Preferential Attachment and salton similarity in directed networks

Question

Preferential Attachment similarity between two nodes in an undirected graph is the degree of the first node multiplied by the degree of the second node. But what about directed graphs? Which degree should one use? In-degree or out-degree?

Edit: although my question is regarding PA, but I'm also concerned about other similarity indices like salton index. Salton index is defined as the number of common neighbors between two nodes divided by the square root of the multiplication of the degrees of the nodes. But the question, what degree to use for directed networks?

$k_x$ and $k_y$ are the degrees of $x$ and $y$. The numerator is just the number of common neighbors between the two nodes. As you can see the same issue here, which degree to use in case of directed graphs?

Answer 1

The simple answer to this is that you would use \begin{equation} s_{x,y}^{PA} = k_{out, x} \times k_{in, y} \end{equation} which corresponds to situation B in the paper linked by Barry Cipra ("Directed Scale-Free Graphs" by Bollobas, Borgs, Chayes, and Riordan), where you choose a source, $x$, and a target, $y$, based on their out- and in-degrees.

Note that this is not necessarily the case if for instance the source node is fixed, which happens in link prediction. Then you could argue that the preferential attachment score only depends on the target in-degree, situation A from the linked paper.

Answer 2

You might take a look at "Directed Scale-Free Graphs" by Bollobas, Borgs, Chayes, and Riordan.