top of page
Search
  • johnmcgaughey255

Hierarchical structures pt. 1

Updated: Mar 29, 2021

Imagine a string, and at the end of the string are two different strings, and at the ends of those two different stings, there are two more attached to each one. These strings go on for a while, with each layer having a number of strings double the previous, exponential. The exact number of strings is not really important for this thought experiment. The set of strings, and the patterns within the stings are the most easy to interpret if we pick the one at the top and dangle the rest from there. We can look at this arranged pattern of strings, where there are more and more strings per layer, and it makes sense what the pattern is. Lets say, instead of deciding to dangle the string from the top string, we choose one from a layer a bit further down. When we hold this string at the top and let the rest of the structure dangle down, it is harder to make sense of the structure as a whole as it dangles down.

Imagine this same structure, but each of the strings is numbered according to its layer. The top string is labeled '1', both of the two strings on the second layer are labeled '2', all four of the strings on the third layer are labeled '3', and so on exponentially. Now, it would make even more sense when we dangle the structure from the top string. We could see order in this structure, as the layers go down, the numbers go up, this can be viewed as a hierarchy. Notice how we can choose to dangle this structure of strings however we like. The structure itself is set in stone, it cannot be changed for the purpose of this thought experiment. I choose to use strings because it was in my mind the easiest way to portray this idea. This structure, this hierarchy, of strings cannot be cut or broken, ripped or teared, but we have the ability to move it around, and select how we want to dangle it. Let's say that instead of picking it up at the string labeled '1' and dangling it from there, we pick up a string labeled '5' or '6', which would be on the 5th or 6th layer. Notice that it still forms a pyramid like structure, but the numbers don't make as much sense now.

You'll have to stick with me through this part. Lay this structure of strings out on a table, and make sure that the top sting, labeled '1', is at the top of this hierarchy. Give up this notion of numbers corresponding to layers, but keep the basic idea of order when dangled a certain way. Take on the idea that each one of these strings has some meaning attached to it, some value of truth. To understand the value of a string at a given layer, you must know the value of the string that it is attached to. The value of a string on layer L is derived from the value of the string right above it on layer L+1. Also, understand that knowing the value of L+1 means you now have access to all of the values of strings directly under it. Let us say that this structure, laid out in order like the labeled example, has a number N of total layers. The idea of picking up a sting and dangling it, should be translated as acceptance of the truth at that given layer. Each string on a given layer has both truths governing it, and provides derivation for truths in subsequent layers. If we take this structure from the 'top' layer (L+(N-1)) and dangle it, every value of the strings can be properly understood, because it is arranged orderly.

Now imagine some capacity factor, some limiting factor for how many strings we can hold up. Let's think of that factor as weight, to reinforce this idea of a physical structure. Let's also go back to the numbered example for simplicity. Each string has a weight of 1 kg. When we do lift this structure off of the ground, the last layer, where there are no more layers connected, is the only layer that can remain on the ground. This last condition is a must have. Man I really wish I could type math in here, but I don't know how to. The weight of the structure as a whole is 1kg + 2kg + 4kg + 8kg... where each term is a layer. Lets say we have 10 layers, the total weight would be 1023kg. Let us say that we get a maximum strength that allows us to lift 200 kilograms. We can only understand that we lift off of the ground. In the final layer, labeled '10', there are 2^9 or 512 different strings. We know that we can ignore this weight, because it is allowed to be left on the ground, so the weight of the structure that we can pick up is 511 kilograms. We can distribute this power to lift 200 kg however we want to across this structure. We could choose to lift many different strings if we want to, but we have to make sure that we do not exceed our limit of 200. We could also choose only one or two to lift across the structure, these would be in a layer closer to the top. Notice that choosing only one or two to dangle the rest from would provide a nice looking, and understandable structure, but it might not cover a wide enough domain. As we can see, this turns into a somewhat optimization problem although it is not entirely evident what we are optimizing.

Again, give up the numerical connections to these strings, and connect the idea of truths. To understand a string is to accept the value of the string above it. And when we arrange these strings in the 'best' order to arrange them. Like the numbers, we can follow a line of reasoning down the hierarchy to the last layer, and every layer is according to the layer above it. Again, we need to have this concept of a capacity. Just as there was a limit to how much you could lift in the numerical example, there is a capacity to how much we can understand and reason. The same 10 layer hierarchy is used in this thought experiment, and most of the same principles still apply. We can choose how to distribute these 200 strings however we want, but only 200 of them can be off of the ground. We can either choose many shallow explanations, or a few deep ones, or anywhere in-between.


7 views0 comments

Recent Posts

See All

Meta-heuristic optimization

These are all drafts, by the way. They are not meant to be perfect or to convey all the information I wish to convey flawlessly. My blogs are just a way for me to get ideas and my thoughts realized as

Dimensionality reduction and model interpretability

I would say that the main purpose of communication is to give a universal understanding of abstract ideas. An abstraction is, for my intents and purposes, a lower dimensional encoding of a higher dime

bottom of page