Another concept of centrality Many centrality measures count walks of different types going through or starting/ending at a given vertex.
I wonder if the following concept of centrality in directed graphs which reflects a somehow geometric aspect of centrality (i.e. being central) has been considered, defined, and under which  name. The following are my terms.

in-centrality $c_{in}$ of $\nu$ = the length of the shortest shortest path from a
  root node (in-degree $0$, e.g. a sensory neuron) to the vertex (distance from the "incoming periphery")
out-centrality $c_{out}$ of $\nu$ = the length of the shortest shortest path from the
  vertex to a leaf node (out-degree $0$, e.g. a motor neuron) (distance from the "outgoing periphery")

The centrality of a node might be defined as $|c_{in} - c_{out}|$.
For layered graphs with $l$ layers we have trivially $c_{in} + c_{out} = l$ for all vertices.
An interesting characteristic of a graph could be the matrix $A = (a_{ij})$ with
$a_{ij}$ the number of nodes with $c_{in} = i$ and $c_{out}=j$. 
For a strictly layered graph this matrix would look like this:
$$\begin{array}{c|c|c|} 
 & \text{0} & \text{1} & \dots & l-1 & l \\ \hline
\text{0} & 0 & 0 & \dots & 0 &a_{l|0} \\ \hline
\text{1} & 0 & 0  & \dots & a_{l-1|1} & 0\\ \hline
\dots & 0 & 0 & \dots & 0 & 0 \\ \hline
l-1 & 0 & a_{1|l-1} & \dots & 0 & 0 \\ \hline
l & a_{0|l} & 0 & \dots & 0 & 0 \\ \hline
\end{array}$$

Main question: Has this concept of centrality been defined and investigated before? Under which name? References are welcome.

Side question and motivation: I am interested in (an approximation of) this matrix for typical brains (considered as directed graphs). First of all: what is $l$ for typical brains, i.e. the maximal in- and out-centralities? 
 A: This measure is not defined in complete graphs as @kbau pointed out in the comments. However this is somewhat unfair because complete graphs are I think not the subject of the question
I do see some value in this measure so I'll share some of my observations with the hope that I can give you some insights on this measure. I will also define a slightly different measure with some very nice properties that as far as I know is not currently studied but related to this one.
As far as I know this has not been studied as a centrality concept and I will attempt to explain why:
In my opinion it does not measure centrality. It doesn't really work if distances can be arbitrary since  extremely high distance connections (very uninteresting connections) can both increase and decrease it's centrality. thus a node that is far away from the rest of the network and that is connected to one leaf/sink node can be either very central of very not central, completely based on how far that sink/leaf node is, mostly independent from the rest of the network. Thus I think with weighted edges it doesn't really work. 
Defining it with uniform distances works better but then you get problems with situations with more than one root/source and more than one sink/leaf. Why this gives problems is that slightly different graphs have completely different centrality for it's nodes, by just by adding one sink/leaf or one source/root or one arc. In general this is not a good sign because it means your conecpt of centrality is very unstable and hard to build theory on since it can change very easily. It is also very hard to look at general structures for the matrix you are interested in , since tweeking the graph a little completely changes the values. consider adding or substracting a sink/root or source/root.
However
with uniform distances it can give some interesting results. especially if we weigh it on the total number of nodes. first of all it has very low computation time since we can only have to find  shortest paths for the node and the sink and source nodes to the node of interest. If we look at the case with one sink and one source we also get a nice measure of how this graph looks. If we calculate  $ \max \frac{|c_{in}−c_{out}|}{|V|}$ we get an idea about how much this graph behaves like a line; and thus the resilience of the graph in respect to the connectivity of the sink and the source. If $\frac{|c_{in}−c_{out}|}{|V|}$ is close to 1 this means that deleting a small number of edges relative to the total number of edges can mean the source and sink are not connected anymore. While if $\frac{|c_{in}−c_{out}|}{|V|}$  is large this connectivity tends to be very resilient. A nice property is that this can be used to compare very different graphs with this property since the measure is scaled.
In a tree (let L denote the number of leafs) then $\frac{|c_{in}−c_{out}| }{|v|-L}$ denotes a similar notion for the most vulnerable leaf. However this is less general and I am pretty sure it can be improved. 
In general your  matrix is something else entirely. the matrix tends to have many values in the higher left corner if the graph is resilient and more values in the lower right corner if it is less stable. 
