Of course they can, and it's no magic.

If you're spanking new to php/mysql, you might want to consider a few warm-ups first before delving into this. Then, you can tackle them individually as smaller challenges (contact list importer, message system, friends list...)

Alternatively, you can do a web search and find a few tool kits that can help you put one together.

Since there are lots of social networks written on php + mysql already I guess it is possible!

My experience tells me that php is used mostly for small and medium size projects, huge projects are written on other environment.

If you designing a website its specification should be taken into account in order to decide which platform and environment to use.

Let say I am developing a large site for a region let say East Africa can they do?.......

3 "words": Facebook, Yahoo, and a little one called Wikipedia (MediaWiki)

PHP and Mysql can do the job but you have to get up to speed with a lot of skills and tricks...

To run an effiecient social network with php you have to get to an intermediate level because this days there are variety of PHP frameworks(symfony,zend, etc) used for web development for ease of management and expansion.

I would say PHP can tackle any project regardless of its scope but your programming level determines how much.

Sometimesyou find companies implementing php,jsp,java,python and other technologies on same portals. they achieve this feats not necessarily because of limitations but due to scope,requirement and availability.

If your social utility site has to start in less than 6-months then you should consider working with experience php coders and also learn from the project.
Good luck.
Herkym!

I heard that AJAX is good!... I dont understand it well. I need to learn it in two months. I am not sure if it is possible. The project is running out of time!.

AJAX isn't a programming language or a platform, it's a methodology.

You should consult some webmaster for this purpose. Besides this there are many software are also available in market for developing social network.

You'd need at least a small team of PHP developers, and at least 6 months without sleep .