---- > [!theorem] Theorem. ([[percolation by non-uniform removal in the configuration model]]) > Consider a [[network percolation|site percolation process]] on [[configuration model]] in which each node remains functioning with **occupation probability** $\phi_{k}$, where $k$ is the node's [[degree]]. > \ > After percolation, the fraction $S$ of nodes in the [[giant cluster]] out of nodes in the original [[network]] is $\begin{align} S= & f_{0}(1) - f_{0}(u), \text{ } \\ u = & 1 - f_{1}(1) + f_{1}(u) \\ f_{0}(z) = & \sum_{k=0}^{\infty} p_{k} \phi_{k}z^{k} \\ f_{1}(z) = & \sum_{k=0}^{\infty} q_{k} \phi_{k+1} z^{k}=\frac{1}{\langle k \rangle } \sum_{k=1}^{\infty} kp_{k} \phi_{k}z^{k-1} = \frac{f_{0}'(z)}{g_{0}'(1)}. \end{align}$ > [!basicexample] Example. (Hub removal) > [!proof]- Proof. ([[percolation by non-uniform removal in the configuration model]]) > As usual, let $u$ be the average [[probability]] that a node is not connected to the [[giant cluster]] via a particular one of its neighbors. If a node has [[degree]] $k$, then the probability it is not connected to the [[giant cluster]] at all becomes $u^{k}$ and hence the [[probability]] it is connected to the [[giant cluster]] *and is functional* is $\phi_{k}(1-u^{k})$. Now using [[law of total probability]], the [[probability]] of belonging to the [[giant cluster]] is $\begin{align} > S= & \sum_{k=0}^{\infty } \overbrace{p_{k}}^{\mathbb{P}(\text{deg }k)} \overbrace{\phi_{k}(1-u^{k})}^{\mathbb{P}(\text{belong }| \text{deg }k)}= \sum_{k=0}^{\infty} p_{k}\phi_{k} - \sum_{k=0}^{\infty} p_{k}\phi_{k}u^{k} \\ > = & f_{0}(1) - f_{0}(u), > \end{align}$ > where $f_{0}(z)=\sum_{k=0}^{\infty}p_{k}\phi_{k}z^{k}$. Note that $f_{0}$ isn't normalized, for $f_{0}(1)=\mathbb{E}[\phi_{k}]$. > > Now we need to find $u$, the probability of not being connected to the [[giant cluster]] via a particular neighbor. My neighbor is not connected to the [[giant cluster]] if it is not occupied or it *is* occupied but none of its other neighbors are connected to the [[giant cluster]]. Assuming the neighbor has [[excess degree]] $k$, the former occurs with [[probability]] $1-\phi_{k+1}$. The latter occurs with [[probability]] $\phi_{k+1}u^{k}$. So given that my neighbor has [[degree]] $k$, I am not connected to the [[giant cluster]] via it with probability $1-\phi_{k+1} + \phi_{k+1}u^{k}$ and now taking the [[law of total probability]] over $k$ we get $u=\sum_{k=0}^{\infty}q_{k}(1-\phi_{k+1}+\phi_{k+1}u^{k})=1-f_{1}(1)+f_{1}(u),$ > where $f_{1}(z)=\sum_{k=0}^{\infty}q_{k}\phi_{k+1}z^{k}$ and we have used $\sum_{k}q_{k}=1$. We can substitute the expression for [[excess degree distribution]] derived in [[configuration model]] to have $f_{1}(z)= \frac{1}{\langle k \rangle }\sum_{k=0}^{\infty}(k+1)p_{k+1}\phi_{k+1}z^{k}=\frac{1}{\langle k \rangle } \sum_{k=1}^{\infty}kp_{k} \phi_{k}z^{k-1}.$ > We can also write $f_{1}(z)=\frac{f_{0}'(z)}{g_{0}'(1)}$. ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```