The number of criminality is
increase (*grammer* +ing= increasing rapidly throughout the world.
--->doesn't sound natural .
is an uncountable Noun (dictionary meaning : criminal quality of a crime/offence.- like a concept).An offence / committed crimes are countable ,therefore either of them can be used with "The number of.." instead of uncountables.
is used to describe "how fast" something happens. E.g.: The stock prices increased rapidly. If you are not able to give a reference to demonstrate how fast the numbers grow but you still would like to point out the importance/significance of the growth in numbers, you may want to use words like : dramatically , significantly , vastly,considerably,greatly,substantially. I tried to rewrite the same sentence below.I hope it helps.
My suggested version is : The number of violent crimes committed
has increased significantly
throughout the world.