Is it possible to calculate the number of dimensions in some discrete space if we have only a complete scheme of all its points and possible transitions between them? There are no regularities, fractals and the like in its organization. We have access only to an array of points and transitions between them. Such computations can be resource-intensive, so I'm especially looking for algorithms that can quickly estimate the dimensionality of the space based on the available data about the points in the space and their adjacencies. I have looked through the material on discreteness of space (mostly discussions), but have not encountered approaches to dimensioning space where linearly independent vectors are not possible.
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What is the relationship of this to physics? What are “transitions” between points? – Ghoster Oct 19 '22 at 23:43
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@Ghoster, Thank you for your reply! On the first question, I believe that discrete space is related to mathematics and physics, and there are interesting hypotheses and studies related to discrete space specifically in quantum physics (e.g., "Discrete spacetime, quantum walks and relativistic wave equations" Leonard Mlodinow and Todd A. Brun, https://arxiv.org/abs/1802.03910) – Oct 21 '22 at 10:22
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@Ghoster, on the question 2, discrete space is a space in which points form discontinuous sequences, that is, some points are isolated and some are not (so motion between them is possible). In the aforementioned work, this difference in possible transitions was described with "quantum walks," which "provide a discrete model of particle dynamics. In such a walk, a particle may be located at any of the vertices of a graph or lattice, and its state evolves in a sequence of discrete time steps. In each step, the particle may move to one of the neighboring vertices (connected by an edge)." – Oct 21 '22 at 10:32