On a lighter note, here is a very interesting blog piece, bringing some sort of similarity to the giant component behaviour on a pile of staple pins. This is sort of a layman view of the Erdos Renyi explanation.

In one version of the Erdős-Rényi process, one start with a set of n isolated vertices and then keep adding random edges one at a time;  More specifically, at each stage one choose two vertices at random from among all pairs that are not already connected, then draw an edge between them. It turns out there’s a dramatic change in the nature of the graph when the number of edges reaches n/2 . Below this threshold, the graph consists of many small, isolated components; above n/2, the fragments coalesce into one giant component that includes almost all the vertices. “The Birth of the Giant Component” was later described in greater detail in an even bigger paper–it filled an entire issue of Random Structures and Algorithms (1993, 4:233–358)–by Svante Janson, Donald E. Knuth, Tomasz Luczak and Boris Pittel.

Ofcourse Joel Spencer has written a beautiful article on AMS commemurating the golden anniversity of the giant component.