As recently suggested, the brain activity displays multidimensional features (1). Recent work has shown that the appropriate isometric space for embedding complex networks (and in particular the neural multidimensional ones, such as the human connectome) is not the ﬂat Euclidean space, but a negatively curved hyperbolic space (2,3).
Indeed, hyperbolic space has the property that power-law degree distributions, strong clustering and hierarchical community structure emerge naturally when random graphs are embedded in hyperbolic space. It is therefore logical to exploit the structure of the hyperbolic space for useful embeddings of complex networks. It has been demonstrated that, when applied to the task of classifying vertices of complex networks, hyperbolic space embeddings signiﬁcantly outperform embeddings in Euclidean space (4).
Quote as: Tozzi A. 2019. Embeddings of connectome graphs in hyperbolic spaces. (electronic response to: Revealing the Hippocampal Connectome through Super-Resolution 1150-Direction Diffusion MRI. JMaller JJ, Welton T, Middione M, Callaghan FM, Rosenfeld JV, Grieve SM. 2019. Scientific Reports, 9: 2418).
- Tozzi A. 2019. The multidimensional brain. Physics of Life Reviews. doi: https://doi.org/10.1016/j.plrev.2018.12.004. In press.
- Sengupta B, Tozzi A, Coray GK, Douglas PK, Friston KJ. 2016. Towards a Neuronal Gauge Theory. PLOS Biology 14 (3): e1002400. doi:10.1371/journal.pbio.1002400.
- Tozzi A, Peters JF, Jaušovec N. 2018. EEG dynamics on hyperbolic manifolds. Neurosci Lett, 683: 138-143. https://doi.org/10.1016/j.neulet.2018.07.035.
- Chamberlain BP, Clough J, Deisenroth MP. 2017. Neural Embeddings of Graphs in Hyperbolic Space. arXiv:1705.10359