What are Markov chains?

Question

I'm currently reading some papers about Markov chain lumping and I'm failing to see the difference between a Markov chain and a plain directed weighted graph.

For example in the article Optimal state-space lumping in Markov chains they provide the following definition of a CTMC (continuous time Markov chain):

We consider a finite CTMC $(\mathcal{S}, Q)$ with state space $\mathcal{S} = \{x_1, x_2, \ldots, x_n\}$ by a transition rate matrix $Q: \mathcal{S} \times \mathcal{S} \to \mathbb{R}^+$.

They don't mention the Markov property at all, and, in fact, if the weight on the edges represents a probability I believe the Markov property trivially holds since the probability depends only on the current state of the chain and not the path that lead to it.

In an other article On Relational Properties of Lumpability Markov chains are defined similarly:

A Markov chain $M$ will be represented as a triplet $(S, P, \pi)$ where $S$ is the finite set of states of $M$, $P$ the transition probability matrix indicating the probability of getting from one state to another, and $\pi$ is the initial probability distribution representing the likelyhood for the system to start in a certain state.

Again, no mention of past or future or independence.

There's a third paper Simple O(m logn) Time Markov Chain Lumping where they not only never state that the weights on the edges are probabilities, but they even say:

In many applications, the values $W(s, s')$ are non-negative. We do not make this assumption, however, because there are also applications where $W(s, s)$ is deliberately chosen as $-W(s, S \setminus \{s\})$, making it usually negative.

Moreover, it's stated that lumping should be a way to reduce the number of states while maintaining the Markov property (by aggregating "equivalent" state into a bigger state). Yet, to me, it looks like it's simply summing probabilities and it shouldn't even guarantee that the resulting peobabilities of the transitions to/from the aggregated states are in the range $[0,1]$. What does the lumping actually preserve then?

So, there are two possibilities that I see:

• I didn't understand what a Markov chain is, or
• The use of the term Markov chain in those papers is bogus

Could someone clarify the situation?

It really looks like there are different communities using that term and they mean widely different things. From these 3 articles that I'm considering it looks like the Markov property is either trivial or useless, while looking at a different kind of papers it looks fundamental.