I am frustrated of looking for a simple explanation of this term $\phi$-divergence, but I cannot find any. Therefore I would be really grateful if somebody could introduce a reference or write a definition (even qualitative for it). Here is the paper that I first noticed the term in:
Jiménez, R., and J. E. Yukich. "Statistical distances based on Euclidean graphs." Recent Advances in Applied Probability. Springer US, 2005. 223-239.