How can we calculate the number of dimensions in a discrete space if we only have a complete scheme of all its points and possible transitions between them (or data about the adjacency of points)? Such a scheme can be very confusing and far from the clear two- or three-dimensional space we know. We can observe it, but it is stochastic and there are no regularities, fractals or the like in its organization. We only have access to an array of points and transitions between them.
Such computations can be resource-intensive, so I am especially looking for algorithms that can quickly approximate the dimensionality of the space based on the available data about the points of the space and their adjacencies.
I would be glad if you could help me navigate in dimensions of spaces in my computer model :-)
Applications are made with some set of conditions only, you may design or can think your own different algorithm as per need but a perfect answer or any such applications with random choices do not exist. In short from design point of view question is false or not asked as per needs.
Thank you for your answer! Did I understand correctly that my question is too general and therefore cannot have a universal answer - an algorithm for all possible cases?
Can I then ask another question: do you know any approaches (algorithms) to calculating dimensionality for discrete space, or can we talk about calculating dimensionality only for linear continuous space?
Anil Kumar Jain The description of discrete spaces is found in physical works, e.g. "Discrete spacetime, quantum walks and relativistic wave equations" by Leonard Mlodinow and Todd A. Brun, https://arxiv.org/abs/1802.03910. But I have not seen any attempt to quantify the dimensionality of such spaces. This is exactly what I am looking for.
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