The Information Flow Problem on Clock Networks

The information flow problem on a network asks whether $r$ senders, $v_1,v_2, \ldots ,v_r$ can each send messages to $r$ corresponding receivers $v_{n+1}, \ldots ,v_{n+r}$ via intermediate nodes $v_{r+1}, \ldots ,v_n$. For a given finite $R \subset \mathbb{Z}^+$, the clock network $N_n(R)$ has edge $v_iv_k$ if and only if $k>r$ and $k-i \in R$. We show that the information flow problem on $N_n(\{1,2, \ldots ,r\})$ can be solved for all $n \geq r$. We also show that for any finite $R$ such that $\gcd(R)=1$ and $r = \max(R)$, we show that the information flow problem can be solved on $N_n(R)$ for all $n \geq 3r^3$.