Time homogeneous Markov chain for Axelrod's model I am reading paper Axelrod's model of dissemination of culture , I am unable to understand the transition probabilities of time homogeneous Markov chain for this model. Can some one please explain it clearly how the expression 
$$ p_{ij} = \frac{1}{L^2 h_{ij}} \sum_{r\in H_{ij}} \frac{n_{kr}}{f} \frac{1}{f-n_{kr}} \qquad  (i \neq j)$$
is written in the paper.
 A: As described on your link, we are trying to compute the probability that we will transition between two fully described states.  So, by $i$ and $j$ they mean a complete specification of all of the attributes for each of the agents.
Now, the probability that we will transition is determined by the rules of the simulation.  In each time step we (1) choose a random agent ($k$).  (2) choose a random neighbor ($r$), (3) decide whether they interact, with probability $n_{kr}/f$ (their cultural similarity), and (4) if they interact, have $k$ adopt one of $r$'s features that differ (of which there are $f - n_{kr}$ options) at random and adopts its trait.
So, assuming we are looking at a particular complete specification $i$ and $j$ that differ only in a particular agent, $k$ having changed the trait of one of his features, the only way we could have gotten from $i$ to $j$ in one step is if we (1) were lucky enough to choose the $k$ guy out of all of the guys on the lattice.  This happens with probability $1/L^2$.  (2) we choose a neighbor that had the correct trait for the feature of interest.  Here we don't necessarily know how many such neighbors there are, but let's denote the number of neighbors we could steal the train in question from $h_{ij}$ and use $H_{ij}$ to represent the set of neighbors, and $N$ to denote the total number of neighbors.   (3) Even if we select one of the correct neighbors, they have to interact, which only happens with probability $  n_{kr} / f $. (4) Even if they interact, we need to choose the right feature to copy, since this is a choice of the correct one out of $ f- n_{kr}$ options, we will make this choice with probability $1/(f - n_{kr})$.
So, schematically we have
$$ \begin{align*}
P_{\text{state}_i, \text{state}_j}  &= P(\text{ choosing agent } k) \times P(\text{ choosing valid neighbor}) \\ & \quad \times P(\text{they interact}) \times P(\text{choosing the right feature})  \\
 &= \left( \frac{1}{L^2} \right)\left( \frac{1}{N} \sum_{r \in H_{ij}} \right) \left( \frac{n_{kr}}{f} \right) \left( \frac{1}{f - n_{kr}} \right) \\
 &= \frac{1}{L^2 N} \sum_{r\in H_{ij}} \frac{ n_{kr}}{f} \frac{ 1 }{ f - n_{kr} } 
\end{align*}
$$
which is as the page reports, though I think they may have goofed, because in order to have a transition, we need to select one of the neighbors that have the shared trait, and this will depend on the number of neighbors $N$, so I think they accidentally have $h_{ij}$ instead of $N$ in their transition probabilities.
