I was reading papers about percolation theory in which I was confused by the expression "self-consistent equation", for example in Temporal percolation in activity-driven networks. I read some references and googled articles. I still do not understand what is meaning of "self-consistent equation", how do I write one, and when should I use one. Any references?
Percolation in random networks can be studied by applying the generating function approach developed in Ref. [31], which is valid assuming the networks are degree uncorrelated. Let us define $G_0(z)$ and $G_1(z)$ as the degree distribution and the excess degree distribution (at time $T$) generating functions, respectively, given by [2].
$$G_0(z)=\sum_k P_T(k)z^k, G_1(z)=\frac{G_0^\prime(z)}{G_0^\prime(1)}$$
The size $S$ of the giant connected component is then given by
$$S=1-G_0(u)$$
where u, the probability that a randomly chosen vertex is not connected to the giant component, satisfies the self-consistent equation
$$u=G_1(u)$$
Another paper:
Random graphs with arbitrary degree distributions and their applications
$H_1(x)$ must satisfy a self-consistency condition of the form $$H_1(x)=xq_0+xq_1H_1(x)+xq_2[H_1(x)]^2+...$$
Suppose we randomly choose a link and find an arbitrary node $u$, by following this link in an arbitrary direction. The probability that the node $u$ has degree $k'$ is
$$\frac{P(k^\prime)k^\prime}{\sum_kP(k)k}=\frac{P(k^\prime)k^\prime}{\langle k \rangle}$$ For this node $u$ to be part of the giant component, at least one of its other $k-1$ out going links (other than the link we first picked) must lead to the giant component.
By calculating this probability, we can write out the self-consistent equation for $x$:
$$x=\sum_k\frac{P(k)k}{\langle k\rangle} [1-(1-x)^{k-1}]$$
