Giant component and phase transition

Percolation is the simplest fundamental model in statistical mechanics that exhibits phase transitions signaled by the emergence of a giant connected component (or GCC, it is a connected component that contains a finite fraction of the nodes as the network size $N \rightarrow \infty$, i.e. it has an "extensive" scaling, in the language of Statistical physics). The parameter that controls the existence of a GCC is the occupation probability, $p$ (or the "attach probability" $q=1-p$), the critical value at which the transition happens is called the percolation threshold

In particular, the transition is often a continuous transition (2nd oder) with a critical point. Behaviour at this point is thus an example of Critical phenomena, and at this point the system is self-similar (see Fractals), and as a consequence, many quantities follow Power laws. See section 12.2, and exercise 12.12, as well as exercise 2.13 of this book.

Percolation threshold

For random site percolation on a configuration model graph:

$p_c = \frac{1}{g'_1(1)} = \frac{\langle k \rangle}{\langle k^2 \rangle - \langle k \rangle}$

where $g_1(z)$ is the generating function of the excess degree distribution. See Newman's book.

Note that even if there is a GCC, its size may be small, so a full understanding of the network's resilience should include the dependence of the size of the GCC with $p$.

Critical phenomena in percolation