I wonder if the following statistical description of the layer architecture of finite graphs has been considered before and where I can find some references (especially under which name).

Consider a huge (but finite) directed graph with a big set of roots $\{\rho_j\}$ having in-degree 0.

Let $d(\nu_1,\nu_2)$ be the length of the shortest path from node $\nu_1$ to node $\nu_2$ and $\infty$ when there is no path from $\nu_1$ to $\nu_2$.

A graph has a **strictly layered** architecture when for each node $\nu$ there is a unique number $\mathcal{l}$ (the level of the node) such that $d(\rho,\nu)=\mathcal{l}$ or $=\infty$ for all root nodes $\rho$. Call the set of all nodes having level $\mathcal{l}$ the **layer** $\mathcal{l}$.

Strictly layered graphs prohibit "short cuts" from the root layer (layer 0) to a given node $\nu$, but permit different paths from a root node $\rho$ to $\nu$. Furthermore they permit inner layer connections and arbitrary cycles (because these can be ignored when considering shortest paths).

^{Note that inner layer connections or cycles may possibly occur only inside or between a few layers, i.e. only inside layer 3 or by nodes from layer 4 being linked back to nodes from layer 2. You may notice that I have neural networks in mind.}

For a given node $\nu$ let $p(x)$ be the number of root nodes $\rho$ such that $d(\rho,\nu)=x$. Normalize such that $\sum_x p(x)=1$, making $p(x)$ a probability distribution. Call $p(x)$ the **layer distribution** of $\nu$. Note that $0 \leq x \leq L$, the maximal distance in the graph.

Consider a set $\mathcal{L} = \{f_{\sigma_i}^{l_i}(x)\}_{0\leq i < n}$ of normal distributions with mean $l_i$, $0 \leq l_i \leq L$, and standard deviation $\sigma_i$. Call $\mathcal{L}$ a **layer scheme**.

When for each node $\nu$ of a given graph $G$ the layer distribution $p(x)$ overlaps strongly enough with one of the elements/distributions of the layer scheme $\mathcal{L}$, one could say that the layer scheme $\mathcal{L}$ reflects the (fuzzy) layer structure of the graph $G$.

This last part is intentionally left vague because I don't know yet how to specify "overlaps strongly enough" and "reflects the fuzzy layer structure". But I am confident, that both can be made precise and quantified.

What seems clear to me (but maybe I'm wrong) is:

being "strictly layered" means "having a layer scheme" $\mathcal{L}$ with all standard deviations $\sigma_i = 0$

there is an ordering of possible layer schemes $\mathcal{L}$, making some better than others

there are optimal layer schemes describing a given graph

### Questions

- Has this approach been taken before?
- Under which name?
- References?