I have this paragraph in this research paper.
Let $\sigma_{k,k′}^{i,j}$ be the sequence of messages $p_i$ sends to $p_j$ during iteration $k$ of $p_i$ and $k′$ of $p_j$.
What kind of sequence $\sigma_{k,k′}^{i,j}$ is? How this mathematical expression is called ?
Is this a double series ?
It is not a double series. This has nothing to do with summation. I wouldn't even call it calculus (therefore I removed that tag). It's rather discrete mathematics.
Therefore I don't think that this expression has a name in mathematics; it's a notation introduced in this paper just to have some notation for a sequence of messages between processes. It has the indices of the sender and receiver processes. It also has indices of in what iterations the messages are sent and received.