#multigraphic

[retrocgads]

USA 1997

[bubbloquacious]

Hydrogen bomb vs. coughing baby: graphs and the Yoneda embedding

So we all love applying heavy duty theorems to prove easy results, right? One that caught my attention recently is a cute abstract way of defining graphs (specifically, directed multigraphs a.k.a. quivers). A graph G consists of the following data: a set G(V) of vertices, a set G(A) of arrows, and two functions G(s),G(t): G(A) -> G(V) which pick out the source and target vertex of an arrow. The notation I've used here is purposefully suggestive: the data of a graph is exactly the same as the data of a functor to the category of sets (call it Set) from the category that has two objects, and two parallel morphisms from one object to the other. We can represent this category diagrammatically as ∗⇉∗, but I am just going to call it Q.

The first object of Q we will call V, and the other we will call A. There will be two non-identity morphisms in Q, which we call s,t: V -> A. Note that s and t go from V to A, whereas G(s) and G(t) go from G(A) to G(V). We will define a graph to be a contravariant functor from Q to Set. We can encode this as a standard, covariant functor of type Q^op -> Set, where Q^op is the opposite category of Q. The reason to do this is that a graph is now exactly a presheaf on Q. Note that Q is isomorphic to its opposite category, so this change of perspective leaves the idea of a graph the same.

On a given small category C, the collection of all presheaves (which is in fact a proper class) has a natural structure as a category; the morphisms between two presheaves are the natural transformations between them. We call this category C^hat. In the case of C = Q, we can write down the data of such a natural transformations pretty easily. For two graphs G₁, G₂ in Q^hat, a morphism φ between them consists of a function φ_V: G₁(V) -> G₂(V) and a function φ_A: G₁(A) -> G₂(A). These transformations need to be natural, so because Q has two non-identity morphisms we require that two specific naturality squares commute. This gives us the equations G₂(s) ∘ φ_A = φ_V ∘ G₁(s) and G₂(t) ∘ φ_A = φ_V ∘ G₁(t). In other words, if you have an arrow in G₁ and φ_A maps it onto an arrow in G₂ and then you take the source/target of that arrow, it's the same as first taking the source/target in G₁ and then having φ_V map that onto a vertex of G₂. More explicitly, if v and v' are vertices in G₁(V) and a is an arrow from v to v', then φ_A(a) is an arrow from φ_V(v) to φ_V(v'). This is exactly what we want a graph homomorphism to be.

So Q^hat is the category of graphs and graph homomorphisms. This is where the Yoneda lemma enters the stage. If C is any (locally small) category, then an object C of C defines a presheaf on C in the following way. This functor (call it h_C for now) maps an object X of C onto the set of morphisms Hom(X,C) and a morphism f: X -> Y onto the function Hom(Y,C) -> Hom(X,C) given by precomposition with f. That is, for g ∈ Hom(Y,C) we have that the function h_C(f) maps g onto g ∘ f. This is indeed a contravariant functor from C to Set. Any presheaf that's naturally isomorphic to such a presheaf is called representable, and C is one of its representing objects.

So, if C is small, we have a function that maps objects of C onto objects of C^hat. Can we turn this into a functor C -> C^hat? This is pretty easy actually. For a given morphism f: C -> C' we need to find a natural transformation h_C -> h_C'. I.e., for every object X we need a set function ψ_X: Hom(X,C) -> Hom(X,C') (this is the X-component of the natural transformation) such that, again, various naturality squares commute. I won't beat around the bush too much and just say that this map is given by postcomposition with f. You can do the rest of the verification yourself.

For any small category C we have constructed a (covariant) functor C -> C^hat. A consequence of the Yoneda lemma is that this functor is full and faithful (so we can interpret C as a full subcategory of C^hat). Call it the Yoneda embedding, and denote it よ (the hiragana for 'yo'). Another fact, which Wikipedia calls the density theorem, is that any presheaf on C is, in a canonical way, a colimit (which you can think of as an abstract version of 'quotient of a disjoint union') of representable presheaves. Now we have enough theory to have it tell us something about graphs that we already knew.

Our small category Q has two objects: V and A. They give us two presheaves on Q, a.k.a. graphs, namely よ(V) and よ(A). What are these graphs? Let's calculate. The functor よ(V) maps the object V onto the one point set Hom(V,V) (which contains only id_V) and it maps A onto the empty set Hom(A,V). This already tells us (without calculating the action of よ(V) on s and t) that the graph よ(V) is the graph that consists of a single vertex and no arrows. The functor よ(A) maps V onto the two point set Hom(V,A) and A onto the one point set Hom(A,A). Two vertices (s and t), one arrow (id_A). What does よ(A) do with the Q-morphisms s and t? It should map them onto the functions Hom(A,A) -> Hom(V,A) that map a morphism f onto f ∘ s and f ∘ t, respectively. Because Hom(A,A) contains only id_A, these are the functions that map it onto s and t in Hom(V,A), respectively. So the one arrow in よ(A)(A) has s in よ(A)(V) as its source and t as its target. We conclude that よ(A) is the graph with two vertices and one arrow from one to the other.

We have found the representable presheaves on Q. By the density theorem, any graph is a colimit of よ(V) and よ(A) in a canonical way. Put another way: any graph consists of vertices and arrows between them. I'm sure you'll agree that this was worth the effort.

[processes]

The Abelian sandpile model (ASM) is the more popular name of the original Bak–Tang–Wiesenfeld model (BTW). The BTW model was the first discovered example of a dynamical system displaying self-organized criticality. It was introduced by Per Bak, Chao Tang and Kurt Wiesenfeld in a 1987 paper.

The identity element of the sandpile group of a rectangular grid. Yellow pixels correspond to vertices carrying three particles, lilac to two particles, green to one, and black to zero.

Three years later Deepak Dhar discovered that the BTW sandpile model indeed follows the abelian dynamics and therefore referred to this model as the Abelian sandpile model.

The model is a cellular automaton. In its original formulation, each site on a finite grid has an associated value that corresponds to the slope of the pile. This slope builds up as "grains of sand" (or "chips") are randomly placed onto the pile, until the slope exceeds a specific threshold value at which time that site collapses transferring sand into the adjacent sites, increasing their slope. Bak, Tang, and Wiesenfeld considered process of successive random placement of sand grains on the grid; each such placement of sand at a particular site may have no effect, or it may cause a cascading reaction that will affect many sites.

Dhar has shown that the final stable sandpile configuration after the avalanche is terminated, is independent of the precise sequence of topplings that is followed during the avalanche. As a direct consequence of this fact, it is shown that if two sand grains are added to the stable configuration in two different orders, e.g., first at site A and then at site B, and first at B and then at A, the final stable configuration of sand grains turns out to be exactly the same. When a sand grain is added to a stable sandpile configuration, it results in an avalanche which finally stops leading to another stable configuration.

Dhar proposed that the addition of a sand grain can be looked upon as an operator, when it acts on one stable configuration, it produces another stable configuration. Dhar showed that all such addition operators form an abelian group, hence the name Abelian sandpile model. The model has since been studied on the infinite lattice, on other (non-square) lattices, and on arbitrary graphs (including directed multigraphs). It is closely related to the dollar game, a variant of the chip-firing game introduced by Biggs.

[thecrazyneographist]

Tips for making an orthography?

I should probably make a PSA about asks of this sort, it's hard to answer and I'd rather not spend a whole post about it rather than say it in DMs or @ the person who asked at @conlangcrab, yet here I am talking to an anon without such possibility.

Depends on what you mean and what you need.

An orthography per definition is a set of conventions for spelling phonetic information via graphemes (the symbols of a given script). Some orthographies are phonetically consistent, others are not.

Here's an example for an English orthography I made just now:

Ysc-Orthosci /aɪ̯s ʊɹˈθʊsɪ/

An orthography depends on A) Phonetic inventory of the language and B) The script's limitations.

The Latin alphabet is only 26 letters, and boy there are more sounds than there are letters. There are three ways to solve this problem:

Positional phonetic alteration, when a letter is pronounced differently depending on its position in the word.

Diacritics, additional markings on the letter itself that alter the pronunciation, like carons, macrons, circumflexes, umlauts et cetera.

Digraphs/multigraphs, when several letters are seen as one phonetic unit. Think English "th" in "the"; There's one sound, but two letters to represent it.

The more letters there are, the easier is the spelling due to them covering more sounds - that allows for greater correspondence between what's written and its pronunciation. Some languages, like the Slavic family, written Hawaiian, Japanese/Korean, have high correspondence, others have not (English, and lord have mercy, French).

That is applicable, though, only if the script is representing phonetic values directly and isn't hieroglyphic. In Chinese, a single symbol can be read several different ways in one dialect, not talking about other dialects.

In conclusion: The more letters you have, the easier the spelling will be. But that will take away some of the spice of a language; I find French, German, and English spelling to be quite fabulous, and Gaelic? Just marvelous, with all the tricky rules of writ and pronunciation.

But please, if you are sending asks like that either ask them in DMs or without an anonymous option; I would prefer not to make a whole script just to answer an ask, since the amount of posts on this blog is equal to the amount of neographies I've created in my life.

[planerider-ryn]

Spite is the best motivator -> I got annoyed at networkx for not properly displaying multigraphs, started coding my own, ended up building a full symbolic creator of series-parallel graphs in python

[langnerdtobi]

What is everyone's favorite phoneme?

Mine is definitely ɬ (Welsh Ll, also the sound in the tl multigraph in Nahuatl - /tɬ/)

[multigraphicsprint]

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[multigraphic]

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[multigraphics1999]

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[laika-the-bitch]

digital typefaces should treat multigraphs as single characters for the purposes of line breaks that split words, just my too sense

[vacuously-true]

People are so focused on gender functions, always talking about even and odd gender functions, injectivity and surjectivity. Well what about other gender relations? What about the empty gender relation? What about the universal gender relation? What about the identity gender relation? What about reflexivity, transitivity, and symmetry? What about gender equivalence relations? What about fucked up gender relations that are best represented by graphs and digraphs? What about fucked up multiset gender relations that are best represented by multigraphs? What about fucked up gender relations that can't even be represented with finite information? What then? Your odd invertible polynomial ℝ -> ℝ gender function ain't shit. Only talk to me if your gender relation is at least as cool as the mathematics genealogy project digraph. If your gender can be represented as a finite planar simple graph, dni, you're part of the problem.

Gender non-convergent

[bubbloquacious]

So defining directed multigraphs as presheaves over a category V ⇉ A is pretty cool. I think you can get undirected multigraphs very similarly. You add an endomorphism p: A -> A satisfying p ∘ p = id_A, p ∘ s = t, and p ∘ t = s (where s and t are the parallel morphisms V -> A). You can get graphs with labelled vertices by adding an object L and a morphism ℓ: L -> A. That one's interesting cause you get one more representable presheaf.

[willisdayjrwad2]

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[pedestriansteppers]

Photo-Multigraphs: The Mirror and the Camera

In 1893, H. P. Ranger was granted Patent No. 505,127 for a “Mirror For Use In Photography.” This was a device comprised of two adjustable mirrors set at an angle. When a subject was placed in front of it, his or her image was reflected in each mirror and that reflection was again reflected, resulting in five or more figures—the number of figures determined by the angle of the mirrors.

We now own a photo-multigraph tintype that is especially interesting because it shows some the studio wherein the image was taken, including a raised platform and large mirrors that would certainly be capable of showing a standing subject. This gives us hope of finding a full-length photo-multigraph in the future.

[abalidoth]

Hmmm. These are all quartic multigraphs but the directed pair is throwing me off a bit. The different colors and the directed pair give Cayley graph vibes.

I guess the directed pair is the only ones where the orientation of the 3-cycles matters... Is this, like, the distinct digraphs you get from looking at the combination of two different permutations?

[changoblanco]

#multigraph #photomultigraph #trickmirror #portraitphotography #lanephotographers #lanestudios https://www.instagram.com/p/CcN_2qgrdlz/?igshid=NGJjMDIxMWI=