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Recent work has shown that sparse graphs containing many triangles cannot bereproduced using a finite-dimensional representation of the nodes, in whichlink probabilities are inner products. Here, we show that such graphs can bereproduced using an infinite-dimensional inner product model, where the noderepresentations lie on a low-dimensional manifold. Recovering a globalrepresentation of the manifold is impossible in a sparse regime. However, wecan zoom in on local neighbourhoods, where a lower-dimensional representationis possible. As our constructions allow the points to be uniformly distributedon the manifold, we find evidence against the common perception that trianglesimply community structure.

Implications of sparsity and high triangle density for graph representation learning