Functional connectomes of neural networks

Tananun Songdechakraiwut; Yutong Wu

REVIEW 4 major objections 5 minor 37 references

Functional connectomes of neural networks

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that a neural network's functional connectome—the correlation graph of hidden-neuron activations—carries topological signatures that distinguish how the network was regularized and which input class it is processing.

desk verdict A scalable topological summary for neural-network functional connectomes with a real runtime win, but the empirical validation leans on clustering purity without significance tests or a raw-activation control. read the letter →

arxiv 2412.15279 v2 pith:YBNT7BK6 submitted 2024-12-18 cs.NE cs.AIcs.LGq-bio.NC

Tananun Songdechakraiwut , Yutong Wu This is my paper · ORCID

classification cs.NEcs.AIcs.LGq-bio.NC MSC 55N3162R4068T07

keywords functionalconnectomeneuralnetworkinterpretabilitypersistentgraphhomologyWassersteindistancetopologicaldataanalysiscentroid-basedclusteringregularizationbrain-inspiredcomputing

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 argues that a neural network's behavior can be read from the topology of its functional connectome, a graph whose edges are correlations between hidden neurons' activations across a set of inputs. The authors show that analyzing this brain-inspired representation with persistent graph homology separates networks trained under different regularization strategies and separates the internal processing of different input classes, all in near-linear time without thresholding. If the claim holds, it gives researchers a scalable, threshold-free way to probe what a trained network has learned and how it processes information, borrowing tools from human brain connectomics.

What carries the argument

Persistent graph homology, a threshold-free topological summary that records births of connected components and deaths of cycles as an edge-weight threshold rises; for a complete weighted graph it reduces to the sorted maximum-spanning-tree edge weights (births) and the sorted non-MST edge weights (deaths). The paper's mechanism is the closed-form Wasserstein distance on these sorted vectors, $W_{p,B}(G^{(1)},G^{(2)})=\lVert\mathbf{b}_{G^{(1)}}-\mathbf{b}_{G^{(2)}}\rVert_p$ and analogously for deaths, which makes the Wasserstein barycenter a coordinate-wise average and enables Lloyd-style centroid clustering in $O(n\log n)$ time.

What would settle it

Permute the class labels of the functional dataset across many trials and recompute Top's clustering purity; if purity stays well above chance, the class separation reflects input statistics rather than the claimed class-specific activation structure. Alternatively, run the same clustering on raw activation vectors; if they match Top's purity, no topological signal is needed.

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Extended reading notes

Core claim

The central claim is that topological signals measured by persistent homology on neural-network functional connectomes characterize neural network function. A functional connectome is the complete graph on hidden neurons with edge weights given by the absolute Pearson correlation of their activation vectors over a functional dataset. Persistent graph homology tracks connected components and cycles across all correlation thresholds, and the paper presents closed-form Wasserstein distance, barycenter, variance, and gradient statistics for these summaries. Empirically, clustering these summaries separates regularization strategies (batch norm, dropout, L2, vanilla) with high purity and separates per-class connectomes at levels far above chance, with exact computation for thousands of nodes in about a second.

Load-bearing premise

The load-bearing premise is that unsupervised cluster purity against predefined classes is evidence that the topology captures meaningful functional mechanisms; if the clusters are separable by trivial activation statistics or by the particular functional dataset chosen, the interpretability claim does not follow.

Editorial extensions

If this is right

Regularization strategy leaves a detectable topological fingerprint: clustering connectomes from batch norm, dropout, L2, and vanilla training separates the four strategies with high purity, and each pairwise comparison against vanilla reaches purity near 1.0 in most datasets.
Different input classes are processed through distinct functional mechanisms: per-class functional connectomes cluster with purity 0.5–0.6 across ten classes, far above the 0.1 random baseline.
Exact topology is computable at scale: Top computes exact Wasserstein distances for connectomes with thousands of nodes and millions of edges in roughly one second, while conventional persistent-homology baselines stall at a few hundred nodes.
Closed-form Wasserstein statistics provide a gradient-based tool: because barycenter, variance, and distance gradients are analytic, the representation can plug into centroid clustering and, potentially, other gradient-optimized machine learning objectives.
The method works beyond toy settings: it remains effective on the convolutional CIFAR-10 network by restricting analysis to the final fully-connected layers.

Reading between the lines

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

Beyond the paper: the sorted birth and death vectors are a compressed, threshold-free fingerprint of a network's activation geometry; a natural test is whether these fingerprints track generalization, calibration, or adversarial robustness across training runs and seeds.
Beyond the paper: because the Wasserstein barycenter and variance are closed-form, one could build topological analogues of PCA or regression on connectomes, or monitor how topology evolves during training, none of which the paper demonstrates.
Beyond the paper: the purity-based validation does not rule out that simple activation statistics (e.g., mean or variance of activations) drive the clusters; a permutation test or a baseline on raw activation vectors would test whether topology adds signal beyond those statistics.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague, quick take: this paper does something genuinely useful and mostly sound, but the headline interpretability claim is under-validated. The authors adapt their prior persistent graph homology framework to neural-network functional connectomes, and the main win is computational: they compute exact Wasserstein distances between connectomes in O(n log n), and the runtime plot shows their Top method handling graphs with thousands of nodes in a second, where conventional persistence baselines die at a few hundred. That's a real contribution, and it is new relative to Zhang et al., which thresholded and stayed cubic.

The math is fine, as far as I can tell. The birth/death sets from persistent graph homology reduce to sorted edge weights in the MST and its complement, so the equal-cardinality sorted matching gives exact Wasserstein distances. They cite the proofs from their earlier papers rather than reproducing them, which is acceptable for an application paper. The two studies—clustering by regularization strategy and by input class—are reasonable demonstrations, and the code and data are linked.

Here is where I lower my enthusiasm. The central claim that topological signals characterize neural network function rests entirely on clustering purity. The paper advertises permutation tests as a benefit but runs none. The purity comparison to a nominal 0.1 random baseline is weak because k-means on random labels will beat 1/K; without a permutation test, values like 0.46–0.57 in Study 2 are not established as significant. The stress-test note also lands: there is no raw-activation baseline. The Adj baseline uses the same correlation matrices without the topological decomposition, and Top does not consistently beat it—in Study 2, SWK and WD often outperform Top, and Adj is sometimes not far behind. So the unique contribution of the topology itself is not nailed down.

That said, the paper is not overreaching in every sentence; in Study 2 they present Top as one of several effective topological methods, not the clear winner. The scalability claim is solid, and the framework could be immediately useful for monitoring training or comparing models. The gaps are fixable: add permutation tests, add a baseline on raw activation vectors or simple correlation statistics, and temper the interpretability language.

I would send this to review. It deserves referee time, but I would expect major revision. The method is valuable, the runtime advantage is real, and the empirical weaknesses are addressable rather than fatal. A reader working on interpretability or TDA will get something out of it.

Referee Report

4 major / 5 minor

Summary. This paper proposes a framework for characterizing neural network function by constructing functional connectomes (absolute Pearson correlations between neuron activations) and summarizing their topology with persistent graph homology. The authors derive closed-form expressions for Wasserstein distances, barycenters, variances, and gradients of birth/death values, and use centroid-based clustering (Top) to group networks by regularization strategy (Study 1) and by input class (Study 2). Experiments on MNIST, Fashion-MNIST, and CIFAR-10 compare Top with six baselines and include a runtime study. The paper claims that topological signals effectively characterize neural network functions and that Top offers scalable exact computation.

Significance. If the central claim holds, the paper offers a threshold-free and scalable representation of network behavior, which would be a useful addition to interpretability tools. The strengths include a clean definition of functional connectomes, exact near-linear-time computation, no fitted parameters in the representation, public code, and a runtime comparison showing orders-of-magnitude speedup over conventional persistence diagram methods. However, the empirical validation does not yet establish that the topological decomposition itself, rather than low-order activation statistics or the clustering algorithm, is responsible for the reported separability; the paper also overstates the performance of Top relative to baselines. These issues are fixable with additional controls and corrected claims.

major comments (4)

[Section 3, Cluster analysis and Study 2] The comparison of purity scores (0.46–0.57) as "significantly better than the 0.1 score expected if clustering was made randomly" is statistically invalid because k-means with K=10 always produces purity greater than 1/K on label-random data, and the reported values are not tested against a permutation null. The Potential Impact section advertises permutation tests, but none is performed. Please add permutation or label-shuffling tests and report p-values or null distributions for the purity scores.
[Section 3, Tables 1 and 2] The statement "In most settings, the proposed Top method outperforms other baselines" is not supported by the tables. For example, in Table 2, Fashion-MNIST Vanilla SWK achieves 0.64 vs Top 0.53; Fashion-MNIST Batch Norm SWK 0.60 vs Top 0.49; MNIST Batch Norm WD 0.50 vs Top 0.46; and in Table 1, MNIST All SWK 0.85 vs Top 0.78 and CIFAR-10 All WD 0.98/SWK 0.96 vs Top 0.88. The paper should either provide a statistical comparison or reframe the contribution as scalability, not clustering accuracy.
[Section 2 and Section 3, Cluster analysis] The clustering signal could be carried by trivial activation statistics or by the global distribution of pairwise correlations rather than by the topological birth/death decomposition, but no control on raw activation vectors is provided. The Adj baseline uses the same correlations and is much worse than Top, which is encouraging, but a k-means baseline on raw per-neuron activation vectors (or on the full correlation vector) is needed to support the claim that the "deeper understanding of underlying mechanisms" comes from topology. Please add such a control.
[Section 3, Method comparison and Tables 1-2] Many cells in Tables 1 and 2 are reported without standard deviations (e.g., Table 1: WD 0.75, SWK 0.85; Table 2: several entries), and no significance tests are used to compare methods across the 20 trials. Since the tables are the primary evidence for the clustering claims, the missing uncertainty quantification makes it impossible to assess whether the observed differences are reliable.

minor comments (5)

[Section 2, Persistence Statistics] The claim that the exact p-Wasserstein distance equals the Lp distance between sorted birth/death vectors is cited to prior work but not derived; a brief proof sketch or an explicit statement of the equal-cardinality and no-diagonal conditions would make the paper more self-contained.
[Section 2, Eq. (5) and surrounding text] The notation W_p,B appears in text but the equation uses W_p,B(G(1),G(2)) with a semicolon in one place; unify the notation throughout.
[Section 2, Functions of Neural Networks] The sentence "As is common in machine learning, since we know a computable formula..." is awkward and should be rewritten for clarity.
[References] In the reference to Xiao et al., the author name is typeset as "V ollgraf"; this should be "Volgraf". Also, the in-text citation to "Songdechakraiwut et al. 2023" in the persistent graph homology paragraph should be cross-checked against the reference list to ensure the intended source is unambiguous.
[Section 3, Figure 2] The figure caption says "Persistence diagrams and statistics for each strategy" but the figure itself is not included in the submitted text; ensure the figure is legible and the shaded regions are clearly labeled as Wasserstein standard deviation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the framework is an unsupervised empirical application of an externally published topological statistic; no fitted parameter is relabeled as a prediction.

full rationale

The paper's derivation chain is not circular. The functional connectome is explicitly defined as an absolute Pearson-correlation matrix of neuron activation vectors (equations for rho_jk and G_jk), and the persistent-graph-homology representation is the sorted list of MST edge weights (births) and non-MST edge weights (deaths), with Wasserstein statistics given as closed-form L2 distances on those sorted vectors. Every quantity is computed from the trained network's activations on the functional dataset; no parameter is fitted to class labels, regularization labels, or purity targets. The clustering results in Tables 1 and 2 are empirical observations that could have been low and are not forced by the construction; for example, Top does not uniformly beat the Adj baseline. The main theoretical formulas are imported from the authors' prior publications (Songdechakraiwut and Chung 2023; Songdechakraiwut et al. 2022, 2023), but these are externally published, parameter-free mathematical results whose assumptions do not include the paper's empirical claims, so the self-citations are independent support rather than circular load-bearing. The absence of permutation tests and the possibility that low-order activation statistics drive the separation are validity or robustness concerns, not definitions or fitted-input reductions, and therefore do not constitute circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No new physical or mathematical entities are postulated. The framework relies on three imported modeling assumptions: the persistent graph homology theorem from the authors' prior work, the Pearson correlation and absolute-value convention from connectomics, and the well-generalized network assumption. The only user-chosen numbers are K and p, neither fitted to optimize the reported results.

free parameters (2)

Number of clusters K = 4 (Study 1), 10 (Study 2)
Set to the number of predefined classes rather than learned; this makes purity evaluation meaningful but is a user-chosen input the clustering results depend on.
Wasserstein exponent p = 2
The clustering objective uses squared 2-Wasserstein distances; p=2 is standard but not fitted, and results would change with p.

assumptions (4)

domain assumption Exact Wasserstein distance between persistent graph homology summaries equals Lp distance between sorted birth and death value vectors (Songdechakraiwut and Chung 2023).
This theorem is imported from prior work and not proved in the paper; the entire clustering and barycenter computation depends on it.
domain assumption Functional synergy between neurons is captured by absolute Pearson correlation of activation vectors.
The paper adopts the fMRI connectomics convention, but this discards the sign of correlations and only measures linear dependence.
domain assumption The trained networks are well-generalized, so functional connectomes reflect general mechanisms rather than memorization.
The framework assumes k-fold grid search yields optimal hyperparameters; training details are deferred to the supplementary material.
domain assumption All connectomes in a comparison have the same number of neurons, so birth and death sets have equal cardinality and sorted matching is valid.
The closed-form Wasserstein formula requires equal cardinality; the paper fixes architectures per dataset, but this limits generality.

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Cite this review

Pith. "Pith review of Functional connectomes of neural networks." pith.science (2026). https://pith.science/paper/YBNT7BK6

@misc{pith2026241215279,
  author       = {Pith},
  title        = {Pith review of: Functional connectomes of neural networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YBNT7BK6}},
  note         = {Machine review of arXiv:2412.15279}
}

read the original abstract

The human brain is a complex system, and understanding its mechanisms has been a long-standing challenge in neuroscience. The study of the functional connectome, which maps the functional connections between different brain regions, has provided valuable insights through various advanced analysis techniques developed over the years. Similarly, neural networks, inspired by the brain's architecture, have achieved notable success in diverse applications but are often noted for their lack of interpretability. In this paper, we propose a novel approach that bridges neural networks and human brain functions by leveraging brain-inspired techniques. Our approach, grounded in the insights from the functional connectome, offers scalable ways to characterize topology of large neural networks using stable statistical and machine learning techniques. Our empirical analysis demonstrates its capability to enhance the interpretability of neural networks, providing a deeper understanding of their underlying mechanisms.

Figures

Figures reproduced from arXiv: 2412.15279 by the authors.

**Figure 1.** A schematic for extracting persistent graph homology, representing the topology of neural-network-derived functional [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Statistics of the functional dataset used in Study 1. Left: Sample means of the functional connectomes, averaged within [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Average runtime of each method for computing [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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Reference graph

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ENTRY address archivePrefix author booktitle chapter edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.a...
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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address archivePrefix author booktitle chapter edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.a...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

[3] [3]

Adams, H.; Emerson, T.; Kirby, M.; Neville, R.; Peterson, C.; Shipman, P.; Chepushtanova, S.; Hanson, E.; Motta, F.; and Ziegelmeier, L. 2017. Persistence images: a stable vector representation of persistent homology. Journal of Machine Learning Research, 18(8): 1--35

work page 2017

[4] [4]

Biagetti, M.; Cole, A.; and Shiu, G. 2021. The persistence of large scale structures. Part I. Primordial non-Gaussianity . Journal of Cosmology and Astroparticle Physics, 2021(04): 061

work page 2021

[5] [5]

Bullmore, E.; and Sporns, O. 2009. Complex brain networks: graph theoretical analysis of structural and functional systems. Nature Reviews Neuroscience, 10(3): 186--198

2009

[6] [6]

Carriere, M.; Cuturi, M.; and Oudot, S. 2017. Sliced Wasserstein kernel for persistence diagrams. In International Conference on Machine Learning (ICML), 664--673

work page 2017

[7] [7]

Edelsbrunner, H.; and Harer, J. L. 2022. Computational Topology: An Introduction. American Mathematical Society

work page 2022

[8] [8]

Forgy, E. W. 1965. Cluster analysis of multivariate data: efficiency versus interpretability of classifications. Biometrics, 21: 768--769

work page 1965