TL;DR
It is proved that ( 𝑑 + 1 ) -dimensional random dot product graphs generalize 𝑑 -dimensional random ball graphs up to constant factors and vice versa.
This work compares ball-and weighted graphs and gives a complete comparison in the one-dimensional case, and shows that the minimal dimension for ball graphs can at most be one larger than the minimal dimension for weighted graphs (weighted dimension), but also that the weighted dimension can exceed the ball dimension...
The findings illustrate that geometric insights grounded in hyperbolic geometry can offer powerful tools for understanding, embedding, and visualizing complex graph structures.
This approach demonstrates that models trained on small-scale random graphs learn to extract universal distance-preserving features, achieving robust generalization to large-scale, real-world networks that match or exceed the fidelity of classical, exact landmark-based embeddings.
A continuum of degree-normalized spectral embeddings that includes these commonly used choices as special cases is studied, and a row-wise central limit theorem is established under a random dot product graph model for this family of embeddings.
The novel Radial Graphlet Distribution Distance is effective, and comparable in performance to state-of-the-art methods, and the easy-to-compute Joint Degree Distance is a viable alternative to graphlet-based distances, especially for measuring randomness in sets of very large networks.
It is shown that there exist graphons that interpolate between Riemannian manifolds and weighted geometric graphs, and a monotonicity inequality is established which reveals an implicit relationship between numerous combinatorial parameters and geometric quantities on graphons.
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