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arxiv: 2601.10333 · v2 · submitted 2026-01-15 · ❄️ cond-mat.dis-nn

Computer Generation of Disordered Networks with Targeted Structural Properties

Florin Hemmann , Vincent Glauser , Ullrich Steiner , Matthias Saba This is my paper

Pith reviewed 2026-05-16 14:27 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn
keywords disordered networksbond switchingcoordination numberstructural colororder metricsneural network predictionbiophotonic structuresnetwork generation
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The pith

A modified Wooten-Weaire-Winer algorithm uses maximum bond repulsion to generate disordered networks of arbitrary coordination number and targets specific order metrics via a feedforward neural network.

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

The paper extends the established bond-switch method for creating disordered networks by adding a maximum bond repulsion term in the strain energy. This change removes the previous limit to coordination numbers of four or less and allows networks with any average number of connections per node. Disorder level and character are tuned by changing the bond-bending stiffness and the temperature schedule during the switching process. A set of order parameters measured in both real and reciprocal space quantifies the resulting structures. A simple neural network is trained to map the input parameters directly to these metrics, so that networks with chosen properties can be produced without repeated trial runs. The method is shown to reproduce the statistical features of four real biophotonic networks known for structural color.

Core claim

We introduce a maximum bond repulsion term into the strain energy of the Wooten-Weaire-Winer algorithm, enabling generation of spatial networks with arbitrary coordination number. The degree and type of disorder are controlled by the bond-bending force constant and the temperature profile during bond-switch moves. Structural characteristics are quantified by a collection of order metrics in direct and reciprocal space, and a feedforward neural network is trained to predict these metrics from the algorithm inputs, permitting efficient targeted network generation. As a demonstration, four disordered biophotonic networks that exhibit structural color are statistically reproduced.

What carries the argument

Maximum bond repulsion term added to the strain energy, combined with a feedforward neural network that maps bond-bending constant and temperature profile to a vector of real-space and reciprocal-space order metrics.

If this is right

  • Networks with coordination numbers higher than four become accessible for scattering and localization studies.
  • Targeted structural color can be obtained by feeding desired order-metric values into the trained neural network and running the algorithm once.
  • Systematic variation of the bending stiffness and temperature schedule produces families of networks whose disorder can be classified along a single continuous parameter axis.
  • The same workflow can be applied to other wave-transport problems in which both real-space connectivity and reciprocal-space correlations control the physics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The neural-network surrogate could be inverted to perform direct design of network topology for prescribed optical response without iterative simulation.
  • The method supplies a controllable test-bed for studying how specific real-space and reciprocal-space correlations separately affect Anderson localization or photonic band gaps.
  • Extension to time-dependent or active networks is possible by replacing the static strain energy with a time-varying repulsion term.

Load-bearing premise

The chosen order metrics in direct and reciprocal space are sufficient to capture all structural features that matter for the wave-scattering behavior the networks are intended to model.

What would settle it

Generate networks using the reported input parameters and compare their pair-correlation functions, structure factors, or higher-order correlation measures against the original biophotonic samples using metrics not included in the training set; systematic mismatch would falsify the claim that the method reproduces the targeted structures.

Figures

Figures reproduced from arXiv: 2601.10333 by Florin Hemmann, Matthias Saba, Ullrich Steiner, Vincent Glauser.

Figure 1
Figure 1. Figure 1: A Monte Carlo move is shown for a 2D, three-valent network. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: In our extended WWW algorithm, we introduce disorder to networks by successively [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: a A section of the skeletonized, disordered photonic network of the Pachyrhynchus congestus mirabilis weevil gives rise to blue structural color (PCM blue). b The periodic ctn network has similar coordination number statistics as PCM blue, as shown in c. d and e show sections of the StV green and StV blue biological networks, respectively. f The periodic bcumod network has coordination number statistics si… view at source ↗
Figure 4
Figure 4. Figure 4: We applied the extended WWW algorithm to the periodic diamond [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: a The number of accepted Monte Carlo moves in the ctn network increases with increasing maximal heating temperature Tmax and with decreasing heating gradient ∆T. The black markers correspond to networks with ten or fewer accepted moves and show that the melting transition begins at Tmax ≳ Tmelt. All values of β ∈ [0, 10] are considered. b The isotropy metric hb effectively measures the melting transition o… view at source ↗
Figure 6
Figure 6. Figure 6: There is an inverse correlation between the bond bending force constant [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Disorder in the primitives of networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The homogeneity metrics introduced in Section [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The isotropy and topology metrics of Sections [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Example of a three-layer (Nlayers = 3) feedforward neural network. The input layer is counted as 0. The black circles represent neurons and the gray lines depict the data flowing to the right. Our neural network has Nin = 3 inputs: β, Tmax, and ∆T. The number of outputs Nout equals the number of order metrics or PCs that the network is trained on. The depicted network has Nout = 2 outputs. Between input a… view at source ↗
Figure 11
Figure 11. Figure 11: Training the neural network using ten PCs representing 42 order metrics yields [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Neural network predictions of the order metrics describing primitives for networks [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Neural network predictions of the order metrics describing homogeneity for networks [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Neural network predictions of the order metrics describing isotropy and topology for [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The coefficient of determination R2 measures the accuracy of a neural network’s predictions. It indirectly quantifies the variance in the order metrics of the generated networks when using the same generation algorithm inputs. between two networks with the same functional form to quantify their structural similarity, dorder = " N Xmetric i=1 wi [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: a The network with the shortest order metric distance to the PCM blue biophotonic network (Figure 3a) was generated from an initial ctn network. b The order metric values of the network generated in a (small light blue markers) closely match those of the PCM blue biophotonic network (large dark blue markers). The black bars represent the order metrics ranges achieved by modifying the initial ctn network w… view at source ↗
Figure 17
Figure 17. Figure 17: a The generated network bcuStV green has the smallest order metric distance to the StV green biological network (Figure 3l). It was generated from an initial bcumod network. b The generated network bcuStV blue is the most similar to StV green (Figure 3e). c The order metric values of networks bcuStV green (a, small olive green markers) and bcuStV blue (b, small light blue markers) closely match the biopho… view at source ↗
Figure 18
Figure 18. Figure 18: a The network with the smallest order metric distance to the StA orange biological network (Figure 3e) was generated from an initial pcumod network. b The order metric values of the network in a (small red markers) closely match those of the StA orange biophotonic network (large orange markers). The black bars represent the range of order metrics achieved by modifying the initial pcumod network with at le… view at source ↗
Figure 19
Figure 19. Figure 19: Illustration of the melting transition for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p035_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The number of accepted Monte Carlo moves for the initial [PITH_FULL_IMAGE:figures/full_fig_p035_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: The network primitive metrics for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p036_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: The homogeneity metrics for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p036_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: The isotropy and topology metrics for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p037_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Illustration of the melting transition for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p038_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: The number of accepted Monte Carlo moves for the initial [PITH_FULL_IMAGE:figures/full_fig_p038_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: The network primitive metrics for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p039_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: The homogeneity metrics for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p039_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: The isotropy and topology metrics for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p040_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Illustration of the melting transition for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p041_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: The number of accepted Monte Carlo moves for the initial [PITH_FULL_IMAGE:figures/full_fig_p041_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: The network primitive metrics for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p042_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: The homogeneity metrics for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p042_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: The isotropy and topology metrics for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p043_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: Illustration of the melting transition for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p044_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: The number of accepted Monte Carlo moves for the initial [PITH_FULL_IMAGE:figures/full_fig_p044_35.png] view at source ↗
Figure 36
Figure 36. Figure 36: The network primitive metrics for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p045_36.png] view at source ↗
Figure 37
Figure 37. Figure 37: The homogeneity metrics for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p045_37.png] view at source ↗
Figure 38
Figure 38. Figure 38: The isotropy and topology metrics for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p046_38.png] view at source ↗
Figure 39
Figure 39. Figure 39: Illustration of the melting transition for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p047_39.png] view at source ↗
Figure 40
Figure 40. Figure 40: The number of accepted Monte Carlo moves for the initial [PITH_FULL_IMAGE:figures/full_fig_p047_40.png] view at source ↗
Figure 41
Figure 41. Figure 41: The network primitive metrics for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p048_41.png] view at source ↗
Figure 42
Figure 42. Figure 42: The homogeneity metrics for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p048_42.png] view at source ↗
Figure 43
Figure 43. Figure 43: The isotropy and topology metrics for networks generated from the initial [PITH_FULL_IMAGE:figures/full_fig_p049_43.png] view at source ↗
Figure 44
Figure 44. Figure 44: Absolute values of the PCA loadings with the mean Steinhardt local bond order [PITH_FULL_IMAGE:figures/full_fig_p050_44.png] view at source ↗
read the original abstract

Disordered spatial networks describe structures and interactions across multiple length scales. The scattering and interference of waves within these networks result in structural phase transitions, localization, diffusion, and band gaps. Studying these phenomena requires efficient numerical methods for generating disordered networks with specific structural properties. The Wooten-Weaire-Winer algorithm is an established method that introduces disorder into an initial network through a series of bond switch moves. However, the strain energies that govern this evolution are conventionally limited to three-dimensional networks with coordination numbers of no more than four. We here introduce a maximum bond repulsion to produce networks with an arbitrary coordination number. We control the degree and type of disorder by adjusting the bond-bending force constant in the strain energy and the temperature profile. The effects of these variables are quantified through a list of order metrics that capture both direct and reciprocal space. A feedforward neural network predicts the structural characteristics from the algorithm inputs, enabling efficient targeted network generation. As a case study, we statistically reproduce four disordered biophotonic networks that exhibit structural color. This work presents a versatile method for generating disordered networks with tailored structural properties. It will provide new insights into structure-property relations.

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

1 major / 2 minor

Summary. The paper extends the Wooten-Weaire-Winer (WWW) algorithm by introducing a maximum bond repulsion term to generate disordered networks with arbitrary coordination numbers. Disorder type and degree are controlled by tuning the bond-bending force constant in the strain energy and the temperature profile during bond-switch moves. A set of direct- and reciprocal-space order metrics quantifies the resulting structures, and a feedforward neural network is trained to predict these metrics from the algorithm inputs, enabling efficient targeted generation. The method is demonstrated in a case study by statistically reproducing four real disordered biophotonic networks known to exhibit structural color.

Significance. If the chosen order metrics prove sufficient proxies for the structural features governing wave interference, the approach supplies a practical, tunable generator for disordered networks with prescribed coordination and disorder characteristics. The neural-network acceleration step is a clear efficiency gain for iterative design workflows in biophotonics and related fields where structure-property mapping is central.

major comments (1)
  1. [Case study] Case study section: the claim that the generated networks statistically reproduce the four biophotonic networks (and thereby their structural color) rests solely on agreement of the selected order metrics; no direct optical observable such as reflectance spectrum, photonic density of states, or localization length is compared between real and synthetic networks. This gap is load-bearing because the central utility asserted is reproduction of wave-scattering behavior.
minor comments (2)
  1. [Abstract] Abstract and methods: quantitative error bars, cross-validation statistics, and explicit comparison against alternative network-generation algorithms are not reported, making it difficult to assess the neural-network prediction accuracy and the incremental benefit of the new repulsion term.
  2. [Methods] The manuscript does not specify the neural-network architecture details (layer sizes, activation functions, training-set size, or regularization) or the precise definition of the order metrics, both of which are needed for reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment below and have revised the manuscript to strengthen the presentation of the case study.

read point-by-point responses

  1. Referee: [Case study] Case study section: the claim that the generated networks statistically reproduce the four biophotonic networks (and thereby their structural color) rests solely on agreement of the selected order metrics; no direct optical observable such as reflectance spectrum, photonic density of states, or localization length is compared between real and synthetic networks. This gap is load-bearing because the central utility asserted is reproduction of wave-scattering behavior.

    Authors: We agree that direct comparison of optical observables would provide stronger validation of the wave-scattering reproduction. The order metrics were selected because prior literature has established their correlation with the structural features that govern interference and color in these specific biophotonic networks. To address the referee's concern, the revised manuscript now includes additional calculations of reflectance spectra and photonic density of states for both the real and generated networks, demonstrating quantitative agreement. We have also added a brief discussion clarifying the link between the chosen metrics and optical behavior. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic extension plus NN surrogate on generated data remains self-contained

full rationale

The paper extends the established WWW bond-switch algorithm by adding an explicit maximum bond repulsion term to allow arbitrary coordination numbers, then tunes disorder via the bond-bending force constant and temperature schedule. Structural features are measured with a fixed list of direct- and reciprocal-space order metrics that are computed independently from the generated networks. A feedforward neural network is trained in the standard supervised manner on input-parameter to metric pairs produced by the algorithm itself, then used only as a fast surrogate for targeting. No equation reduces a reported metric or prediction to a quantity defined by the same fitted parameters, no uniqueness theorem is imported from self-citation, and the central claim (statistical reproduction of four external biophotonic networks via the chosen metrics) rests on direct numerical comparison rather than definitional closure. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The approach rests on the domain assumption that a modified strain-energy function with bond repulsion produces physically plausible networks and that the selected order metrics are sufficient proxies for the structural properties that govern wave phenomena.

free parameters (2)
  • bond-bending force constant
    Tuned to set the degree of disorder; value chosen per run
  • temperature profile
    Controls the type and extent of disorder during bond switches
axioms (1)
  • domain assumption Network evolution under the strain energy with added bond repulsion yields topologically valid disordered networks
    Invoked to justify the core generation step

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