Valued graphs are much more complicated to analyze than simple 1/0 links. For betweenness, you are basically asking how removing a node would affect the distances among all the other nodes, which is easier to conceptualize when links are either present or absent.

My intuitive feelings is that a simple 1/0 directed graph would not be much different from a symmetrical tie, since the calculation of distances is still straightforward.

High betweenness centrality means that the node would be on shortest path between any pair node in the network. Since its valued graph, it will not count as a simple graph (0/1), the shortest path will become the cheaper cost to travel.  So it is clear in your case that high betweenness centrality will easily visible 

Andry suggestion of the analogy to travel cost (i.e., the traveling salesman problem) is useful. It suggests that for a valued network, betweenness should correspond to the idea that removing a point will create the greatest increase in travel distance for the remaining nodes.

There are two full Betweenness measures (many random walk derivatives) -- One is the original by Freeman(1979) which works with only non-directed networks.  The second is an adaptation of the first by Borgatti and White specifically for directed networks -- http://www.steveborgatti.com/research/publications/borgatti%20-%20Betweenness%20centrality%20measures%20for%20directed%20graphs.pdf

Since Borgatti is the developer of Netdraw and UCInet this directed network betweenness option should be in one/both of the software packages!  Check the help files or call user support.  We include both versions Betweenness in InFlow.  

Hope this tutorial (including the source code details) will work... 

https://youtu.be/l7_RNYay1qM

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