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Effect of local topological changes on resistance in spatially embedded disordered networks

Chenxi Wang, Charles Emmett Maher, and Katherine A. Newhall*

  • *Contact author: knewhall@unc.edu
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Phys. Rev. E 114, 024305 – Published 10 August, 2026

DOI: https://doi.org/10.1103/kqkn-mymm

Abstract

Disordered materials occur naturally and also provide a broader design space than ordered or crystalline structures. We investigate a two-dimensional disordered network metamaterial constructed from a Delaunay triangulation of an underlying point cloud. Small perturbations in the point cloud induce discrete topological changes. One such change we identify is a Delaunay flip, in which two neighboring Delaunay triangles that form a convex quadrilateral structure with their common edge being one of the two quadrilateral diagonals exchange this diagonal for the other diagonal. These topological changes can cause substantial jumps in the effective resistance measured diagonally across the network, when the change is located near the source or the sink node. The jumps are explained analytically by showing that the change in effective resistance from edge removal or addition depends on the voltage drop across that edge. However, Delaunay flips have less impact on global resistance measurements and in larger networks. These local topological changes are relevant for finite-sized samples and experimentally measurable properties such as electrical transport. Global characterizations of the network disorder or topology lack the location-specificity of our observed effects on network transport, and thus may be inadequate for predicting certain experimentally measurable transport properties in disordered network metamaterials, highlighting the importance of localized regions in material design.

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References (64)

  1. C. P. Broedersz and F. C. MacKintosh, Modeling semiflexible polymer networks, Rev. Mod. Phys. 86, 995 (2014).
  2. P. S. Stewart and S. H. Davis, Dynamics and stability of metallic foams: Network modeling, J. Rheol. 56, 543 (2012).
  3. A. A. Moreira, C. L. N. Oliveira, A. Hansen, N. A. M. Araújo, H. J. Herrmann, and J. S. Andrade, Fracturing highly disordered materials, Phys. Rev. Lett. 109, 255701 (2012).
  4. C. Obrero, M. Tirfe, C. Lee, S. Saptarshi, C. Rock, K. E. Daniels, and K. A. Newhall, Electrical transport in tunably disordered metamaterials, Phys. Rev. E 112, 035505 (2025).
  5. H. Wang, M. A. G. Cunha, J. C. Crocker, and A. J. Liu, Mechanosensitive remodeling sustains rigidity homeostasis and elastic memory in actin cortex models, bioRxiv (2025).
  6. F. Barrows, G. Zhang, S. Anand, Z. Chen, J. Lin, A. Desai, S. Martiniani, and F. Caravelli, A unifying approach to self-organizing systems interacting via conservation laws, Advanced Intelligent Discovery 0, e202500190 (2026).
  7. P. H. Nardelli, N. Rubido, C. Wang, M. S. Baptista, C. Pomalaza-Raez, P. Cardieri, and M. Latva-aho, Models for the modern power grid, Eur. Phys. J.: Spec. Top. 223, 2423 (2014).
  8. M. Barthelemy, Spatial Networks: A Complete Introduction: From Graph Theory and Statistical Physics to Real-world Applications (Springer International, Cham, 2022).
  9. S. Fulco, M. K. Budzik, H. Xiao, D. J. Durian, and K. T. Turner, Disorder enhances the fracture toughness of 2D mechanical metamaterials, PNAS Nexus 4, pgaf023 (2025).
  10. Y. Jiao, Acoustic-edge and thermal scaling in disordered hyperuniform networks: A first principles theory, Phys. Rev. Lett. 136, 137402 (2026).
  11. S. Torquato and D. Chen, Multifunctional hyperuniform cellular networks: Optimality, anisotropy and disorder, Multifunct. Mater. 1, 015001 (2018).
  12. A. Montiel, T. Nguyen, C. L. Rountree, V. Geertsen, P. Guenoun, and D. Bonamy, Effect of architecture disorder on the elastic response of two-dimensional lattice materials, Phys. Rev. E 106, 015004 (2022).
  13. M. Skolnick and S. Torquato, Effective transport and mechanical properties of two-phase materials across the order-disorder spectrum, Acta Mater. 290, 120921 (2025).
  14. A. T. Chieco and D. J. Durian, Quantifying the long-range structure of foams and other cellular patterns with hyperuniformity disorder length spectroscopy, Phys. Rev. E 103, 062609 (2021).
  15. E. Newby, W. Shi, Y. Jiao, S. Torquato, and R. Albert, From point patterns to networks: To what extent does the Delaunay triangulation reproduce key spatial and density information? J. Complex Netw. 13, cnaf040 (2025).
  16. C. E. Maher and K. A. Newhall, Characterizing the hyperuniformity of disordered network metamaterials, Phys. Rev. E 111, 065420 (2025).
  17. L. Siedentop, G. Lui, G. Maret, P. M. Chaikin, P. J. Steinhardt, S. Torquato, P. Keim, and M. Florescu, Stealthy and hyperuniform isotropic photonic band gap structure in 3D, PNAS Nexus 3, pgae383 (2024).
  18. D. R. Reid, N. Pashine, J. M. Wozniak, H. M. Jaeger, A. J. Liu, S. R. Nagel, and J. J. de Pablo, Auxetic metamaterials from disordered networks, Proc. Natl. Acad. Sci. USA 115, E1384 (2018).
  19. M. Shen, M. A. Reyes-Martinez, L. A. Powell, M. A. Iadicola, A. Sharma, F. Byléhn, N. Pashine, E. P. Chan, C. L. Soles, H. M. Jaeger, and J. J. de Pablo, An autonomous design algorithm to experimentally realize three-dimensionally isotropic auxetic network structures without compromising density, npj Comput. Mater. 10, 113 (2024).
  20. K. Moody, M. Li, C. E. Maher, K. Lee, T. Horn, K. A. Newhall, R. Hurley, K. E. Daniels, and C. Rock, A methodology for manufacturing 3D disordered metamaterials using laser powder bed fusion from granular packings and hyperuniform point clouds, engrXiv (2026).
  21. D. Chen, Y. Zheng, L. Liu, G. Zhang, M. Chen, Y. Jiao, and H. Zhuang, Stone–Wales defects preserve hyperuniformity in amorphous two-dimensional networks, Proc. Natl. Acad. Sci. USA 118, e2016862118 (2021).
  22. P. Alonso Ruiz, Power dissipation in fractal Feynman-Sierpinski AC circuits, J. Math. Phys. 58, 073503 (2017).
  23. J. P. Chen, L. G. Rogers, L. Anderson, U. Andrews, A. Brzoska, A. Coffey, H. Davis, L. Fisher, M. Hansalik, S. Loew, and A. Teplyaev, Power dissipation in fractal AC circuits, J. Phys. A: Math. Theor. 50, 325205 (2017).
  24. A. Muranova, Effective impedance over ordered fields, J. Math. Phys. 62, 033502 (2021).
  25. A. Muranova, On the effective impedance of finite and infinite networks, Potent. Anal. 56, 697 (2022).
  26. A. Muranova, On the notion of effective impedance, Operators Matrices 14, 723 (2020).
  27. A. Muranova and R. Schippa, Eigenvalues of the normalized complex Laplacian on finite electrical networks, in Current Research in Mathematical and Computer Sciences V, edited by A. Lecko and J. Rättyä (University of Warmia and Mazury, Olsztyn, 2025), pp. 109–124.
  28. A. Muranova and W. Woess, Networks with complex weights: Green function and power series, Mathematics 10, 820 (2022).
  29. C. R. Paul, Analysis of Multiconductor Transmission Lines, 2nd ed. (Wiley-IEEE Press, New York, 2008).
  30. S. H. Strub and L. Böttcher, Modeling deformed transmission lines for continuous strain sensing applications, Meas. Sci. Technol. 31, 035109 (2020).
  31. L. Böttcher and M. A. Porter, Dynamical processes on metric networks, SIAM J. Appl. Dyn. Syst. 24, 2848 (2025).
  32. M. Bartkowiak and G. D. Mahan, Nonlinear currents in Voronoi networks, Phys. Rev. B 51, 10825 (1995).
  33. R. G. Morris and M. Barthelemy, Transport on coupled spatial networks, Phys. Rev. Lett. 109, 128703 (2012).
  34. B. He, A. Ye, S. Chi, P. Mi, Y. Ran, L. Zhang, X. Zou, B. Pu, Q. Zhao, Z. Zou, D. Wang, W. Zhang, J. Zhao, M. Avdeev, and S. Shi, CAVD, towards better characterization of void space for ionic transport analysis, Sci. Data 7, 153 (2020).
  35. N. Masuda, M. A. Porter, and R. Lambiotte, Random walks and diffusion on networks, Phys. Rep. 716-717, 1 (2017).
  36. P. Doyle and J. Snell, Random Walks and Electric Networks, Carus Mathematical Monographs Vol. 22 (American Mathematical Society, Providence, RI, 1984).
  37. I. García-Redondo, C. Landi, S. Percival, A. Skeja, B. Wang, and L. Zhou, Effective resistance in simplicial complexes as bilinear forms: Generalizations and properties, arXiv:2511.10749.
  38. M. J. Gu, S. Lin, X. F. Xu, C. R. Wang, B. H. Wu, and J. C. Cao, Random-resistor network modeling of resistance hysteresis of vanadium dioxide thin films, J. Appl. Phys. 132, 015301 (2022).
  39. P. H. Winterfeld, L. E. Scriven, and H. T. Davis, Percolation and conductivity of random two-dimensional composites, J. Phys. C 14, 2361 (1981).
  40. G. R. Jerauld, J. C. Hatfield, L. E. Scriven, and H. T. Davis, Percolation and conduction on Voronoi and triangular networks: A case study in topological disorder, J. Phys. C 17, 1519 (1984).
  41. G. R. Jerauld, L. E. Scriven, and H. T. Davis, Percolation and conduction on the 3D Voronoi and regular networks: A second case study in topological disorder, J. Phys. C 17, 3429 (1984).
  42. M. A. Klatt, J. Kim, and S. Torquato, Cloaking the underlying long-range order of randomly perturbed lattices, Phys. Rev. E 101, 032118 (2020).
  43. S. Lloyd, Least squares quantization in PCM, IEEE Trans. Inf. Theory 28, 129 (1982).
  44. L. Papadopoulos, M. A. Porter, K. E. Daniels, and D. S. Bassett, Network analysis of particles and grains, J. Complex Netw. 6, 485 (2018).
  45. L. A. Torres-Sánchez, G. T. Freitas de Abreu, and S. Kettemann, Analysis of the dynamics and topology dependencies of small perturbations in electric transmission grids, Phys. Rev. E 101, 012313 (2020).
  46. S. Gnutzmann, H. Schanz, and U. Smilansky, Topological resonances in scattering on networks (graphs), Phys. Rev. Lett. 110, 094101 (2013).
  47. E. Berthier, M. A. Porter, and K. E. Daniels, Forecasting failure locations in 2-dimensional disordered lattices, Proc. Natl. Acad. Sci. USA 116, 16742 (2019).
  48. A. Sanner, L. Michel, A. Lingua, and D. S. Kammer, Less is more: Removing a single bond increases the toughness of elastic networks, Int. J. Fract. 249, 78 (2025).
  49. E. Young and M. A. Porter, Dynamical importance and network perturbations, Phys. Rev. E 110, 064304 (2024).
  50. V. Tejedor, O. Bénichou, R. Voituriez, and M. Moreau, Response to targeted perturbations for random walks on networks, Phys. Rev. E 82, 056106 (2010).
  51. J. Sherman and W. J. Morrison, Adjustment of an inverse matrix corresponding to a change in one element of a given matrix, Ann. Math. Stat. 21, 124 (1950).
  52. S. Hong, M. A. Klatt, G. Schröder-Turk, N. François, and M. Saadatfar, Dynamical arrest of topological defects in 2D hyperuniform disk packings, EPJ Web Conf. 249, 15002 (2021).
  53. M. A. Klatt, J. Lovrić, D. Chen, S. C. Kapfer, F. M. Schaller, P. W. A. Schönhöfer, B. S. Gardiner, A.-S. Smith, G. E. Schröder-Turk, and S. Torquato, Universal hidden order in amorphous cellular geometries, Nat. Commun. 10, 811 (2019).
  54. A. Ghosh, S. Boyd, and A. Saberi, Minimizing effective resistance of a graph, SIAM Rev. 50, 37 (2008).
  55. M. E. J. Newman and M. Girvan, Finding and evaluating community structure in networks, Phys. Rev. E 69, 026113 (2004).
  56. M. Newman, Networks (Oxford University Press, Oxford, 2018).
  57. D. J. Klein and M. Randić, Resistance distance, J. Math. Chem. 12, 81 (1993).
  58. I. Gutman and B. Mohar, The Quasi-Wiener and the Kirchhoff indices coincide, J. Chem. Inf. Comput. Sci. 36, 982 (1996).
  59. S. Torquato, Physics of sphere packings, Nat. Rev. Phys. 8, 261 (2026).
  60. J. V. Raj, X. Sun, C. E. Maher, K. A. Newhall, and M. A. Porter, Local geometric and transport properties of networks that are generated from hyperuniform point patterns, arXiv:2511.21082.
  61. S. Torquato, Random Heterogeneous Materials, edited by S. S. Antman, L. Sirovich, J. E. Marsden, and S. Wiggins, Interdisciplinary Applied Mathematics Vol. 16 (Springer, New York, 2002).
  62. C. Obrero, https://github.com/DMREF-networks/config_generation.
  63. K. Newhall, github.com/dmref-networks/config_resistance, doi:10.5281/zenodo.18472236.
  64. C. Obrero, M. Tirfe, C. Lee et al., Data from: Electrical transport in tunably-disordered metamaterials [Dataset], Dryad, 2025, https://doi.org/10.5061/dryad.d2547d8bd.

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Phys. Rev. E 114, 024305– Published 10 August, 2026
  • Received 4 February 2026
  • Accepted 16 July 2026
Article part of CHORUS
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