I don't think you need a network to begin cleaning up dirty corners.
Just pick up garbage you see on the street when you walk around. It has an immediate impact, is easy and doesn't cost anything. Other people may see it and may begin doing that as well. I'm surprised more people don't do.
If you want, you can make a facebook group telling people you're going to be picking up garbage at a certain time, if anyone would like to join.
Garbage on the street has a very negative externality as people are less mindful of tossing garbage if there is already garbage on the street. It also strips people of dignity about where they live and can have other negative side-effects to the neighborhood or even promote lawlessness.