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How to map coordinates in causal and branchial separation?

Posted 2 months ago
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In the two paper on the relativistic and quantum properties of the Wolfram models it's not specifically explained how the branching section of the multiway causal graph should be mapped, but only its consequences once a specific metric (from a Minkowskian norm) has been added. The huge problem with this explanation is that there is no single, uniquely acceptable coordinate system for mapping, so changing the coordinates allows you to get whatever you want.

In particular, this is not evident in the paper due to the fact that the examples used have a spatial graph composed of a straight line of points for which a trivial preferential ordering is available.

Is it possible to have an example of how the mapping of coordinates should be done on a causal or branchial graph? In particular, if I have two points of a causal graph, how can I calculate their Minkowskian distance if I have to arbitrarily choose the coordinates of the two points?

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Is this thread a duplicate of https://community.wolfram.com/groups/-/m/t/1950038 ?

Anyway, as I explained in your other thread regarding this topic, layered digraph embedding for acyclic graphs is a well-defined optimization problem, and hence the embedding of both causal graphs and multiway evolution graphs into integer lattices is well-defined (so the resultant norm is, as well). Please do let me know if you think I missed something important here, but I believe this is explained in my two papers :)

Sorry for the necroposting, but I spent this month trying to reread everything possible to understand what I was missing, and unfortunately the situation has improved very little. I realized that actually the missing part in my mind is contained in the formalism of the optimization problem concerning layered digraph embedding for acyclic graphs.

But this was not enough to satisfy my doubts, in particular:

  1. There is hardly any material on the mathematical details of the optimization problem mentioned, and that very little that is found in the bibliography of the relativity paper is studied above all for digraphs and not for hypergraphs (if I understood correctly while reading).

  2. Minkowsky's distance is defined before the curvature work mentioned in https://www.wolframphysics.org/questions/spacetime-relativity/why-do-you-get-a-euclideanriemannian-metric-as-opposed-to-a-taxicab-metric-induced-on-your-hypergraphs/ and this, in particular, leads me to consider that without a rigorous demonstration that there is only one well-defined embedding (and for this, I need the rigorous explanation of the optimization problem) we go back to the problem that I had previously that this Minkowskian distance cited on page 13 of the paper on relativity suffers from strong arbitrariness. The only other possible explanation that came to me is that that "distance" is NOT defined on the Z ^ n grid on which the hypergraph was embedding, and also, in this case, the thing is not at all obvious and explicit.

In summary, my problems would be almost certainly solved once there is a thorough and formal explanation of the optimization problem mentioned here (obviously for hypergraphs and not for digraphs).

Certainly this month has solved many problems for me, especially your answers scattered here and there on the forum, but still, without a formal and explicit explanation of the optimization problem, it is difficult for those who do not have a degree in computer science or at least in sciences dealing exhaustively structures such as hypergraphs, to fully understand the mathematical formulation of the theory.

Actually, I would settle for a link to a book, I tried to read the pages

[21] G. di Battista, P. Eades, R. Tamassia, I. G. Tollis (1998), "Flow and Upward Planarity", Graph Drawing: Algorithms for the Visualization of Graphs: 171-213. Prentice Hall. ISBN 978-0-13-301615-4.53

[22] G. di Battista, F. Frati (2012), "Drawing Trees, Outerplanar Graphs, Series-Parallel Graphs, and PlanarGraphs in Small Area", Thirty Essays on Geometric Graph Theory, Algorithms and Combinatorics, Springer, 29: 121 -165.

but unfortunately, apart from some examples concerning digraphs and a lot of useful graph notions, they have not satisfied my doubts.

I also apologize in advance, because it is not unlikely that it is something simpler than it appears and I simply cannot understand it.

Thanks a lot for the enormous patience and I apologize again for the necroposting!

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