  Finding suitable embeddings for connectomes (spatially embedded complex
networks that map neural connections in the brain) is crucial for analyzing and
understanding cognitive processes. Recent studies have found two-dimensional
hyperbolic embeddings superior to Euclidean embeddings in modeling connectomes
across species, especially human connectomes. However, those studies had
limitations: geometries other than Euclidean, hyperbolic, or spherical were not
considered. Following William Thurston's suggestion that the networks of
neurons in the brain could be successfully represented in Solv geometry, we
study the goodness-of-fit of the embeddings for 21 connectome networks (8
species). To this end, we suggest an embedding algorithm based on Simulating
Annealing that allows us to embed connectomes to Euclidean, Spherical,
Hyperbolic, Solv, Nil, and product geometries. Our algorithm tends to find
better embeddings than the state-of-the-art, even in the hyperbolic case. Our
findings suggest that while three-dimensional hyperbolic embeddings yield the
best results in many cases, Solv embeddings perform reasonably well.
