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Scaling - an idea to extract network features

Posted 8 months ago
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I am thinking about the heuristic rules we can apply in the Wolfram Physics Project. And I'd like to share my idea with you guys.

What if we cluster some closely related nodes (a sub-network) as a new node in a larger scale network (Maybe iterating the k-means algorithm with different k on a different scale)? The network may be nested somehow on a different scale. But in a way, it may be fractal-like.

On the one hand, we can simplify the original network by this. On the other hand, the computational universe may resemble a fractal (in a sense, our real world is a fractal). Many more interesting properties may emerge after it, e.g. self-organizing. And the cross-scale feedback seems to be more obvious in this structure.

Welcome to discuss, and thanks in advance!

POSTED BY: Ce Luo
Answer
6 Replies

The k-means algorithm does not apply here. K-Means is useful for vector quantization. Our nodes are not vectors, and are quantifiable. Let me explain, k-means is define by this: enter image description here

The problems are:

We have no - defined between nodes. It is not a vector space/field.

You suppose more mathematically structure than we have.

We also have no || . || defined since our nodes are not in some metric space.

Posted 8 months ago

Thanks a lot for your insight! Those are really what I missed. It seems that we may need to somehow define a coordinate system in the computational universe. Do you have some ideas to solve the problem? I think we may need to define a kind of coordinate system that relies on some characters of the network itself. Or... can we set a standard to measure all the universes?

POSTED BY: Ce Luo
Answer

It seems that we may need to somehow define a coordinate system in the computational universe. Do you have some ideas to solve the problem?

Your question doesn't make that much sense. This is mostly due to the loose way you used terms with strict mathematical definition before. What attributes do you hope your coordinate system might have?

Or... can we set a standard to measure all the universes?

What do you mean by standard, what do you mean by measure and what do you mean by all the universes?

Posted 8 months ago

I am sorry for the unclarity. Actually, I just wanted to find a way to package a sub-network as a node and see what properties will the new net have.

For the standard, I just wanted to define a distance unit. And for the universes, I mean universes generated by different rules.

Thanks.

POSTED BY: Ce Luo
Answer

I am sorry for the unclarity. Actually, I just wanted to find a way to package a sub-network as a node and see what properties will the new net have.

So you are having problems finding tooling to reshape graphs?

The tooling of WFPP is not really suitable for that as it based on the SetReplace primitive which match a fixed pattern of free variables against a (hyper-) graph. Consider this pattern:

enter image description here

You see there are some nodes which have 2, 4, 6 neighbors. It is not a possible to write a rule in this framework which takes an arbitrary sub network and replaces it with a node and tie up all the out going connections to the new node. In fact you can't write a procedure which replaces any node creates a new node and ties it up in the same way as it was before.

The language of hyper-graph modifications in the WFPP is quiet restrictive. Your use case is to advanced for the current tooling, other places of the Wolfram language working on Graphs, not Hypergraphs are more suited for you. Furthermore i have not seen much interest in graph numbers other than dimensional of a local embedding.

And for the universes, I mean universes generated by different rules.

It is not only rules that matter but also initial state.

Posted 7 months ago

Thanks a lot! My idea seems to be very naive.

I will reconsider its effect. I will come back to you for further discussion when I make it more practicable (if you don't mind).

But it may be, as you said, not so interesting. If so, thank you very much for your time and help to solve my puzzle!

POSTED BY: Ce Luo
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