What is betweenness centrality?

The betweenness centrality score of a vertex measures - how well is the vertex positioned on the shortest paths connecting other vertices or, in other words, "in-between" how many vertices is our vertex sitting and - how unique is the position of the vertex with respect to those paths

Consider the network shown on the picture below:

The size ofthe vertices is proportional to the values of their betweenness centrality scores (BC).

First, notice that many of the actors on this network  {1,3,7,9,10} have BC=0. This is precisely because these actors do not belong to any shortest path (hence do not sit between any pair of vertices).

Actor 5 has a modest scores of BC=0.037. He does sit on shortest paths from actor 7 and actors {1,2,3,4}, however his position is not unique, because there are alternative shortest paths from 7 to {1,2,3,4} which bypass actor 5 (these are paths going through actors 6 and 8). the same can be said about position of actor 6.

Finally, actors 2, 4 and 8 are the actors with the highest centrality scores (4 is more central than the other two), because if we are to traverse the network from a vertex on the left part of the diagram to any vertex on the right we have no other option than to pass through (some of) those guys.

[All credits to my co-author Bulat Sanditov, TEM]