We often say something/someone plays a ... role in something. But what if the subject is in a plural form? For example dogs. Can I say "dogs play a big role in the security" or should I say "dogs play big roles in the security" In some article, people even say "dogs play big role" Which one is correct or more naturally used?
They are both correct, but have different meanings.
"Dogs play big roles..." <-- This says that there are multiple different and important roles that are fulfilled by dogs in general.
"Dogs play a big role..." <-- This means that dogs in general play a common important role.
When you use 'plural', you still count them as one entity, don't you? That said, if one dog plays a big role in securing something, many dogs collectively play the same role.
Thus, it's fine to say...
Dogs play a big role in...
As far as your link is concerned, it's headlinese where authors take all freedom to eat up the articles! :)