Topology is the study of relationships between objects in a network. It is a subset of set theory, which is an aspect of ontology.
More precisely, you are describing a ‘sociogram’.
So far as I understand, your question doesn't actually ask for anything that I would consider to concern the resolution or building of puzzles, but we could easily devise a form of logic puzzle which presented you with various people, qualified as nodes in a network, and their relationships to other people.
As an example, let's say that you are given a list of people. With each person you are given one typified relationship to one other person in that same list. You are tasked to classify each person in one of two sets: whether a person is eligible or ineligible to wed a select person X.
First, you would need to construct a topological graph — i.e. a sociogram — which allowed you to deduce the unknown relationships between persons.
Then, based on whatever rules the puzzle states for us as to what criteria make two people ineligible to wed, the puzzle can be solved if you can map every node to one and only one of those two classes for any arbitrarily selected person X — or, at the very least, to a only one certain person X given in the conditions of the puzzle.
Because I didn't really answer your question, I'll list where further information can be obtained by researching:
- General topology
If you research ‘network topology’ you are more likely to encounter the specialist usage of the word ‘network’ — pertaining to computer and signal technology — rather than the generalist, mathematical one.
- Graph theory
- Knot theory
- Mathematical puzzles
- Disentanglement, a.k.a. topological, puzzles
E.g. the renowned tavern puzzles of topologically interlocked pieces of forged iron or bronze.
- “Three utilities problem”
You have three houses arranged as points on a straight line. On a parallel Euclidean line are three other points representing Water, Electricity, and Gas — or Heating Oil or Steam or whatever you please. Can each house be connected to each utility service by pipes and wires which neither cross nor leave the plane containing the two parallel lines?