# Identifying middlemen in a social network

I have a data set and I want to examine an hypothesis in there and probably Network Analysis should prove or reject my theory.

I have a list of products and a group of people who give the product to each other. Anyone can sell it to another in any price he want (hopefully someone will buy it).

So, imagine a chain of products transfers between the people. Except the sell price, one will also take a percentage of the sale only of the next one who will re-sell the product. And also, if I sell for the first time a product, I will take a percentage forever for every sale.

Example:

A -> B -> C

Person A will take the money from sale to Person B. Person B will take the money of the sale to person C and Person A will take a percentage of this sale as the previous owner and the creator of the chain. If person C sell the product to someone else, only person B will take the previous percentage, but person A will take the percentage of the creator.

I believe that there are people that try to exploit the system and create small or bigger circles to take advantage of these percentages of next sales to "hide" themselves.

Is there any way to identify those people?

The closest concept I can think of in network science is betweenness centrality. Suppose you're interested in studying some node $u$. First, for every pair of nodes in your network (excluding $u$), you identify all possible shortest paths. Then, you count how many of them are passing through $u$. Some variants normalize this number in various ways, but, basically, that's it. The highest the value, the more central $u$.

In your case, there's some kind of attenuation effect, so maybe you should also look at the matrix-based centralities, such as the Eigenvector centrality, Katz centrality, PageRank centrality, Alpha centrality, HITS scores, and there are certainly others. With Katz centrality, in particular, the centrality of a node depends mainly on the centrality of its 1st order neighbors, then a bit less of its 2nd order neighbors, and so on.

You can find tens of centrality measures in the literature. I'm not sure if one of them corresponds exactly to what you're looking for, but it's certainly worth having a look there, first. There's a classic paper by Freeman, and more recent reviews, such as (Landherr et al. 2010).

There is a paper discussing the same problem:

From its abstract:

[...] A system is topocratic if the compensation and power available to an individual is determined primarily by her position in a network. [...] individual compensation is based on the number of shortest paths that go through them in the network. [...] the distribution of payoffs is meritocratic only if the average degree of the nodes is larger than a root of the total number of nodes.

So in short:

• the quantity you are asking is betweenness centrality (as Vincent have already mentioned),
• you can exploit only some systems, where there are not many connections.