All I know about topology is coming from Numberphile videos of Cliff Stoll (klein bottles, donuts...) and I am learning about network topology only because I'm an electrical engineering student; so I might have got something wrong.
Background
In topology, I know that we can transform a mug into a donut.
There are bus and star topologies in Network Topology. They are, in practice, different:
But when they are drawn in a more abstract way...:
... I start to see a similarity. Like this:
My Questions
1- Does the transformation that I am imagining brake the rules of topology at any step?
2- If it doesn't, can we say that a "star network" is homotopic to a "bus network"?
3- If we cannot say that, does that mean the topology of "network topology" may not be the exact same term used in maths?
Well, no they don't "break" the rules but these network topologies (and I imagine most of them) are graphs that are trees specifically (i.e. they have no loops) and any tree is contractible to a single point. So while they are homotopy equivalent (not homotopic), it's not a particularly useful relation. The word topology here is probably used in the general sense of "shape" or "connectivity".
Your next guess might be to consider homeomorphisms, but the bus example might illustrate why this too is not a useful relation. If I slide two of the connections on the bus so that two nodes meet the bus at the same point, it really represents the same network but they are no longer homeomorphic.
If you really wanted to try and use topology to deal with this, you would maybe have to deal with homotopy equivalences that send distinct nodes to distinct nodes. This might work, but then your "rules" change. As far as I can tell from a cursory Google search, this isn't really done, probably because graph theory could offer better tools anyway.
Basically, homotopy equivalences are too coarse and homeomorphisms might be too fine.