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arxiv: 2606.25826 · v1 · pith:MWZW7TA4new · submitted 2026-06-24 · ❄️ cond-mat.dis-nn · cs.SI· math-ph· math.MP· q-bio.NC

Weight geometry governs functional memory in complex systems

Elka\"ioum M. Moutuou , Habib Benali This is my paper

Pith reviewed 2026-06-25 19:49 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cs.SImath-phmath.MPq-bio.NC
keywords complex systemsfunctional memoryweight geometrynetwork topologyinformation flownull modelsdynamical organizationsmultiscale analysis
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The pith

Real interaction strengths organize functional memory at greater hierarchical depth than random weight assignment on the same topology.

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

The paper shows that in thirty-four networks spanning multiple domains and scales, real interaction strengths organize functional memory at greater hierarchical depth than random weights on the same topology. This is measured via a thermodynamic description of multiscale information flow that tracks memory distribution across path lengths. Memory organization collapses to four recurrent types, pointing to low-dimensional structure in how these systems function. Comparisons to null models show that weight geometry, mesoscale structure, and directionality each play distinct roles, with weights being primary for memory depth.

Core claim

Using a thermodynamic description of multiscale information flow, the authors quantify memory distribution across path lengths in thirty-four networks. They find that real interaction strengths organize memory at greater hierarchical depth than random weight assignment, and that this organization collapses onto four recurrent dynamical organisations. Weight geometry governs memory depth, mesoscale structure shapes it across scales, and directionality modulates sensitivity to perturbation, establishing that weights carry dynamical structure beyond binary topology.

What carries the argument

weighted transport geometry, the mechanism that organizes functional memory at greater hierarchical depth than random weights on identical topology

Load-bearing premise

The thermodynamic description of multiscale information flow accurately quantifies functional memory and the null models that selectively perturb weighted transport geometry, mesoscale structure, and directionality correctly isolate their distinct contributions.

What would settle it

Replicating the memory-depth calculation on the same thirty-four networks but finding that random weight assignment produces equal or greater hierarchical depth than the real weights.

Figures

Figures reproduced from arXiv: 2606.25826 by Elka\"ioum M. Moutuou, Habib Benali.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
read the original abstract

Complex systems, from gene regulatory networks to neural circuits and transportation infrastructures, exhibit rich functional behaviour that topology alone does not capture. Here we show that functional memory exhibits a universal organisational regularity: in every biological, ecological, social, and technological domain studied, real interaction strengths organise memory at greater hierarchical depth than random weight assignment on the same topology, across thirty-four networks spanning several orders of magnitude in size and density. Using a thermodynamic description of multiscale information flow, we quantify how memory is distributed across path lengths and show that functional memory organisation collapses onto four recurrent dynamical organisations, revealing an intrinsically low-dimensional structure. Comparing each network against null models that selectively perturb weighted transport geometry, mesoscale structure, and directionality reveals that these ingredients contribute distinct and non-equivalent roles: weight geometry systematically governs memory depth, mesoscale structure shapes memory organisation across scales, and directionality modulates the sensitivity of the cascade to structural perturbation. The same comparison provides an operational criterion for whether network weights encode genuine functional interaction structure. These results establish weighted transport geometry as a primary organiser of functional memory and show that weighted interactions carry dynamical structure that binary topology alone cannot recover.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript claims that across 34 networks from biological, ecological, social, and technological domains, real interaction strengths organize functional memory at greater hierarchical depth than random weight assignment on the same topology. Using a thermodynamic description of multiscale information flow, memory distribution across path lengths is quantified and shown to collapse onto four recurrent dynamical organizations. Null models selectively perturbing weighted transport geometry, mesoscale structure, and directionality demonstrate that these factors play distinct roles, with weight geometry governing memory depth, leading to the conclusion that weighted transport geometry is a primary organizer of functional memory beyond binary topology.

Significance. If the thermodynamic measure is established as a valid proxy for functional memory, the work would be significant for showing that weighted interactions encode dynamical structure not recoverable from topology alone, across a large and diverse set of networks. The scale of the study (34 networks spanning orders of magnitude in size and density) and the use of multiple targeted null models to isolate contributions are strengths that would support broader implications for understanding functional organization in complex systems.

major comments (3)
  1. [Abstract] Abstract: The central claim that weight geometry 'governs functional memory' equates differences in the thermodynamic multiscale information-flow measure (deeper hierarchical organization under real weights) with governance of functional memory. No independent validation is provided that this measure tracks actual functional outcomes (e.g., robustness, response specificity, or empirical memory tasks) in any of the 34 networks, so the null-model results demonstrate only a difference in the chosen dynamical statistic rather than functional governance.
  2. [Abstract] Abstract: The statement that functional memory organisation 'collapses onto four recurrent dynamical organisations' is load-bearing for the low-dimensional structure conclusion, yet the abstract (and by extension the manuscript) provides no detail on the classification procedure, distance metric, or statistical controls used to identify these four organizations or confirm their recurrence across domains.
  3. [Abstract] Abstract: The operational criterion for whether network weights encode 'genuine functional interaction structure' is derived from the same null-model comparisons; without a shown link between the thermodynamic quantities and functional performance, this criterion lacks external grounding and cannot be evaluated as operational.
minor comments (1)
  1. The abstract is information-dense; consider splitting the description of the null-model results and the four organizations into separate sentences for improved readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below, proposing targeted revisions to the abstract and discussion to improve precision and clarity while preserving the manuscript's core findings.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that weight geometry 'governs functional memory' equates differences in the thermodynamic multiscale information-flow measure (deeper hierarchical organization under real weights) with governance of functional memory. No independent validation is provided that this measure tracks actual functional outcomes (e.g., robustness, response specificity, or empirical memory tasks) in any of the 34 networks, so the null-model results demonstrate only a difference in the chosen dynamical statistic rather than functional governance.

    Authors: The thermodynamic measure is grounded in a maximum-entropy model of multiscale information flow and serves as a quantitative proxy for memory organization. The null-model results establish that real weights produce deeper hierarchies than randomized weights on the same topology. We agree that the manuscript does not include direct empirical validation against specific functional performance metrics across the 34 networks. We will revise the abstract and relevant discussion sections to frame the measure explicitly as a proxy and to qualify the 'governs' language as indicating systematic influence on the chosen dynamical statistic. revision: partial

  2. Referee: [Abstract] Abstract: The statement that functional memory organisation 'collapses onto four recurrent dynamical organisations' is load-bearing for the low-dimensional structure conclusion, yet the abstract (and by extension the manuscript) provides no detail on the classification procedure, distance metric, or statistical controls used to identify these four organizations or confirm their recurrence across domains.

    Authors: The classification into four recurrent organizations is performed via clustering of normalized memory-distribution profiles, using a Euclidean distance metric with permutation-based statistical controls to confirm recurrence across domains; full details appear in the Methods and Supplementary Information. We will revise the abstract to include a concise description of this procedure. revision: yes

  3. Referee: [Abstract] Abstract: The operational criterion for whether network weights encode 'genuine functional interaction structure' is derived from the same null-model comparisons; without a shown link between the thermodynamic quantities and functional performance, this criterion lacks external grounding and cannot be evaluated as operational.

    Authors: The criterion is defined internally by whether real weights produce memory depth that deviates significantly from the randomized-weight null model. We will revise the abstract sentence to state explicitly that the criterion is based on this thermodynamic null-model comparison. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected; derivation relies on external null-model comparisons.

full rationale

The provided abstract and context describe a thermodynamic measure of multiscale information flow applied to real weights versus null models that randomize weights on fixed topology. This yields an empirical contrast (real weights produce greater hierarchical depth) rather than any reduction of the output to the input by definition, fitting, or self-citation. No equations or steps are quoted that equate the claimed 'governance' result to a fitted parameter or prior self-citation; the null-model design supplies an independent benchmark. The paper is therefore self-contained against its stated controls, consistent with the default expectation that most papers are not circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies insufficient detail to enumerate free parameters, axioms, or invented entities; the thermodynamic description of information flow is invoked but not specified.

pith-pipeline@v0.9.1-grok · 5745 in / 1105 out tokens · 30086 ms · 2026-06-25T19:49:08.070530+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

57 extracted references · 5 canonical work pages

  1. [1]

    This convention makes columns correspond to sources and rows to targets; self-loops are permitted

    Memory propagators and the thermalization cascade A directed network is modelled as a weighted directed graphG= (V, E)with adjacency matrixA, whereA ij ≥0encodes the interaction strength from nodejto nodei. This convention makes columns correspond to sources and rows to targets; self-loops are permitted. Two representations are used depending on the netwo...

  2. [2]

    We therefore estimate the relative Frobenius error in equation (7) using randomized trace estimation [31, 32]

    Randomized estimation of effective memory depth For large networks, explicitly forming the infinite-memory resolventR (∞) t becomes computationally expensive and numerically ill-conditioned near the critical inverse temperatureβ c. We therefore estimate the relative Frobenius error in equation (7) using randomized trace estimation [31, 32]. Letz∈R N be a ...

  3. [3]

    The spectrum records how the total thermalization time is distributed across memory depths

    Hierarchical memory spectra and Wasserstein comparison The hierarchical memory spectrum of a network is the probability distribution µG(l) = τl−1 −τ l τ0 , l≥1,(11) normalized so that X l≥1 µG(l) = 1. The spectrum records how the total thermalization time is distributed across memory depths. Large tail mass indicates that thermalization remains distribute...

  4. [4]

    Thermodynamic protocol Recall that the framework distinguishes two timescales. Microscopic probability flow is described by the propagatorsT (n) t , whereas a slow macroscopic timescale governs the progressive suppression of memory depth through the inverse-temperature parameterβ(t). At fixed macroscopic time, the propagators evolve under quasi-static the...

  5. [5]

    Null model ensembles Three null ensembles are used as counterfactual perturbations of each empirical network. Each ensemble selectively disrupts a distinct structural ingredient while preserving the others, allowing the contribution of mesoscale organisation, weight geometry, and directionality to functional memory organisation to be separated quantitativ...

  6. [6]

    Probabilistic classification of functional memory organisation To summarize the diversity of functional memory organisation across the empirical networks, we per- formed an unsupervised Gaussian mixture analysis on thermodynamic observables derived from the ther- malization cascade. The resulting latent components were interpreted along the two complement...

  7. [7]

    A.The Architecture of Complexity, 457–476 (Springer US)

    Simon, H. A.The Architecture of Complexity, 457–476 (Springer US)

  8. [8]

    Newman, M. E. The structure and function of complex networks.SIAM review45, 167–256 (2003)

    2003

  9. [9]

    Bianconi, G.et al.Complex systems in the spotlight: next steps after the 2021 nobel prize in physics.Journal of Physics: Complexity4, 010201. 27

    2021

  10. [10]

    Moutuou, E. M. & Benali, H. Kubo-martin-schwinger states of path-structured flow in directed brain synaptic networks.Physical Review E112

  11. [11]

    Moutuou, E. M. & Benali, H. Brain functions emerge as thermal equilibrium states of the connectome.Phys. Rev. Res.7, 033156 (2025). URLhttps://link.aps.org/doi/10.1103/jmqh-bqnc

    work page doi:10.1103/jmqh-bqnc 2025

  12. [12]

    & Virasoro, M.Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications(WORLD SCIENTIFIC)

    Mezard, M., Parisi, G. & Virasoro, M.Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications(WORLD SCIENTIFIC)

  13. [13]

    & Cugliandolo, L

    Vincent, E., Hammann, J., Ocio, M., Bouchaud, J.-P. & Cugliandolo, L. F.Complex Behaviour of Glassy Systems, chap. Slow dynamics and aging in spin glasses, 184–219 (Springer Berlin Heidelberg)

  14. [14]

    & Ulanowicz, R

    Baird, D. & Ulanowicz, R. E. The seasonal dynamics of the chesapeake bay ecosystem.Ecological Monographs 59, 329–364

  15. [15]

    Villani, C.Optimal Transport(Springer Berlin Heidelberg)

  16. [16]

    S., Milo, R., Mangan, S

    Shen-Orr, S. S., Milo, R., Mangan, S. & Alon, U. Network motifs in the transcriptional regulation network of escherichia coli.Nature Genetics31, 64–68

  17. [17]

    M.et al.Large-scale mapping of human protein–protein interactions by mass spectrometry.Molecular Systems Biology3

    Ewing, R. M.et al.Large-scale mapping of human protein–protein interactions by mass spectrometry.Molecular Systems Biology3

  18. [18]

    Stelzl, U.et al.A human protein-protein interaction network: A resource for annotating the proteome.Cell122, 957–968

  19. [19]

    Han, H.et al.Trrust: a reference database of human transcriptional regulatory interactions.Scientific Reports5

  20. [20]

    R., Chen, B

    Varshney, L. R., Chen, B. L., Paniagua, E., Hall, D. H. & Chklovskii, D. B. Structural properties of the caenorhabditis elegans neuronal network.PLOS Computational Biology7, 1–21 (2011). URLhttps: //doi.org/10.1371/journal.pcbi.1001066

    work page doi:10.1371/journal.pcbi.1001066 2011

  21. [21]

    Cook, S.et al.Whole-animal connectomes of both caenorhabditis elegans sexes.Nature571, 63 (2019)

    2019

  22. [22]

    & Meinertzhagen, I

    Ryan, K., Lu, Z. & Meinertzhagen, I. A. The cns connectome of a tadpole larva of ciona intestinalis (l.) highlights sidedness in the brain of a chordate sibling.eLife5

  23. [23]

    URLhttps://www.science.org/doi/abs/10.1126/science.add9330

    Winding, M.et al.The connectome of an insect brain.Science379, eadd9330 (2023). URLhttps://www.science.org/doi/abs/10.1126/science.add9330. https://www.science.org/doi/pdf/10.1126/science.add9330

    work page doi:10.1126/science.add9330 2023

  24. [24]

    Milo, R.et al.Superfamilies of evolved and designed networks.Science303, 1538–1542

  25. [25]

    A., Labandeira, C

    Dunne, J. A., Labandeira, C. C. & Williams, R. J. Highly resolved early eocene food webs show development of modern trophic structure after the end-cretaceous extinction.Proceedings of the Royal Society B: Biological Sciences281, 20133280

  26. [26]

    Williams, R. J. & Martinez, N. D. Simple rules yield complex food webs.Nature404, 180–183

  27. [27]

    A., Williams, R

    Dunne, J. A., Williams, R. J. & Martinez, N. D. Food-web structure and network theory: The role of connectance and size.Proceedings of the National Academy of Sciences99, 12917–12922

  28. [28]

    & Aspuru-Guzik, A

    Mohseni, M., Rebentrost, P., Lloyd, S. & Aspuru-Guzik, A. Environment-assisted quantum walks in photosyn- thetic energy transfer.The Journal of Chemical Physics129

  29. [29]

    & Vespignani, A

    Barrat, A., Barth ´elemy, M., Pastor-Satorras, R. & Vespignani, A. The architecture of complex weighted net- 28 works.Proceedings of the National Academy of Sciences101, 3747–3752

  30. [30]

    & Saram ¨aki, J

    Holme, P. & Saram ¨aki, J. Temporal networks.Physics Reports519, 97–125 (2012). URLhttps://www. sciencedirect.com/science/article/pii/S0370157312000841. Temporal Networks

    2012

  31. [31]

    Kivel ¨a, M.et al.Multilayer networks.Journal of complex networks2, 203–271 (2014)

    2014

  32. [32]

    M., Ali, O

    Moutuou, E. M., Ali, O. B. K. & Benali, H. Topology and spectral interconnectivities of higher-order multilayer networks.Frontiers in Complex Systems1(2023). URLhttps://www.frontiersin.org/articles/ 10.3389/fcpxs.2023.1281714

    work page doi:10.3389/fcpxs.2023.1281714 2023

  33. [33]

    H.Logical Depth and Physical Complexity, 227–258 (Oxford University PressOxford)

    Bennett, C. H.Logical Depth and Physical Complexity, 227–258 (Oxford University PressOxford)

  34. [34]

    Measures of complexity: a nonexhaustive list.IEEE Control Systems Magazine21, 7–8 (2001)

    Lloyd, S. Measures of complexity: a nonexhaustive list.IEEE Control Systems Magazine21, 7–8 (2001)

    2001

  35. [35]

    & Vinodchandran, N

    Antunes, L., Fortnow, L., van Melkebeek, D. & Vinodchandran, N. Computational depth: Concept and applica- tions.Theoretical Computer Science354, 391–404

  36. [36]

    B.Thermodynamics and an introduction to thermostatistics(Wiley, New York, NY , 1985)

    Callen, H. B.Thermodynamics and an introduction to thermostatistics(Wiley, New York, NY , 1985). URL https://cds.cern.ch/record/450289

    1985

  37. [37]

    A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines

    Hutchinson, M. A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines. Communications in Statistics - Simulation and Computation19, 433–450

  38. [38]

    & Toledo, S

    Avron, H. & Toledo, S. Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix.J. ACM58, 8:1–8:34

  39. [39]

    Bouchaud, J. P. Weak ergodicity breaking and aging in disordered systems.Journal de Physique I2, 1705–1713

  40. [40]

    & Bouchaud, J.-P

    Monthus, C. & Bouchaud, J.-P. Models of traps and glass phenomenology.Journal of Physics A: Mathematical and General29, 3847–3869

  41. [41]

    Bochner’s method for cell complexes and combinatorial ricci curvature.Discrete and Computational Geometry29, 323–374 (2003)

    Forman, R. Bochner’s method for cell complexes and combinatorial ricci curvature.Discrete and Computational Geometry29, 323–374 (2003)

    2003

  42. [42]

    & Jost, J

    Weber, M., Saucan, E. & Jost, J. Characterizing complex networks with forman-ricci curvature and associated geometric flows.Journal of Complex Networks5, 527–550

  43. [43]

    P., Mohanraj, K., Jost, J., Saucan, E

    Sreejith, R. P., Mohanraj, K., Jost, J., Saucan, E. & Samal, A. Forman curvature for complex networks.Journal of Statistical Mechanics: Theory and Experiment2016, 063206

  44. [44]

    & Weber, M

    Saucan, E., Samal, A. & Weber, M. Discrete curvatures and network analysis.MATCH Communications in Mathematical and in Computer Chemistry80(2018)

    2018

  45. [45]

    M., Scotognella, F

    Montepietra, D., Bellingeri, M., Ross, A. M., Scotognella, F. & Cassi, D. Modelling photosystem i as a complex interacting network.Journal of The Royal Society Interface17, 20200813

  46. [46]

    Milo, R.et al.Network motifs: Simple building blocks of complex networks.Science298, 824–827

  47. [47]

    Rossi, R. A. & Ahmed, N. K. The network data repository with interactive graph analytics and visualization. In AAAI(2015). URLhttps://networkrepository.com

    2015

  48. [48]

    KONECT – The Koblenz Network Collection

    Kunegis, J. KONECT – The Koblenz Network Collection. InProc. Int. Conf. on World Wide Web Companion, 1343–1350 (2013). URLhttp://dl.acm.org/citation.cfm?id=2488173

    2013

  49. [49]

    Peixoto, T. P. The netzschleuder network catalogue and repository

  50. [50]

    Thompson, R. M. & Townsend, C. R. Impacts on stream food webs of native and exotic forest: 29 An intercontinental comparison.Ecology84, 145–161 (2003). URLhttps://esajournals. onlinelibrary.wiley.com/doi/abs/10.1890/0012-9658%282003%29084%5B0145% 3AIOSFWO%5D2.0.CO%3B2. https://esajournals.onlinelibrary.wiley.com/doi/pdf/10.1890/0012- 9658%282003%29084%5B0...

    work page doi:10.1890/0012-9658 2003

  51. [51]

    H., Zhang, S., Clauset, A

    Wapman, K. H., Zhang, S., Clauset, A. & Larremore, D. B. Quantifying hierarchy and dynamics in us faculty hiring and retention.Nature610, 120–127

  52. [52]

    Adamic, L. A. & Glance, N. The political blogosphere and the 2004 u.s. election: divided they blog. In Proceedings of the 3rd international workshop on Link discovery, KDD05, 36–43 (ACM)

    2004

  53. [53]

    & Krevl, A

    Leskovec, J. & Krevl, A. SNAP Datasets: Stanford large network dataset collection.http://snap. stanford.edu/data(2014)

    2014

  54. [54]

    Ruiz-Garc ´ıa, M.et al.Triadic influence as a proxy for compatibility in social relationships.Proceedings of the National Academy of Sciences120

  55. [55]

    & Panzarasa, P

    Opsahl, T. & Panzarasa, P. Clustering in weighted networks.Social Networks31, 155–163

  56. [56]

    C., Webster, C

    Freeman, L. C., Webster, C. M. & Kirke, D. M. Exploring social structure using dynamic three-dimensional color images.Social Networks20, 109–118

  57. [57]

    directed highway

    Kumar, S., Spezzano, F., Subrahmanian, V . & Faloutsos, C. Edge weight prediction in weighted signed networks. InData Mining (ICDM), 2016 IEEE 16th International Conference on, 221–230 (IEEE, 2016). 30 Supplementary Information S1.Minimal example: memory compression in a single-node network c v Single-node network withkself-loops 101 103 105 k 0.0 0.5 1.0...

    2016