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Dimensionality of Network Topologies

  1. Apr 25, 2018 #1
    A poster (@Mr_Phil_Osophy ) encountered an article on the network topology of the brain:
    Unfortunately, he confused the dimensionality of neuron interconnections with physical dimensionality - and posted it in the Physics->"Beyond the Standard Model thread". Apparently, his misguided speculation was not well received by moderation and the thread was soon deleted.

    However, the dimensionality of network topology is an interesting Computer Science topic where it bears directly on the design of multi-core processing. It is discussed extensively in this paper:
    http://prod.sandia.gov/techlib/access-control.cgi/2008/080069.pdf

    Since that's probably a bit heady for some posters and lurkers, I have resurrected parts of that original thread here.

    As I explained to @Mr_Phil_Osophy, the article in question deals with topological dimensions, not physical ones.
    https://www.sciencealert.com/science-discovers-human-brain-works-up-to-11-dimensions

    For example, if I have 4 nodes and each one is connected to every other, I have a topological pyramid, a 3-dimensional structure. It doesn't matter whether those four nodes are on the same plane.

    Similarly, if I have 12 nodes and each one is connected to every other one, I have a total of 66 connection lines and an 11-dimensional structure. But that's 11 topological dimensions, not 11 physical dimensions.

    The topology of a computer network is often a key indicator of its capabilities. Moving from pyramids to cubes, consider: 8 processors arranged as the vertices of a cube, each with a direct communications path to three of the other processes in the same way that the edges of a cube connect each corner point to three other corner points.

    In such a topology, there are many data processing operations (such as sorting or FFT's) where this arrangement of processors can cooperate very efficiently. And if that topology is expanded to 2048 processors in with the topology of an 11-dimensional hyper-cube, then those same data processing can be efficiently shared among 2048 processors instead of just 8.

    As an example, here is a link to an article describing a highly efficient sort algorithm based on just such a structure:
    http://oldweb.ltu.bg/jmsd/files/articles/20/20-22_D_Gichev.pdf

    I should also add, that although this article was published in 2002, sort algorithms based on the cubic topology have been in the literature since at least the 1980's.

    Our protagonist, @Mr_Phil_Osophy, then expressed some confusion over how such a topology would support the types of applications required by the brain - prompting this explanation:

    ------------------
    You are confusing hardware and software. I'm not sure what the topological dimensionality of a Turing machine would be. You could probably design one that is limited to 2 dimensions. But that would not keep it from processing 3D, 4D, 5D, or more data structures. That would only be a matter of its programming.

    What this article (https://www.sciencealert.com/science-discovers-human-brain-works-up-to-11-dimensions) is describing is the interconnection of neurons in the brain - and more particularly, in the human frontal cortex. In the simplest of cases (which is not a realistic case), every neuron would be connected to a single information bus line. That would allow every neuron to talk to every other neuron, but only one pair could exchange data at a time. That would be a 1D topology. It could still perform the same operations as an 11-D topology, but it would take much, much longer.

    Here is the topology of the vertices of a 5-D cube, presented in 2-D form:
    [​IMG]

    In the brain, each of those colored dots would represent a neuron and each line would represent direct data path between two neurons. With the 2-D representation, you might have problems with crossing lines "shorting out", but add just a scratch of the third dimension to it and this 5-D topology can easily be contained. Of course, when it is flatten out like this, it is hardly recognizable as any sort of cube.

    But it has the same interconnections as the points of a 5-D cube. Thus it is topologically a 5-D hyper-cube.
    --------------------

    Finally, @Mr_Phil_Osophy inquired as to the meaning of the term "clique" as used in the article of interest.

    That article talks about neuron "groups" forming cliques. A bit of interpretation is needed there. I believe the "groups" they are talking about are physical grouping - not just topological. So they are saying that in many cases, neurons that are physically close to each other are also tightly interconnected with each other and less tightly connected to more distant neurons. And they are calling those "cliques".

    In the process of computing the topological dimensionality of the connections, they discovered these cliques by themselves rated high topological dimensionality (say 5D or 6D). These cliques were then connected to many, many other cliques in a topology that compounded the dimensionality, thus adding another 5D or 6D to the total.

    Imagine four cubes, each a 3-D clique. Then connect each corner point of each cube to the corner points on two other cubes. That would give you the 5-D topology in the figure I posted earlier.
     
  2. jcsd
  3. Apr 25, 2018 #2
    whaaaa? Why did they delete it? I thought they moved things around if they were mis-categorised?

    Thank you for making this subject clearer for me. Sorry if I came across a bit simple. I'm just incredibly interested in this stuff and so I guess I got compelled to jump in the deep end before I could swim. I think I can see the mistakes that I have made, and will certainly read more into this, so thank you for sharing more information and clearing things up.

    What would you recommend I study in order to get a better understanding of this kind of stuff? Would I start with topology or are there other things I need to become familiar with before moving onto topology?
     
  4. Apr 25, 2018 #3

    berkeman

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    Staff: Mentor

    It's being discussed by the Mentors at the moment...
     
  5. Apr 25, 2018 #4
    oh ok. Tell them I apologise if its blatantly obvious that I'm extending beyond my reach. Again, I'm currently very interested in the subject and so just trying to gauge where I need to investigate in order to get a better understanding.
     
  6. Apr 25, 2018 #5

    berkeman

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    Staff: Mentor

    No worries. The paper you referenced appears to be acceptable, but your thread title was a bit misleading. You did clarify in your text that it concerned mathematical dimensions, not spatial. Hopefully we can allow your thread or merge it into this one...
     
  7. Apr 25, 2018 #6
    Ok, thank you. If not, its not a problem. I think @.Scott helped clear the problem up quite a lot for me.

    I guess I had one major question I was trying to explore: is the scientific community taking extra spacial dimensions seriously at all?

    This video by Carl Sagan is something I had in mind when I was reading the article I posted in the thread:


    Along with the ideas put across in the film 'Flatland'.

    I guess the main thing I wanted to explore is the possibility of extra spatial dimensions, and if it is completely ridiculous or not to explore such ideas.

    The article doesn't seem to have been the best way of raising that question in the end. Which is my fault for misunderstanding what they were explaining in the article in the first place.
     
  8. Apr 25, 2018 #7
    To add to the above.

    The picture I was trying to paint, was that if we are the flat landers in Carl Sagan video, that our minds (if we were two dimensional creatures) would be structures that grow 'up' and 'down' into three dimensions, while our major sense organs existed in the 2 dimensional reality.

    So moving up to our reality, that we, in 3 dimensions, could possibly have minds that have structures which branch out into other dimensions (4th, 5th etc - spacial dimensions that is, not necessarily temporal).

    Again, I understand now that it isn't what the original article I posted was attempting to say... But is it possible to explore this extra dimensionality more or is that a completely bonkers position to hold/explore?
     
  9. Apr 25, 2018 #8
    Let's differentiate between minds and brain. The mind is metaphysics/philosophy and thus not proper topic for the threads you have posted. If you mean "brain", which was what the article was referring to, it is subject to the same Physics as other matter in the universe.

    So, No. Our brains aren't going to do anything special like that.

    And ... that type of speculation is entirely fictional and is the type of thing that will get threads deleted. It would also not fit into "Dimensionality of Network Topologies" or computer science.

    Physicists are familiar with Hilbert space (https://en.wikipedia.org/wiki/Hilbert_space), but only in terms of addressing real Physics.
     
  10. Apr 25, 2018 #9
    Ok, sorry.
     
  11. Apr 25, 2018 #10
    Yeah I was using mind as synonymous with brain here. I wasn't referring to anything metaphysical. Just trying to explore the possibility of such things being the case, or whether the scientific community knew enough to be able to right that off completely.

    Also, I wasn't suggesting the fictional story was a non-fictional one. I just wanted to see what people had to say on the matter from a scientific perspective. I wasn't trying to imply the brain had super powers or anything like that. Just, again, the things Carl Sagan was covering in his video sparked an interest in me I wanted to explore, and I wasn't sure where to post that exactly.
    EDIT: being new to the community and trying to get myself acquainted with how things work here.
     
    Last edited: Apr 25, 2018
  12. Apr 25, 2018 #11
    Can I ask, if its completely unscientific, why is it something that Carl Sagan explored at all?

    His fields of expertise were

    Surely he wouldn't have entertained the idea if it was completely ridiculous/unscientific?
     
  13. Apr 25, 2018 #12
    Right now, I'm not on a computer that can show video's. But as I recall, Sagan used flat-landers to introduce topics involving multi-dimensions. He never suggested that any of that was real or possible.
     
  14. Apr 25, 2018 #13
    No but he also never suggested it wasn't. He left it quite open and never shut it down as complete poetry or story telling. He was exploring something very interesting, and as far as I could make it, was open to the possibility.
     
  15. Apr 25, 2018 #14
    He even makes reference to the fictional creature trying to explain extra dimensions to other 2 dimensional creatures and how they would mock him for it, pat him on the back and ask him to show it too them while dismissing it as ludicrous. I think that moment was rather telling towards the possibility that he was open to it. He didn't explain it in any way that suggested those who doubted the fictional extra dimensional traveller were more rational for doubting him. While explaining extra dimensions he painted them (the doubters) as more naive for assuming they have an understanding of spacial dimensions based on what they experience empirically.
     
  16. Apr 25, 2018 #15
    I am sure flat-landers were only introduced to help explain some ideas.

    Physicists do entertain the notions of additional dimensions - but only in the context of providing a more unified or simplified model of their observations. An example of this would be membrane theory (https://en.wikipedia.org/wiki/M-theory).
    So the point is not there are no additional dimensions.

    The issue is a distinction between hypothesis and modelling versus other speculation. You seem to want to give brains a fundamental physical attribute that other materials do not have. If that's what you want to do, you should lead with specific measured properties of the brain that are not observed in other matter - and cite the source of this according to the rules of the forum. Then some hint of a model would be nice - what are the basics that allow the brain to do this - and how we could detect this being done in other situations. Then explain how it would be simpler to explain these special brain properties as extra dimensions than with other descriptions.
     
  17. Apr 25, 2018 #16
    He was also interested in entertainment and in popularizing science.

    As an ambassador of science, then, he was disposed toward exposition of some ideas that, while they generally had some relationship to science, were not necessarily framed in a scientific manner.

    As a comparatively recent member of Physics Forums, I observe that PF tries to maintain a balance between not alienating people, and keeping things reasonably rigorously scientific.

    There are many websites available in which wilder speculations are explored.

    The discipline level here, though, helps to ensure an important venue for people who are willing to accept some constraints in order to have more productive discussions on matters of science.
     
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