2009· International Journal of Computational Mathematical Ideas· 0 citations
TL;DR
Graph machine learning tailored to planar graphs is developed, with an emphasis on the mathematical intuition that connects the topology of a plane embedding to the spectral and combinatorial structure ex- ploited by learning algorithms.
Abstract
Planar graphs form a structurally rich yet computationally tractable class of graphs that arise
naturally in image analysis, geographic information systems, circuit layout, and molecular chemistry. This
paper develops graph machine learning tailored to planar graphs, with an emphasis on the mathematical
intuition that connects the topology of a plane embedding to the spectral and combinatorial structure ex-
ploited by learning algorithms. We first recall that planarity forces sparsity through Euler’s formula and
small vertex separators through the Lipton and Tarjan theorem, and we explain why these two facts to-
gether make planar learning problems well conditioned. We then treat three learning primitives in a unified
way: spectral partitioning through the Fiedler vector of the graph Laplacian, semi-supervised classification
through harmonic extension of labels, and similarity through diffusion kernels. Throughout we develop
the electrical network interpretation, in which the harmonic solution is a potential and effective resistance
is a learned distance, because this picture is especially transparent on planar graphs. Four algorithms are
presented with complexity analysis, and their behaviour is illustrated on plane-embedded meshes and grids.
The paper is intended as a mathematically motivated entry point for researchers who wish to learn on data
whose relational structure can be drawn in the plane without crossings.
A coloured graph carries a discrete attribute on each vertex, and when the underlying graph is
planar the interaction between that attribute and the topology of the plane embedding becomes especially
rich. This paper, a companion to the author’s study of learning on planar graphs, proposes four novel
algorithms that make colour a first class citizen of graph machine learning on planar structures. We begin
from two classical facts, that colour refinement in the sense of Weisfeiler and Leman produces the coarsest
equitable partition of a graph, and that every planar graph is four colourable, and we show that each of
these facts yields a concrete learning primitive on planar inputs. The first algorithm turns colour refinement
into a permutation invariant feature map whose stable colouring is computed in near linear time on planar
graphs. The second, chromatic block propagation, uses a proper four colouring to schedule harmonic label
propagation as at most four fully parallel sub sweeps, because each colour class is an independent set with
no internal coupling. The third builds a chromatic diffusion kernel that fuses colour refinement histograms
with heat flow on the graph. The fourth uses colour classes as an independent set backbone for multilevel
coarsening of the Laplacian. Each algorithm is analysed for correctness and cost, the intuition connecting
equitable partitions to the Laplacian spectrum is made explicit, and behaviour is illustrated on coloured
plane graphs and grids. The unifying message is that on planar coloured graphs the chromatic structure is
not a nuisance to be hashed away but a schedule, a feature, and a coarsening all at once.
Satyanarayana Sanakkayala· International Journal of Com...· 0 citations
Finite groups are rigid algebraic objects, whose Cayley graphs expose a rich network geometry through which group-theoretic structure can be measured, compared, and learned. In this paper, a dataset of $131{,}406$ Cayley graphs is constructed, covering all groups of order at most $767$ except order $512$, recording exact algebraic labels for group properties together with a broad collection of graph, cycle, distance, and spectral statistics. This census aims to provide novel benchmarks for studying how finite-group properties are reflected in Cayley graph observables. It also yields new enumerative contributions: alongside recovering known OEIS sequences for standard group classes, new sequences for monolithic groups and for groups generated by at most three, four, and five elements are contributed to the OEIS. The accompanying network analysis identifies several empirical regularities and formulates testable conjectures, including relationships involving square clustering, Cayley graph diameter, average graph disorder, and spectral eigengaps of nilpotent groups. Finally, a comparison between classical models, an MLP, and graph neural network architectures is performed for predicting algebraic group properties directly from Cayley graph data. The results show that engineered graph statistics are highly informative, while GNNs, especially GIN and in some fixed-order settings GCN, can recover substantial structural signal directly from the graph. Such that graph-aware architectures show phases of optimality on these group-theoretic graph representations.
Rashid Barket, Enrico Grimaldi, Yacoub Hendi et al.· 0 citations
The findings illustrate that geometric insights grounded in hyperbolic geometry can offer powerful tools for understanding, embedding, and visualizing complex graph structures.
†. SalouaNaama, Kave Salamatian, M. Crovella· 0 citations
Planar graphs are central to graph drawing, with extensive results on planar layouts and related structures. Every planar graph admits a planar straight-line drawing, and algorithms can guarantee additional geometric or combinatorial properties. However, it is unclear which algorithms work best in practice. Even for small graphs with near-perfect manual drawings, standard algorithms might produce poor spacing, distorted faces, or small angles. We present an experimental evaluation of planar graph drawing algorithms on a large benchmark collection of small and medium-sized planar graphs (\(10\)--\(400\) vertices). The study compares established algorithms from the graph drawing literature, practical force-directed and pressure-based heuristics, and new optimization-based methods that directly improve visual properties such as edge-length uniformity, face-area balance, and angular resolution. The results show that no evaluated algorithm is best across all aesthetic criteria, and optimizing one visual property often worsens another. Directly optimizing visual criteria improves targeted scores, and score-guided combination of several methods gives the best aggregate results, but no simple algorithm emerges as a clear universal default. Designing a simple, robust algorithm that performs well across graph families and aesthetic criteria therefore remains an open practical problem.
Understanding the geometry of complex networks is critical for effective modeling and analysis across domains. While discrete notions of Ricci curvature have emerged as powerful tools for characterizing both local and global network structure, existing formulations are largely confined to undirected networks with real-valued weights. This limits the use of curvature-based analysis of directional and complex-weighted relations that arise naturally in many applications, from social and biological systems to quantum and signal-processing networks. In this work, we introduce a principled extension of Ollivier's Ricci curvature to complex-weighted graphs, which encompasses directed graphs as a special case. We establish fundamental theoretical properties of this new notion, including relations to the magnetic Laplacian and combinatorial upper and lower bounds that relate curvature to cycle structure in local neighborhoods. We further develop computational methods for curvature estimation and demonstrate their utility in community detection on directed networks.
Yu Tian, Eleanor P. Wiesler, Melanie Weber· 0 citations
Structured Connection Graph Learning (SCGL), a block-coordinate algorithm that combines closed-form updates, manifold projections, and spectral constraints, and converges to stationary points of the resulting nonconvex problem, is developed.
Leonardo Di Nino, Gabriele D’Acunto, Sergio Barbarossa et al.· 0 citations