The edge is not a permaculture idea. It's a systems theory concept (permaculture = systems theory + farming, ERE = systems theory + personal finance). I just use permaculture because many are familiar with it.
Good primers on systems theory are:
http://www.amazon.com/Thinking-Systems- ... 1603580557
http://www.amazon.com/Introduction-Gene ... 932633498/
http://www.amazon.com/Permaculture-Prin ... 646418440/
For management stuff, see Peter Senge.
For advanced stuff, see Gregory Bateson.
Books on cybernetics are also useful.
An edge is any discontinuity in a quantity. A table edge is a discontinuity in height. Discontinuities in most systems drive change iff there's a pathway, e.g. the ball can roll off the table and fall down unless a barrier prevents it.
In an ecosystem, as described above, the edge hold species from either side (desert and water) but also new species in the interface (river bank). Mature forests are actually relative deserts. If you want to attract wildlife, like deer, you need to create edges. This is done by planting food. Deer will come from the forest to the field and eat at the edges. The edge attracts/creates a population of deer that doesn't exist in either of the other cultures (forest or field on their own).
Another example of an edge are fronts in weather systems (between high pressure and low pressure). The dynamics manifest itself as rain and wind.
Another example is the shock front of an explosion. Things get destroyed at the front, not behind it(*), nor in front of it.
(*) Unless you want to get really technical!
In academia much of the current focus is interdisciplinary. Again, that's a study in edge interfaces. That is interesting.
In cities, edges are created when artists move into poor neighborhoods and gentrify them.