What is a weighted or probabilistic automaton? I'm developing a program that has some entities (things) that are "classified" according to "relevancy". Sort of like search engine (think PageRank).

Therefore, I'm looking to implement an automaton, which I think should handle "value hierarchies". What I mean is that each "next possible state" should have a value that determines how "relevant" it is relative to the state from which are to be moved to it.

Is this called a weighted automaton?
Or a probabilistic automaton?



This is what I want:


(From: http://engineering.flipboard.com/2014/10/summarization/)

To be able to have the states that can be "moved to" have different "relevancy values".



My application reminds of this:


(notice the "sub contexts" centered at the blue texts)

(From: http://shadow.cios.org:2222/www/vcceguide/images/full-screen/vcce-menunew-conceptanalysis-1c3b-tripleconceptnetwork-conceptmenusearch.png)

I think it's an automaton that (in my program) would be going through those "sub contexts" (moving from blue text to blue text, based on user input). But what kind of automaton?
The setting you suggest is not clear enough to determine what kind of an automaton you are looking for.

A short explanation regarding the types of automata:

A weighted automaton is typically an automaton with a weight function on the edges (or states). Then, a run is assigned a weight according to the weight it traverses. There are many different semantics for the value of a run. These include the maximum/minimum weight, the sum, a discounted sum, an average, etc. Finally, nondeterminism is resolved with some semantics. Typically max/min.

The important thing is that a weighted automaton defines a function $f:\Sigma^*\to \mathbb{R}$, instead of a Boolean language (which can be thought of as a function $g:\Sigma^*\to \{0,1\}$).

Probabilistic automata can be thought of as a sort of weighted automata, but a distinction is usually made (mainly because weighted automata are often defined over a tropical semi-ring). In probabilistic automata, the "weights" (probabilities) are multiplied along a run, and the sum of the (accepting) runs is taken. 

If you elaborate or demonstrate the scenario you are thinking of, perhaps I can modify the answer to suggest an appropriate model.