (for a random walk on the network). The algorithm takes the eigendecomposition, drops the kernel, forms the weighted basis $E'=\\{\\lambda_i e_i : \\lambda_i \\neq 0\\}$, and uses the entries of these weighted eigenvectors as coordinates for each node. A cosine-similarity distance is then computed between node pairs, with the distance set to $\\infty$ for pairs that are not in each other's Markov blanket (parents, children, and co-parents). The OPTICS clustering routine turns this distance matrix into macro-nodes, and the algorithm scans distance thresholds to find the clustering with the highest effective information. This machinery works, the paper argues, because the kernel dimension of the adjacency matrix quantifies degeneracy: distributions that collide after one step correspond to kernel vectors, so the spectrum outside the kernel carries exactly the information needed to decide which subgraphs can be grouped into a better scale.","core_discovery":"On the paper's own terms, the central discovery is that a spectral approach to searching the space of network coarse-grainings outperforms previously used search heuristics. The algorithm embeds each node using the nonzero eigenvectors of the transition matrix weighted by their eigenvalues, computes cosine-similarity distances between nodes (setting to infinity any pair not in each other's Markov blanket), and clusters the distance matrix with OPTICS across a range of distance thresholds to maximize effective information gain. In preferential attachment networks, the spectral method runs orders of magnitude faster than the greedy and gradient-descent baselines and finds equal or larger EI increases. The paper interprets this success as a consequence of a linear-algebra fact it demonstrates: degenerate dynamics—many states converging to the same future state—correspond exactly to vectors in the kernel of the adjacency matrix, so removing the kernel and using the remaining spectrum isolates the topological structure that carries scale information. It concludes that macroscales should be modeled explicitly because they can be peaks of efficiency, centrality, and information transmission.","pith_inferences":["The kernel-dimension link, if it holds beyond preferential attachment, suggests that hub-dominated and attractor-heavy networks (star-like topologies, hierarchical systems) are the natural candidates for causal emergence, so the method could transfer to biological or social networks with such structure.","The Markov-blanket restriction is testable: on very small networks one can enumerate all partitions and check whether any merge outside a blanket raises EI; a positive case would indicate the spectral search has a blind spot fixable by a different distance construction.","The same spectral embedding plus OPTICS could be reused with other scale-quality measures (e.g., other information or dynamics-based objectives), making it a generic template for scale detection rather than a single-purpose algorithm."],"forward_implications":["Spectral clustering makes it practical to find informative macroscales in networks of thousands of nodes, where greedy search becomes prohibitive.","The hardest networks to analyze are those with genuine mesoscales (roughly preferential-attachment exponent between 1 and 2); the spectral method keeps runtime roughly constant there, so it is the recommended tool for such systems.","Causal emergence is not confined to contrived examples: superlinear preferential attachment networks show real EI gains at the macroscale, along with lower entropy rate and higher global efficiency.","Network analyses that stop at the microscale may mislocate control and information transmission: macro-nodes show higher betweenness and eigenvector centrality, suggesting that scale matters for conclusions about network function.","Kernel dimension of the adjacency matrix can serve as a diagnostic of a network's potential for causal emergence, connecting spectral graph theory to the study of scale."],"supporting_citations":[{"why":"supplies the effective information measure and the greedy search baseline, and the µ|π macro-node construction the spectral method is compared with","marker":"[2]"},{"why":"introduces causal emergence as the gain in effective information at a macroscale, the target of the search algorithms","marker":"[4]"},{"why":"establishes that the maximum effective information approximates channel capacity, motivating the search for EImax","marker":"[6]"},{"why":"provides the OPTICS clustering algorithm used to convert the spectral distance matrix into macro-nodes","marker":"[7]"},{"why":"supports the use of higher-order dependencies to keep macro-nodes dynamically accurate","marker":"[8]"},{"why":"used to claim uniform random graphs generally have zero kernel dimension, anchoring the kernel-degeneracy argument","marker":"[13]"},{"why":"used to claim star graphs have kernel dimension n−2 and maximal causal emergence, completing the kernel-scale spectrum","marker":"[14]"},{"why":"supplies the preferential attachment growth model whose exponent α generates the test networks","marker":"[18]"}],"fun_headline_variants":["Spectral clustering finds network scales faster than greedy","Speedy spectral method beats gradient descent for macroscales","Spectrum reveals network scales with record speed","Spectral clustering wins speed race for network coarse-graining"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The algorithms assume that merging two nodes that are not in each other's Markov blankets can never increase effective information, so all candidate groupings are restricted to blanket pairs; if that restriction is false, both the greedy and spectral searches can miss the true maximally informative scale.","fun_headline_variants_meta":{"raw":{"variants":["Spectral clustering finds network scales faster than greedy","Speedy spectral method beats gradient descent for macroscales","Spectrum reveals network scales with record speed","Spectral clustering wins speed race for network coarse-graining"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000206,"raw_usage":{"total_tokens":1397,"prompt_tokens":946,"completion_tokens":451,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":389}},"tokens_in":562,"tokens_out":451,"duration_ms":5483,"temperature":1.0,"reasoning_tokens":389,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:12:47.257642+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run an exhaustive search over all partitions of a small network (12–15 nodes), compute effective information for every partition, and compare the maximum against what the spectral algorithm finds under the Markov-blanket restriction; a single network where a non-blanket merge yields higher EI than any blanket-restricted grouping would falsify the restriction and show the algorithm's search is incomplete.","supporting_citations":[{"cited_title":"The emergence of informative higher scales in complex networks","cited_arxiv_id":"1907.03902","evidence_quote":"supplies the effective information measure and the greedy search baseline, and the µ|π macro-node construction the spectral method is compared with"},{"cited_title":"When the map is better than the territory","cited_arxiv_id":null,"evidence_quote":"establishes that the maximum effective information approximates channel capacity, motivating the search for EImax"},{"cited_title":"Spectra of some simple graphs","cited_arxiv_id":null,"evidence_quote":"used to claim star graphs have kernel dimension n−2 and maximal causal emergence, completing the kernel-scale spectrum"}],"review_version":1}