A single-input-single-output communication channel is to be used repetitively. Denote by $X_i \in \mathcal X$ the input at time $i$ and by $Y_i \in \mathcal Y$ the output at time $i$.

Assume the channel is **not memoryless** and $Y_i$ depends on some of the previous inputs and/or outputs in a particular way (that is the same through all $i$). For example, $Y_i$ might depend on $X_i$ as well as $X_{i-1}$ through transition probabilities $P_{Y|X,X'}(\cdot|\cdot,\cdot)$ for all $i$ (where $X'$ represents the input at the previous transmission). For another example, $Y_i$ might depend on $X_i$ as well as $Y_{i-1}$ through transition probabilities $P_{Y|X,Y'}(\cdot|\cdot,\cdot)$ for all $i$ (where $Y'$ represents the output after the previous transmission).

We study the input-output relations of $n$ channel uses. $\{ X_i \}_{i=1}^n$ are the inputs and $\{ Y_i \}_{i=1}^n$ are the corresponding outputs. We represent the dependency relations of these by a graph with $2n$ nodes (one for each $X_i$ and for each $Y_i$), where there is a directed edge whenever there is a dependency. See the examples in the figures below.

[Figure 1][1]

[Figure 2][2]

[Figure 3][3]

Call this graph the *dependency graph* of the channel.

My question is: **Are there works in the literature where channel capacity or other quantities of the channel are studied in connection with the properties of its dependency graph?**


  [1]: https://i.sstatic.net/TM3sl1XJ.png
  [2]: https://i.sstatic.net/pBPyRUBf.png
  [3]: https://i.sstatic.net/6O3n30BM.png