# Difference between directed community indicators variational parameters in Airoldi et al 2009

In the paper by Airoldi, Blei, Fienberg, Xing, 2008, sometimes I see a distinction between $\phi_{p\rightarrow q}$ and $\phi_{q\leftarrow p}$ and sometimes not (the non-distinction in the generative process) when considering the asymmetric directed graphs.

Does this distinction exist or not?

In other words, in a directed network of $N$ individuals, do we have $N-1$ of $\phi_{p\rightarrow .}$'s and $N-1$ of $\phi_{.\leftarrow p}$'s for each individual $p$?

## 1 Answer

They are certainly distinct: Each $\phi_{p \to q}$ and $\phi_{p \leftarrow q}$ is a variational parameter corresponding to $z_{p \to q}$ and $z_{p \leftarrow q}$. These refer, respectively, to the (unobserved, latent) group membership of $p,q$ when $p$ connects to $q$. From the NIPS paper on the model:

The indicator vector $z_{p \to q}$ denotes the specific block membership of node $p$ when it connects to node $q$, while $z_{p\leftarrow q}$ denotes the specific block membership of node $q$ when it is connected from node $p$.

In other words, the ordering of the nodes in the subscript denotes the direction of the connection: $p$ before $q$ implies $p$ is connecting to $q$. The tail of the arrow faces the node whose membership is referenced.

There exist $2N^2$ of them, which you can see intuitively in Fig. 1: • In the generative process for a link between $p$ and $q$, meaning $p\rightarrow q$, the community indicators drawn are $z_{p\rightarrow q}$ and $z_{q\rightarrow p}$. In the picture however it is $z_{p\rightarrow q}$ and $z_{p\leftarrow q}$. My question is whether $z_{q\rightarrow p}$ is any different from $z_{p\leftarrow q}$ – A.Yazdiha Nov 15 '16 at 9:59
• @A.Yazdiha Why don't you e-mail one of the four authors to ask? – Chill2Macht Nov 18 '16 at 14:28
• @A.Yazdiha Ah, I see I erred in typing my arrows, see my edit. Though, the description in the monks example of the fuller is somewhat at odds with how I interpret the description quoted above, so you may wish to follow William's advice and contact one of the authors. – Sean Easter Nov 22 '16 at 15:09