Distance measure in a weighted, fully connected network

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Christian
Christian 2013-8-6
Hey,
I have a network where the adjacency matrix (A) is a transformation of the correlation coefficients and self-edges are removed
A=0.5*(correlation + 1)
Now I need a distance measure to calculate betweenness and thought about transforming it to a distance d by
d=(1-A)^2
How can I check whether the triangle inequality holds for this distance? All other conditions of a metric are satisfied, but I'm unsure how to prove this. I need the distance to calculate betweenness scores for the network and the applied distance measure spreads the distribution of distance so that I get results other than 0.
However, my problem of proving whether the triangle inequality holds remains.
Thank you, Chris

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