Elegant six-page proof reveals the emergence of random structure
quantamagazine.org
Elegant six-page proof reveals the emergence of random structure
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Re: Elegant six-page proof reveals the emergence of random structure
#2Can someone explain to my why proving random graphing can produce known shapes is important? Because this seems absurdly obvious to any layman. Why is this a complex proof? I’m guessing it’s more that they proved the thresholds for these shapes being formed more than why?
Re: Elegant six-page proof reveals the emergence of random structure
#3Can someone explain to my why proving random graphing can produce known shapes is important? Because this seems absurdly obvious to any layman. Why is this a complex proof? I’m guessing it’s more that they proved the thresholds for these shapes being formed more than why?
Yes, this is all about knowing precisely when the properties are going to emerge. How many edges are needed for the Hamiltonian cycle to be present (with high probability).
Re: Elegant six-page proof reveals the emergence of random structure
#4Can someone explain to my why proving random graphing can produce known shapes is important? Because this seems absurdly obvious to any layman. Why is this a complex proof? I’m guessing it’s more that they proved the thresholds for these shapes being formed more than why?
From article, “The conjecture pertains to far more than random graphs. If true, it holds for random sequences of numbers, for generalizations of graphs called hypergraphs, and for even broader types of systems.”
“Obvious to a layman” is also frequently not at all related to proof, particularly in a mathematical sense.
Your summary of what they’ve proven is off, as well.
Re: Elegant six-page proof reveals the emergence of random structure
#5I wish Quanta was a print publication I could subscribe to. Definitely the types of articles I'd like to sit down and read not on a computer.
Re: Elegant six-page proof reveals the emergence of random structure
#6Can someone explain to my why proving random graphing can produce known shapes is important? Because this seems absurdly obvious to any layman. Why is this a complex proof? I’m guessing it’s more that they proved the thresholds for these shapes being formed more than why?
From article, “The conjecture pertains to far more than random graphs. If true, it holds for random sequences of numbers, for generalizations of graphs called hypergraphs, and for even broader types of systems.” “Obvious to a layman” is also frequently not at all related to proof, particularly in a mathematical sense. Your summary of what they’ve proven is off, as well.
Ah yes all clear now :)
Re: Elegant six-page proof reveals the emergence of random structure
#7Can someone explain to my why proving random graphing can produce known shapes is important? Because this seems absurdly obvious to any layman. Why is this a complex proof? I’m guessing it’s more that they proved the thresholds for these shapes being formed more than why?
The interesting thing a out graph properties (like the emergence of connectivity, giant components, Hamiltonian paths etc.) is that they happen as "phase transitions".
A classic example is cuckoo hashing: You want to know how many edges the random graph of hashes can have before it contains a cycle, since that's when you need to rehash into a larger table. You might expect that this number is "pretty random" in that you sometimes get a cycle with few edges or sometimes have many edges but no cycle. However it turns out that it very predictably happens exactly when the graph gets to a certain size. In the same way as 1000 coin flips very predictably have 450-550 heads.
What's so cool about the theorem is thst it proves _any_ property you can think of has _some_ sudden threshold like that.
Re: Elegant six-page proof reveals the emergence of random structure
#8I wish Quanta was a print publication I could subscribe to. Definitely the types of articles I'd like to sit down and read not on a computer.
I feel like there’s a market for a publishing service that could partner with popular blogs and websites. They could either allow articles to be selected by the user or the partner site, which could then be auto-paginated and printed being being posted to the subscriber. Hell, sell a service where you will curate articles based on used interests. Add some relevant Twitter threads for letters to the editor.
Partner sites could also benefit from some increased subscriber revenue, and ads could actually be relevant again (remember when Scientific American advertisements were kind of cool and industry related?).
Call the prototype deadtreepress.io or something.
(I recognise the environmental implications, but I think paper and postage can be sustainable in some regions. Maybe you could integrate offsets into the price?)
Re: Elegant six-page proof reveals the emergence of random structure
#9I wish Quanta was a print publication I could subscribe to. Definitely the types of articles I'd like to sit down and read not on a computer.
I feel like there’s a market for a publishing service that could partner with popular blogs and websites. They could either allow articles to be selected by the user or the partner site, which could then be auto-paginated and printed being being posted to the subscriber. Hell, sell a service where you will curate articles based on used interests. Add some relevant Twitter threads for letters to the editor. Partner si…
I feel I would want that too even though I got used to reading on screens.
Maybe it could be pdf, I think it may be about some finite amount of content to consume. I guess for me some every 3-12 months ebook with articles that are still relevant and will likely stay relevant for at least a few years would be awesome.
Re: Elegant six-page proof reveals the emergence of random structure
#10I wish Quanta was a print publication I could subscribe to. Definitely the types of articles I'd like to sit down and read not on a computer.
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