Do you have any evidence to support your begging theory? I would guess that the homeless go where they can get shelter without being arrested. But I haven't checked the research.
In Atlanta, there is a homeless shelter that is being closed down (it's currently in court) and the shelter can't relocate within the city because of laws on the size of homeless shelters. They were grandfathered in when the law was passed.
Source: I've spent a significant amount of time working with homeless shelters and on homeless policy.