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Ben McKay
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what What is a self-consistent equation in percolation theory

I was reading papers about percolation theory in which I was confusing about selfconfused by the expression "self-consistent equation. And those papers are discussion about percolation theory.

equation", for example in Temporal percolation in activity-driven networks, and I am confusing about self-consistent equation. After 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. Thank you 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 G0(z)$G_0(z)$ and G1(z)$G_1(z)$ as the degree distribution and the excess degree distribution (at time T$T$) 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, S, 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 $$H_1(x)=xq_0+xq_1H_1(x)+xq_2[H_1(x)]^2+...$$



The simplified self-consistent probabilities method for percolation and its application to interdependent networks

Suppose we randomly choose a link and find an arbitrary node, u $u$, by following this link in an arbitrary direction. The probability that the node, u, $u$ has degree k'$k'$ is

$$\frac{P(k^\prime)k^\prime}{\sum_kP(k)k}=\frac{P(k^\prime)k^\prime}{\langle k \rangle}$$ For this node u$u$ 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$x$:

$$x=\sum_k\frac{P(k)k}{\langle k\rangle} [1-(1-x)^{k-1}]$$

what is self-consistent equation in percolation theory

I was reading papers in which I was confusing about self-consistent equation. And those papers are discussion about percolation theory.

Temporal percolation in activity-driven networks, and I am confusing about self-consistent equation. After I read some references and googled articles. I still do not understand what is meaning of, how do I write one, when should I use. Thank you. Any references?



Percolation in random networks can be studied applying the generating function approach developed in Ref. [31], which is valid assuming the networks are degree uncorrelated. Let us define G0(z) and G1(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 of the giant connected component, S, 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 $$H_1(x)=xq_0+xq_1H_1(x)+xq_2[H_1(x)]^2+...$$



The simplified self-consistent probabilities method for percolation and its application to interdependent networks

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:

$$x=\sum_k\frac{P(k)k}{\langle k\rangle} [1-(1-x)^{k-1}]$$

What is a self-consistent equation in percolation theory

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 $$H_1(x)=xq_0+xq_1H_1(x)+xq_2[H_1(x)]^2+...$$



The simplified self-consistent probabilities method for percolation and its application to interdependent networks

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 $x$:

$$x=\sum_k\frac{P(k)k}{\langle k\rangle} [1-(1-x)^{k-1}]$$

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Nick Dong
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Nick Dong
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what is self-consistent equation in percolation theory

I was reading papers in which I was confusing about self-consistent equation. And those papers are discussion about percolation theory.

Temporal percolation in activity-driven networks, and I am confusing about self-consistent equation. After I read some references and googled articles. I still do not understand what is meaning of, how do I write one, when should I use. Thank you. Any references?



Percolation in random networks can be studied applying the generating function approach developed in Ref. [31], which is valid assuming the networks are degree uncorrelated. Let us define G0(z) and G1(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 of the giant connected component, S, 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 $$H_1(x)=xq_0+xq_1H_1(x)+xq_2[H_1(x)]^2+...$$



The simplified self-consistent probabilities method for percolation and its application to interdependent networks

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:

$$x=\sum_k\frac{P(k)k}{\langle k\rangle} [1-(1-x)^{k-1}]$$