Doubts regarding linking (granularity and density) and atomic notes: "How linked is too linked?"

Knowledge management
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I’m currently learning to build a PKM for notes and knowledge related to my studies (STEM) with the following objectives

  • It should be an evergreen library that I can easily access whenever I’m studying something new that depends on things I’ve already studied
  • It should allow me to explore my knowledge and gain new insight as I’m researching or working on tasks
  • It has a well-defined set of objective rules and structure (that I actually keep written as a note)

To achieve this, I’ve looked at the Zettelkasten method and the Evergreen Notes method, and I really ended up liking the idea of interconnected notes.
As I am writing notes and actively applying the core principles, I’m left with a couple of doubts about atomic notes and linking best practices.

Core questions I’ve been trying to answer myself before linking:

  • “Am I linking this note just because I’m saying its name, or does this link actually add a good amount of meaning?”
  • “Will this be useful to my future self?”/“Will my future self need this link to expand or understand better?”
    After writing this, I’m also starting to feel afraid of “thinking too much about graph view/backlinks rather than knowledge itself”, so please point me out of this is the case.

Problem 1: How atomic is too atomic

As an example, consider the notes “Matrices” and “Addition operator for matrices”.
If “Addition operator for matrices” appears only in the “Matrices” note (see “Implied links” problem), wouldn’t it be right to merge those two notes as one note even if they express two different ideas and concepts?

Discussion on problem 1 (How atomic is too atomic)

  • Creating only one note would make reading easier and would keep my vault less cluttered
  • Avoids, for some part, the “Implied links” problem
  • The notes can be split when I talk about something very specific about addition
    • All notes linked to “Matrices” need to be reviewed (see “Implied links”)
    • I can still embed the addition note into matrices for ease of reading

This violates the core principle of atomic notes. I am mixing different ideas together and creating extra work for linking and reviewing (whenever I will want to link only the specific note or split them).

Problem 2: Implied links

As an example, consider the notes “Matrices”, “Determinant of a matrix” (determinant is something that I defined exclusively for matrices) and “Addition operator for matrices”.
When it comes to writing, it usually goes something like this:

  • “Consider the [[Determinant of a matrix|determinant]] of the [[Matrices|matrix]] $A$ …”
  • “Let $A$ be a [[Matrices|matrix]] that has a non-zero [[Determinant of a matrix|determinant]] …”
  • “[[Addition operator for matrices|Add]] the [[Matrices|matrix]] $A$ to …”

Would it be right to assume that if the related context is obvious, links can (should) be avoided (as they add no extra information or value)?

  • If I am talking about the determinant of a matrix, then I am obviously talking about matrices
    • If I’m also discussing or making use of something very specific to “matrices” and unrelated to “determinant” this, of course, does not apply
    • “Consider the [[Determinant of a matrix|determinant of the matrix]] $A$ …”
    • “Let $A$ be a matrix that has a non-zero [[Determinant of a matrix|determinant]] …”
  • The fact that “matrices can be added” is one of the fundamental things I would talk about in the matrices note, so I assume it’s obvious that if I talk about matrices, then I must know how to add them.
    • “Add the [[Matrix|matrix]] $A$ to …”

Discussion on problem 2 (Implied links)

Linking, without the assumption for context, seems too redundant and not meaningful: it does not add (that much) extra value.

It also mixes notes in a way that actually ends up mixing useful data with unrelated data:

  • Both local and global graph view start to get those “MOC like” shapes where data about “manipulating matrices” is mixed with “usage of matrices”, if you get my point
  • If I am looking at the “Matrices” note, it’s very likely that I am more interested in manipulation or notes strictly related to matrices than the usage of matrices themselves

This violates the core principle of densely linked notes:

  • The context (thought to be “obvious”) can be forgotten/lost
  • I am creating distance between my notes, which can be easily felt by going through links in the text rather than graph view or backlinks.
    • As an example (from “Yet another example for implied links”), consider writing a note only linking to “Vector subspace” but it turns out what’s actually needed/what I’m looking for is something linked to the note “Vectors”
    • A big part of my studies is “learning to apply what I already know to new things” (and it’s mostly done by generalizing ideas for bigger contexts). I am afraid this can get in the way
      I also have no way to objectively define “obvious”.

Yet another example for implied links

Consider the notes “Vectors”, “Base of a vector subspace” (it’s composed by vector), “Vector subspace” (you need a base to make one).
I start writing a note about the R2 space as follows:

  • “Let the [[Vector subspace|vector subspace]] generated by the [[Vectors|vectors]] (…) that form a [[Base of a vector subspace|base]] (…)”
    Since I defined a base to be composed by vectors, then I would rewrite it as:
  • “Let the [[Vector subspace|vector subspace]] generated by the [[Base of a vector subspace|base]] (…)”
    Since I stated that a vector subspace is generated by a base, the base is a core component of the vector subspace. Then, I would rewrite as:
  • “Let the [[Vector subspace|vector subspace]] generated by the base (…)”

Problem 3: Merge notes

As an example, consider the notes “Matrices”, “Property A of matrices” and “Property B of matrices”.
I’m very interested in matrices that follow both property A and B and I create 20 notes that begin with something along the lines of:

  • “Let $A$ be a [[Matrices|matrix]] that follows both [[Property A of matrices|property A]] and [[Property B of matrices|property B]]”
    Wouldn’t it be right to create a note a “merging note” called [[Matrices that follow A and B]] to group that context?
  • “Let $A$ be a [[Matrices that follow A and B|matrix that follows both property A and property B]]”

Discussion on problem 3 (Merge notes)

Without a merging note, I feel like I have the following issues:

  • Backlinks and graph don’t always seem appropriate or relevant
  • Increases (in my opinion) the work required to move, merge, split and sort linked notes

This violates the core principle of fine-grained associations, and it creates the same “distance” issues as the “Implied links” problem.
I also am afraid of filling my vault with notes that provide no value or knowledge to the system itself.

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Oh, very interesting topic, big questions and broad answers :slight_smile:

Try to say my thoughts in a quick way, then I read yours.

Regarding links, I adopt an heuristic that a link is “well placed” when:

  1. I’ve thought enough to put that link
  2. I’ve found a clear and strong “why” A and B notes connect together using that link
  3. I find a purpose for that link. If I see the purpose now, or if I preview for tomorrow, that link is “useful”

If a link doesn’t satisfy these rules, I don’t put it

from rule 1 follow that I don’t consider “automatic linking” a good practice. It’s a bad practice

Rule 2 and 3 avoid having useless linking.

“usefulness” can be

  • highlight meaningful relationships between notes (A generates Idea B, for example, or a term in A is defined or detailed by B note)
  • helping navigating between notes (I have to move from A to his parent B, for example, or in a wikipedia-like way (but only for very very relevant words) )
  • give a structure to note systems (MOC and index notes, and even using linking instead of tags, I use this approach)

N.B.

  • It’s not an algorithm, its only a personal guideline;
  • Need to be correctly contextualized. It’s valid for me and I apply it in knowledge development contexts, (learning, idea generation, and so on; in a word, zettelkasting), not in other uses of Obsidian (information tracking, information collecting, and so on). Different goals and results to obtain drive different principles.
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I’ve some heuristics for atomic notes, too :slight_smile:

  • Atomic doesn’t automatic mean “short”. Atomic main property is “not breakable further”, like atoms
  • Atomic notes have big advantages, so if I feel I can split, I split
  • Don’t split further if the next split produce a loss of meaning, a difficoult to read or a difficult to follow the reasoning
  • Atomic doesn’t mean renounce to details, examples, and other content if this content makes a good support context for the concept in the note. For knowing more about it, for making it more clear.
    Atomic doesn’t mean arid
  • A good measure of the atomicy of the note is if its title is a good summary of the whole content
  • Notes are not immutable but can change over time, so can be splitted or merged when I feel need of doing that, following the other principles. So, a concept can have his birth in a “parent” note and stay there until emerge the need to move to another note.

Even in this case is not an algorithm, they are principles to follow adapting them in every case
Even in this case, different goals and results to obtain drive different principles.

Not all notes need to be atomic, of course.
I could put this post as a note in my system, The main note could be considered an atomic note (“How to make atomic notes”) but I could also consider it a “molecular note”, made of six principles that I’d surely move in six different notes, detailing hear, leaving only the links to the five principle notes (if the title is descriptive, the link behaves like a connection but also as a summary for the main note).

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Leonardo, a very short answer to your question about matrices.

N.B. I’ts a strong matter of taste.
If I apply my principles to the example, having so many links in that text is not useful for me, so I don’t make all of these links.

Apply the principle of “notes can evolve over time” to your matrices/matrices addition, if today I’m not sure how addition concept will evolve and which role will have, today I adopt the simplest possible solution: matrice addition stays within matrices note (I make a section “operations”). If tomorrow addition will grow enough, or when main matrices note grows too much, I will tranform them separating notes.
I don’t have to front all the issues today. I can front issues when needed.

A thought about backlinks and graph.
For my opinion, obtaining a “beautiful” backlinking or graph is not a purpose. They are tools, not the goal of notetaking system.
Consider that I don’t really use them :slight_smile:

I repeat, my opinions

For the third problem, merge notes, for me the “solution” is, again, in case of uncertainty for tomorrow, today try to prefer the simplest form that fullfills the needs for today. If tomorrow the content will evolve, adapt the structures for fulfill the evolution.

Having “combination” notes can be fine. For example, in my notes I have a set of notes about uses of tags, a set of notes about uses of folders. But where do I write about “Use of tags vs use of folders for note organization”? I create a note with just this thing, then I link to both tag note and folder note.
It’s not an example that perfectly fits with yours, but represents that a relationship between two concepts (in my case is a “versus”) can be reified as a concept itself.

I don’t know if this apply in your case. You have always to check that what you obtain doesn’t become too complex to manage, write and re-read

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I believe the methods you’re looking at were developed for the humanities and social sciences, so may not have tons of value for STEM.

Links are a convenience — a way to quickly reach one note from another, or remind yourself of notes. (In Obsidian they’re also a convenient way to autocomplete names.) You could use no links at all and still access and remind yourself of notes. If you find a place where you wish you had a link, it’s very easy to add one. If you feel you have too many, it’s not too hard to remove some.

I recommend experimenting. Try different methods in different parts of your vaults or at different times and see how they feel.

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Thank you, @andy76 and @CawlinTeffid.

Although I was already careful not to lose meaning (as atomic notes should not lose meaning or redundantly reintroduce context) and applying more thought to linking, one big thing that this reminded me is that I should not be afraid of notes changing over time.
Hence, my rules should be loose and actually allow for long-term support.

While adding or removing a link is very easy, I was actually very much concerned that I would have to revisit basically every note at the same time (it did happen), but waiting until I actually want or need them differently seems way more manageable than trying to predict what I will want or need in the future.

Thanks to this realization, I actually feel like experimenting without fear, following CawlinTeffid’s advice.
I will post in this thread (or create a new one linking to this one) once I have more thoughts on the matter.

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I think this is the answer to my query in the post I just now created. :grin:

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From the answers given by previous posters how then you will manage the process of linking is thinking ? Because what I get is that if the linking is to be done after serious thought then you will be able to do that only on a few occasions. This is also a sphere where I am struggling. One reply is that link learning won’t be as effecient in learning STEM which is different from humanities ? Have you been able to get around to an answer to this ?

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After making this post and reading the answers, I’ve taken some time to think about my method.

I believe that the most helpful thing I’ve learned so far, which might seem obvious, but it’s really not, is that what I write isn’t necessarily set in stone.
Rather, trying to fix knowledge in place is, in fact, an inappropriate way to think.
Just to name a few issues of the many:

  • I can be “wrong”
  • The source can be “wrong”
  • What I linked makes less sense than what I originally thought it would
  • What I dismissed could have actually proved useful in the future
  • What I thought was obvious at the moment of writing is not so obvious at the moment of reading

Taking with this in mind, my workflow has currently evolved to the following:

  • I just write, really.
  • If I am writing in one note about something that can be viewed from different perspectives, I will split it into notes and deal with it when I need to (= I’m writing about something for which the difference is crucial) or want to (= I’m interested in the difference itself).
  • I do use “merge notes” (view original post). After all, there must be a reason why I’m so interested in several things with this thing in common. Perhaps, I can even link it! But I still actually create a merge note (rather than using a link to a non-existing file) whenever I need to or want to (see above).
  • I do use “implied links” (view original post) in one way or another. Although it could be a bit harder to find something later, I can always add a link at that time. Why bother now?
  • I’ve stopped focusing on the graph view and backlinks, and I’ve started asking myself if I can access what I want or will want quickly.

On every note I make, I actually take the freedom to experiment with ideas and see how they affect my vault. If it works, I’ve learned something new.

I also want to start reviewing random notes periodically. Ideally, one per day. Perhaps I can get more insight by reviewing while I am tackling a completely different subject with a completely different mindset.
Sure, it could rely on luck to get a better understanding, but improvement is still improvement regardless.

In the end, it is, in fact, a strong matter of taste. But this “lazy way of thinking” is working for me on most things.

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Thanks for your work on this excellent post. I’ve been wrestling particularly with the idea of atomic notes being short (3-5 sentences according to the zettelkasten method). But your heuristics make a lot of sense to me i.e. always working towards atomicity but not being constrained by an artificial number of sentences if you can flesh out or give colour to an idea.

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I prefer tagging to linking but my approach could be adapted to either links or tags. It is a relatively recent approach, but after three years of experimenting with various approaches, it has given me the most value in terms of flexibility, recall, and discovery.

I like atomic notes in the sense of atomic concepts, but the degree of atomicity varies. Let’s use this example to demonstrate:

file name: [[public health implications of increasing concentrations of chlormequat chloride]]

Exposure to chlormequat chloride has been linked with issues related to fertility and fetal development. Increasing concentrations of chlormequat chloride in urine have been observed since 2017, raising public health concerns. #agriculture/agricultural-chemicals/plant-regulators #health/public-health

Temkin, Alexis M., Sydney Evans, Demetri D. Spyropoulos, and Olga V. Naidenko. “A Pilot Study of Chlormequat in Food and Urine from Adults in the United States from 2017 to 2023.” Journal of Exposure Science & Environmental Epidemiology, February 15, 2024, 1–5. A pilot study of chlormequat in food and urine from adults in the United States from 2017 to 2023 | Journal of Exposure Science & Environmental Epidemiology.

You can see that I use nested tags. I use tags because I like clicking tags to search for notes and combining multiple tags to narrow the results. I use nested tags because, in my opinion, nested tags are equal to increasing heading levels in an MOC or Wikipedia page. For example:

The nested tag#agriculture/agricultural-chemicals/plant-regulators in an atomic note is equal to the following in a larger, MOC-type note:
# Agriculture
## Agricultural Chemicals
### Plant Regulators

Using the above example, one could even go further in the MOC

# Agriculture
## Agricultural Chemicals
### Plant Regulators
#### Chlormequat
##### Public Health

and the above would be equal to #agriculture/agricultural-chemicals/plant-regulators/chlormequat/public-health

But I prefer to limit my level of depth to three levels.

Anyways, for me atomic is really just a way of understanding (1) how long I want my notes to be (but remaining flexible as I encounter new information on that topic, e.g., the public health implications of the agricultural chemical known as chlormequat); (2) how much hierarchy I impose on my tags.

If I wanted to collect atomic notes into a longer MOC type of note, I could just add the tags according to my desired level of depth as aliases to a given note or do a tag search and drag the desired results into the note.

Just my two cents.