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Finding the right scale of a network: Efficient identification of causal emergence through spectral clustering

T0 review · 4 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict A useful engineering contribution to causal-emergence search, with an unproven pruning rule and no exact baseline; deserves a serious referee but needs revision. read the letter →

arxiv 1908.07565 v2 pith:XXA2EU25 submitted 2019-08-20 cs.SI physics.soc-ph

Ross Griebenow , Brennan Klein , Erik Hoel This is my paper · ORCID
classification cs.SIphysics.soc-ph MSC 05C8268R1094A17
keywords causalemergenceeffectiveinformationspectralclusteringmacroscalenetworkspreferentialattachmentMarkovblanketskerneldimensionentropyrate
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 7 minor

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.

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 (4)
  1. [Section 2.2.2] 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.
  2. [Section 3.1, Fig. 1C/D] 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.
  3. [Section 3.2, Fig. 2E-G] 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.
  4. [Sections 2.2.2 and 2.2.3] 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.
minor comments (7)
  1. [Section 2.1, Eq. (1)] 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.
  2. [Section 2.2.2] 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.
  3. [Section 2.2.2] 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.
  4. [Section 2.2.1] 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.
  5. [Section 2.2.3] The number of macro-nodes k in the gradient descent relaxation is not specified; the paper should state how k is chosen or optimized.
  6. [Section 3.2, Fig. 2L] 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.
  7. [Section 1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spectral-vs-greedy/vs-gradient comparison is empirical and the EI objective is a fixed input, though one unproved pruning rule creates correctness risk.

full rationale

The paper's central operational claim is that spectral clustering finds higher-EI scales faster than, or at least as fast as, greedy and gradient-descent baselines (Fig. 1). This is an empirical comparison on a common, pre-defined objective: EI is given by Eq. (1) and is not redefined by the algorithm's output. The spectral method's epsilon is chosen by evaluating EI over a grid, but that is standard objective-function optimization, not a prediction that is statistically forced by a fit to the target quantity. The µ|π macro-node construction and the EI metric originate in the authors' prior work, but they are fixed inputs rather than conclusions derived from the algorithm; the algorithmic comparison is self-contained and would remain meaningful even if one disputed the macro-node summary. The main weakness is Section 2.2.2's unproved assertion that grouping non-Markov-blanket pairs cannot increase EI; this restricts the search and could cause all compared methods to miss the true optimum, so it is a correctness/falsifiability gap, not a circular reduction. The kernel-degeneracy derivation supports an intuition about why spectral structure matters but does not inject the target result into the algorithm. No equation or fitted parameter was found that reduces to its own input; the reported network-property changes (entropy rate, efficiency, centrality) are distinct quantities measured after the EI-selected scale is fixed, not restatements of the EI objective itself.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities. Its free parameters are the clustering and optimization hyperparameters, chiefly the spectral method's epsilon sweep and the gradient descent settings. The load-bearing assumptions are inherited from the authors' prior causal emergence framework: the EI objective, the mu|pi macro-node construction, and the unproved Markov blanket restriction. The kernel-degeneracy correspondence is a speculative motivation rather than a proven theorem.

free parameters (3)
  • OPTICS distance threshold epsilon = not stated; selected by sweeping values to maximize EI gain
    The spectral algorithm's output depends on epsilon, and the authors choose the best value per network by checking EI gain, making it a per-network fitted parameter.
  • OPTICS clustering parameters (e.g., min_samples) = not stated
    OPTICS requires additional parameters that affect the clustering and therefore the discovered macroscale; values are not reported.
  • Gradient descent hyperparameters (learning rate, momentum, max iterations, initialization) = not stated
    The performance of the gradient descent baseline depends on these hand-chosen settings, which are not specified in the paper.
assumptions (6)
  • domain assumption Effective information, as defined by Eq. (1), is the correct objective for choosing the most informative scale of a network.
    The entire evaluation relies on EI, inherited from Klein and Hoel [2] and Hoel et al. [4]; the paper does not justify EI against alternative scale-selection criteria.
  • domain assumption Macro-nodes constructed as stationary-distribution-weighted averages (mu|pi type) accurately recapitulate micro-scale random-walk dynamics.
    The paper cites [2] for this claim and uses it to compare network properties across scales; it is load-bearing but not demonstrated here.
  • ad hoc to paper If two nodes are not in each other's Markov blankets, grouping them cannot increase EI.
    Stated without proof in Section 2.2.2; this restriction limits both the spectral and greedy search spaces and is central to the algorithms.
  • ad hoc to paper The kernel dimension of the transition matrix corresponds to degeneracy in the network's dynamics.
    The derivation in Section 2.2.2 is informal and contains an ill-defined beta; the paper uses this correspondence to explain why spectral analysis works.
  • domain assumption Clusters found by OPTICS on the eigenvalue-weighted cosine similarity embedding correspond to macro-nodes that increase EI.
    The paper assumes that spectral embedding captures the structure relevant for causal emergence; no theoretical guarantee is provided.
  • domain assumption Preferential attachment networks are a suitable testbed for evaluating methods for finding informative higher scales.
    The paper uses preferential attachment networks throughout; this is a standard model, but the conclusions may not generalize to all network families.

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Pith. "Pith review of Finding the right scale of a network: Efficient identification of causal emergence through spectral clustering." pith.science (2026). https://pith.science/paper/XXA2EU25

@misc{pith2026190807565,
  author       = {Pith},
  title        = {Pith review of: Finding the right scale of a network: Efficient identification of causal emergence through spectral clustering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XXA2EU25}},
  note         = {Machine review of arXiv:1908.07565}
}
read the original abstract

All networks can be analyzed at multiple scales. A higher scale of a network is made up of macro-nodes: subgraphs that have been grouped into individual nodes. Recasting a network at higher scales can have useful effects, such as decreasing the uncertainty in the movement of random walkers across the network while also decreasing the size of the network. However, the task of finding such a macroscale representation is computationally difficult, as the set of all possible scales of a network grows exponentially with the number of nodes. Here we compare various methods for finding the most informative scale of preferential attachment networks, discovering that an approach based on spectral analysis outperforms greedy and gradient descent-based methods. We then use this procedure to show how several structural properties of these networks vary across scales. We describe how meso- and macroscale representations of networks can have significant benefits over their underlying microscale in terms of information transmission, which include properties such as increase in determinism, a decrease in degeneracy, a lower entropy rate of random walkers on the network, an increase in global network efficiency, and higher values for a variety of centrality measures than the microscale.

Figures

Figures reproduced from arXiv: 1908.07565 by the authors.

Figure 1
Figure 1. Comparison of Methods for Computing Causal Emergence (A) [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Preferential attachment network properties at different scales. (A) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

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