----- > [!proposition] Proposition. ([[percolation in configuration model with exponential distribution]]) > [!proof]- Proof. ([[percolation in configuration model with exponential distribution]]) > **Uniform removal.** > The [[network]] has a [[exponential random variable|geometric distribution]] $p_{k}=(1-a)a^{k}$, where $a<1$ and $1-a$ ensures normalization. Then, as discussed in [[exponential degree distribution configuration model]], we have $g_{0}(z)=\frac{1-a}{1-az}, \ \ g_{1}(z)=(\frac{1-a}{1-az})^{2}.$ > Then by [[percolation by uniform removal in the configuration model]] the average [[probability]] that a node $u$ is not connected to the [[giant cluster]] via a particular neighbor is $u=1-\phi + \phi g_{1}(u)=1-\phi + \phi (\frac{1-a}{1-az})^{2}$ and rearranging we find $u(1-au)^{2} - (1-\phi)(1-au)^{2} - \phi (1-a)^{2} = 0.$ > This is a cubic equation $u$; using [[synthetic division]] we factor as $(u-1)[a^{2}u^{2} + a(a\phi - 2)u + \phi - 2a \phi + 1] = 0.$ > Using the [[quadratic formula]] it is clear that one of the two remaining solutions is greater that 1 for $a<1$ and so cannot be a [[probability]]. The other must be $u$: $u=a^{-1} - \frac{1}{2}\phi - \sqrt{ \frac{1}{4}\phi^{2} + \phi(a^{-1} - 1) }.$ > Now we can plug this into [[percolation by uniform removal in the configuration model|the expression for]] $S$ thus: $\begin{align} > S= & \phi \left[ 1- \frac{2(a^{-1} - 1)}{\phi + \sqrt{ \phi^{2} + 4\phi (a^{-1} - 1) }} \right] \\ > = & \phi\left[ 1- 2(a^{-1} - 1) \frac{\phi - \sqrt{ \phi^{2} + 4 \phi (a^{-1} - 1) }}{\phi^{2} -(\phi^{2} + 4 \phi (a^{-1} - 1))} \right] \\ > = & \frac{3}{2}\phi = \sqrt{ \frac{1}{4} \phi^{2} + \phi(a^{-1} - 1) }. > \end{align}$ > The percolation threshold is $\phi_{c}=\frac{1-a}{2a}.$ > > > ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY 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 > ```