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In the course of his life, the subject receive an experience with which he differentiated and perceive the world. Therefore, we can say that there are categories inside its "term name". Here is a primitive "graph" that I made using the tool Figma for the sake of an example that will be clear to everyone:enter image description here

and so, what can we say looking at this example? The fact that the subject, while living his life, drew conclusions about what is dangerous and what is worth being careful of (I added only fire, blades and weapons - I think it is obvious that in the real world there would be much more meanings in this category), just like he made conclusions about who his family members are. And now the question is: what are the names of such graphs/systems? I know about the axiomatic, formal and conceptual system, but none of them is the same as I described in the question, or do I not understand something?

Aside: While writing this question, I also asked myself other questions: "What can a person not differentiate / categorize?", but I think this is a topic for a separate question.

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You ask about the internal representation within a human of his/her experiences with the external world. An internal relation can be named a model. Accordingly your scetch shows the self-model (the bubble named “i”) together with the world-model of the person, and the embedding of the self-model into the world-model.

This approach has been elaborated by Thomas Metzinger: Being No One: The Self-Model Theory of Subjectivity. The books deals with neuroscicence and the philosophy of mind. It is rather challenging, though its many additional popularizations by the author.

If you ask for a name of your graph: It is an undirected graph. The graph from your Figa-scetch is too unspecific to qualify for more attributes from graph theory.

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