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Graphon as a Bridge between Graphs and Manifolds

  • Dong Zhang
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Abstract

We show that there exist graphons that interpolate between Riemannian manifolds and weighted geometric graphs. Specifically, the graph-to-manifold approximation used in manifold learning can be regarded as the composition of a graph-to-graphon convergence and a graphon-to-manifold convergence in a certain sense. Furthermore, we establish a monotonicity inequality which reveals an implicit relationship between numerous combinatorial parameters and geometric quantities on graphons. Using this inequality, we find relations among conductance, maxcut problem, capacity, and packing radius, as well as their limiting behaviors under graph-to-graphon and graphon-to-manifold convergences; some of these relations are novel even for simple graphs and closed manifolds. We also find that in the graphon-to-manifold limiting process, the purely combinatorial parameters diverge, leading to a ``blowup'' phenomenon; and continuous versions of some graph parameters cannot be realized via Riemannian manifolds, but can be realized via graphons.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026graphon,
  title = {{Graphon as a Bridge between Graphs and Manifolds}},
  author = {Dong Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.20213},
  eprint = {2607.20213},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.20213},
}
RIS
TY  - UNPB
TI  - Graphon as a Bridge between Graphs and Manifolds
AU  - Dong Zhang
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2607.20213
UR  - https://arxiv.org/abs/2607.20213
AB  - We show that there exist graphons that interpolate between Riemannian manifolds and weighted geometric graphs. Specifically, the graph-to-manifold approximation used in manifold learning can be regarded as the composition of a graph-to-graphon convergence and a graphon-to-manifold convergence in a certain sense. Furthermore, we establish a monotonicity inequality which reveals an implicit relationship between numerous combinatorial parameters and geometric quantities on graphons. Using this inequality, we find relations among conductance, maxcut problem, capacity, and packing radius, as well as their limiting behaviors under graph-to-graphon and graphon-to-manifold convergences; some of these relations are novel even for simple graphs and closed manifolds. We also find that in the graphon-to-manifold limiting process, the purely combinatorial parameters diverge, leading to a ``blowup'' phenomenon; and continuous versions of some graph parameters cannot be realized via Riemannian manifolds, but can be realized via graphons.
ER  -