Finite discrete memoryless networks and causal codes

A finite discrete memoryless network (DMN) has a node set $[N]=\{1,\dots ,N\}$ for an integer $N\ge 1$, finite nonempty input alphabets ${\mathcal{X}}_k$ and output alphabets ${\mathcal{Y}}_k$, and one transition probability vector $W(y_1,\dots ,y_N\mid x_1,\dots ,x_N)$ for each input tuple. At each use, conditional on the current input tuple, the output tuple is drawn from this vector independently of earlier uses.

Node $k$ may have an independent uniform message $M_k$ in a finite nonempty set ${\mathcal{M}}_k$ and a destination set $D_k\subseteq [N]$. A length-$n$ causal code consists of encoders $X_{k,i}=f_{k,i}(M_k,Y_k^{i-1})$ and, for each destination $d\in D_k$, a decoder ${\widehat{M}}_{kd}=g_{kd}(Y_d^n,M_d)$; here $Y_k^{i-1}=(Y_{k,1},\dots ,Y_{k,i-1})$, with the empty tuple at $i=1$. The average error is the probability that any required estimate differs from its message. A rate tuple is achievable if a sequence of such codes has error tending to zero and message-set rates tending to at least that tuple; the capacity region is the closure of achievable tuples.