In case the author is reading, some places that could be clarified:
> Scatter a bunch of points in a plane. Connect some of them with lines. That’s all a graph is.
This may mislead the uninitiated reader to think that the definition of a graph includes vertex locations in R^2.
> Over the years, mathematicians worked to reduce the number of edges Hamiltonian graphs had to have.
The article is about showing that certain graph properties guarantee the existence of a Hamiltonian cycle. But the logical structure of this sentence implies the converse: showing that the existence of a Hamiltonian cycle guarantees other properties. But all it guarantees is |E| >= |V|.
If (a) it's hard to find cycles in a graph, and (b) it's not hard to construct a graph with a cycle, that sounds to me like the makings of a trapdoor function that could be useful in cryptography.
Nit: It's only hard to find _hamiltonian_ cycles, finding general cycles is trivial. More generally, it's hard to find long simple cycles (a simple cycle is one with no overlapping vertices visited twice).
I don't know how hard it is to generate graphs that are hard to find simple cycles in though. For example, previous work showed that random graphs are likely to contain hamiltonian cycles (and with this algorithm they can now be found).
Take your favorite hash function, and view it as inducing a directed graph on the set of its possible outputs. Finding a cycle is hard! (except, most hash functions map 0->0 for "nothing up my sleeves" appeal)
But also, it's ridiculously hard to answer simple questions about such graphs, such as listing its nodes.
So annoyingly the paper doesn't explicitly state the constant they got, but from a brief skim it looks like it's, like, 10^15?? I really hope someone can improve that...
I wonder if this can be used to prove if all substantial React apps include cycles--even if the raison d'etre of React is one-way data flow (i.e. no cycles).
Sure, a graph is connected if it has a spanning tree, which requires |E|>=|V|-1. Presumably a graph is only highly connected if it has more edges than this minimal requirement, so we have |E|>=|V|, and this graph must have a loop.
I am not an expert but it looks like the quanta author has invented a definition for "highly connected" because C-expander was too complicated a concept for them (me too). There are older competing definitions for "highly connected" along the lines of a graph where the number of edges is greater then half the number of vertices which is obviously insufficient to guarantee a loop.
Then they have used networks instead of graphs and added a comma, for no apparent reason.
Readit News
> Scatter a bunch of points in a plane. Connect some of them with lines. That’s all a graph is.
This may mislead the uninitiated reader to think that the definition of a graph includes vertex locations in R^2.
> Over the years, mathematicians worked to reduce the number of edges Hamiltonian graphs had to have.
The article is about showing that certain graph properties guarantee the existence of a Hamiltonian cycle. But the logical structure of this sentence implies the converse: showing that the existence of a Hamiltonian cycle guarantees other properties. But all it guarantees is |E| >= |V|.
Though maybe just the proof outline section is enough for a Saturday.
If (a) it's hard to find cycles in a graph, and (b) it's not hard to construct a graph with a cycle, that sounds to me like the makings of a trapdoor function that could be useful in cryptography.
[I'm neither a mathematician nor a cryptographer]
I don't know how hard it is to generate graphs that are hard to find simple cycles in though. For example, previous work showed that random graphs are likely to contain hamiltonian cycles (and with this algorithm they can now be found).
But also, it's ridiculously hard to answer simple questions about such graphs, such as listing its nodes.
See related Neural Cryptography https://en.wikipedia.org/wiki/Neural_cryptography
The quote leaves those two objects unnamed. Are they supposed to be “graphs” and “groups”?
Yes, graphs and groups.
Then they have used networks instead of graphs and added a comma, for no apparent reason.
Dead Comment