Markov chain probability in a complete graph I'm having a hard time understanding the following. Assuming we have a complete graph with $n$ vertices, where initially, one of those has an information. At this point, all we care about is that in each time $t$ a node that has the information chooses randomly one of it's neighboors and sends it the info, only if the receiver does not have it. We don't care how the sending is done.
Let $x_t$ be a random variable representing the number of nodes that have the info at the time $t$. It is indicated that $x_t$ can be modeled by a homogeneous increasing Markov chain. So, if at $t$, if $k$ vertices have he info, then the probability of another vertex receiving the information (using population protocols), that is, to have $k+1$ vertices know it is $\frac{2k(n-k)}{n(n-1)}$. Can someone explain why this stands?

The link to the paper I got this from is this: https://drive.google.com/open?id=0B0RiUAwpE4vnTlFqWmRHdlRYSFp4aDlYcWE5dWxUdWg2ZDA4
and the transition graph of the chain is shown in Figure 2.
The probability that you state is the probability that a random edge of the complete graph connects one of the $k$ nodes to one of the other $n-k$ nodes. This means that in your process, a random edge is chosen, and if exactly one vertex already knows the information, then the other vertex is made aware of it. Information spreads exactly when the chosen edge is of this form, which in your case happens with the probability you mention.