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PreprintCC BYChemRxiv

Structural Description of Two-Dimensional Nanoscale Networks Using Graph Theory and Graph Growth Algorithms

  • Pavel Khavlyuk1,2,3,4,5,6,7
  • Xiongye Xiao3,4,5,6,7
  • Will Stephenson3,8,2
  • Dickson Owuor3
  • Chung Man Lim3,9,2
  • Volodymyr Shamraienko1,2
  • Wenbing Wu1,2,3
  • Haiping Sun9,2
  • Peiyu Zhang10,11
  • Ronja Piehler12,13
  • Rostyslav Lesyuk12,13
  • Christian Klinke12,13,14,15
5 more authors
  • Alexander Eychmüller4,5,6,7
  • Paul Bogdan10,11
  • Xiaoming Mao8,2
  • Nicholas Kotov1,2,3,8
  • Nicholas Alexander Kotov
15 affiliations
  1. 1Department of Chemical Engineering
  2. 2University of Michigan
  3. 3NSF Center for Complex Particle Systems (COMPASS
  4. 4Department of Physical Chemistry
  5. 5TU Dresden
  6. 6Department of Electrical Engineering and Computer Science
  7. 7U. of Tennessee
  8. 8Department of Physics
  9. 9Department of Materials Science and Engineering
  10. 10Department of Electrical and Computer Engineering
  11. 11U. of Southern California
  12. 12Institute of Physics
  13. 13University of Rostock
  14. 14Department of Chemistry
  15. 15Swansea University
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Abstract

Nanostructured networks stand out among other nanostructures for their unique combinations of properties and easily scalable energy-efficient synthesis. Their structural organization is, however, difficult to describe using traditional approaches based on crystallinity. Although revealing distinct structural patterns, the nanoscale networks also have a substantial degree of randomness. Two-dimensional (2D) nanostructured networks self-assembled from nanoparticles present a unique opportunity to develop the methodology for structural description of such materials. We describe nanostructured networks as sets of nodes and edges with corresponding weights taking advantage of the mathematical apparatus of Graph Theory (GT). Being extracted from electron microscopy images, the GT descriptors readily identify nanoscale networks obtained under different chemical conditions. Also important is that among the prior studies the GT description of properties has been extensively discussed, while better understanding of adequate GT descriptors is lagging. All GT descriptors were found to be scale-dependent, with average nodal degree, centrality metrics, multi-fractal dimension, and Ollivier-Ricci curvature following smooth superlinear or sublinear trajectories over several thousands of nodes. Statistical analysis of topological and metric parameters of nodes and edges enables development of topometric graph growth algorithms (TOGGA) that capture the structural uniqueness of each nanoscale network. Similarly to the infinite tiling of unit cells in crystals, TOGGA models enabled the extrapolation of nanoscale network structure, commensurate with both micro- and macroscale samples over 10 million nodes and 20 million edges. The predictions of electric transport through the networks comparable in size to experimental samples enable identification of knowledge gaps for rapid design of conductive nanoscale networks. TOGGA models can be used to map out experimentally realistic structures for inverse design of nanoscale networks.

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Cite

BibTeXRIS
BibTeX
@misc{khavlyuk2026structural,
  title = {{Structural Description of Two-Dimensional Nanoscale Networks Using Graph Theory and Graph Growth Algorithms}},
  author = {Pavel Khavlyuk and Xiongye Xiao and Will Stephenson and Dickson Owuor and Chung Man Lim and Volodymyr Shamraienko and Wenbing Wu and Haiping Sun and Peiyu Zhang and Ronja Piehler and Rostyslav Lesyuk and Christian Klinke and Alexander Eychmüller and Paul Bogdan and Xiaoming Mao and Nicholas Kotov and Nicholas Alexander Kotov},
  howpublished = {ChemRxiv},
  year = {2026},
  month = oct,
  doi = {10.26434/chemrxiv.15010157/v1},
  url = {https://chemrxiv.org/doi/full/10.26434/chemrxiv.15010157/v1},
  note = {CC BY},
}
RIS
TY  - UNPB
TI  - Structural Description of Two-Dimensional Nanoscale Networks Using Graph Theory and Graph Growth Algorithms
AU  - Pavel Khavlyuk
AU  - Xiongye Xiao
AU  - Will Stephenson
AU  - Dickson Owuor
AU  - Chung Man Lim
AU  - Volodymyr Shamraienko
AU  - Wenbing Wu
AU  - Haiping Sun
AU  - Peiyu Zhang
AU  - Ronja Piehler
AU  - Rostyslav Lesyuk
AU  - Christian Klinke
AU  - Alexander Eychmüller
AU  - Paul Bogdan
AU  - Xiaoming Mao
AU  - Nicholas Kotov
AU  - Nicholas Alexander Kotov
PY  - 2026
DA  - 2026/10/09
PB  - ChemRxiv
DO  - 10.26434/chemrxiv.15010157/v1
UR  - https://chemrxiv.org/doi/full/10.26434/chemrxiv.15010157/v1
AB  - Nanostructured networks stand out among other nanostructures for their unique combinations of properties and easily scalable energy-efficient synthesis. Their structural organization is, however, difficult to describe using traditional approaches based on crystallinity. Although revealing distinct structural patterns, the nanoscale networks also have a substantial degree of randomness. Two-dimensional (2D) nanostructured networks self-assembled from nanoparticles present a unique opportunity to develop the methodology for structural description of such materials. We describe nanostructured networks as sets of nodes and edges with corresponding weights taking advantage of the mathematical apparatus of Graph Theory (GT). Being extracted from electron microscopy images, the GT descriptors readily identify nanoscale networks obtained under different chemical conditions. Also important is that among the prior studies the GT description of properties has been extensively discussed, while better understanding of adequate GT descriptors is lagging. All GT descriptors were found to be scale-dependent, with average nodal degree, centrality metrics, multi-fractal dimension, and Ollivier-Ricci curvature following smooth superlinear or sublinear trajectories over several thousands of nodes. Statistical analysis of topological and metric parameters of nodes and edges enables development of topometric graph growth algorithms (TOGGA) that capture the structural uniqueness of each nanoscale network. Similarly to the infinite tiling of unit cells in crystals, TOGGA models enabled the extrapolation of nanoscale network structure, commensurate with both micro- and macroscale samples over 10 million nodes and 20 million edges. The predictions of electric transport through the networks comparable in size to experimental samples enable identification of knowledge gaps for rapid design of conductive nanoscale networks. TOGGA models can be used to map out experimentally realistic structures for inverse design of nanoscale networks.
ER  -