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