This question concerns a method of drawing graphs and a graph characteristic about which I want to learn more. Consider a connected directed graph with at least one node with in-degree 0 and one node with out-degree 0 (let's call it *input-output-graph*). Draw the input nodes equally spaced on layer 0: [![enter image description here][1]][1] Each node has a well-defined minimal distance $d_{in}$ to the input layer 0. Draw the nodes with $d_{in}=n$ on layer $n$: [![enter image description here][2]][2] Now draw all the edges that connect adjacent layers: [![enter image description here][3]][3] And finally, draw all the other edges: [![enter image description here][4]][4] Each edge $(v_0,v_1)$ has a unique and well-defined length $\lambda$ with respect to the layers it connects. Let $v_0$ be a node on layer $n_0$ and $v_1$ a node on layer $n_1$. $$\lambda((v_0,v_1)) = n_1 - n_0$$ Note, that by construction there cannot be an edge with $\lambda(e)>1$ The characteristic I have in mind is nothing but the distribution $d$ of the lengths $\lambda$: $$d(l) = \text{number of edges $e$ with $\lambda(e)=l$}$$ It is obvious that for strictly layered input-output-graphs (esp. trees) $d(1) = \text{number of edges}$ and $d(l)=0$ für $l\neq 1$. But what about small-world or random input-output-graphs? Would the characteristic be a good one to distinguish them? > **Question**: Has this characteristic been defined before, and under which name? [1]: https://i.sstatic.net/F2rkv.png [2]: https://i.sstatic.net/6LAzT.png [3]: https://i.sstatic.net/HV7sm.png [4]: https://i.sstatic.net/g1iCK.png