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.
Homeless people congregate where there are services that are available to them. Predominantly, this means shelter and food. So homeless people will know when they can get food, from where, and this generally means places with high population density (it's much easier to walk from place to place in the city than in the suburbs!). In response to this, cities have passed a lot of NIMBY homeless shelter laws in their urban centers.
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.
Having done the "voluntarily homeless" thing, I can say that getting a full night's sleep without being harassed by cops with a hard-on for authority, and keeping clean enough to participate in society, are the common issues.
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.