Problem $1$: Percolation on complex networks Consider a random uncorrelated complex network, completely defined by its degree distribution P$($k$).$ Le us define u as the probability that a randomly chosen edge does not point to the infinite percolation cluster, and P

$\backslash$infty

$$

$($p$)$ as the order parameter, that is$,$ the probability that a randomly chosen belongs to the percolating cluster. $(1)$ The existence of a giant component, scaling as the size of the network, can be shown to be given by the Molloy$-$Reed criterion,

$$k$$

$$k

$2$

$$

$$

$>=2.$ In a network percolation processes, the infinite percolating cluster can be identified as the giant component of the subgraph formed by the occupied nodes and their mutual connections. The degree distribution P

p

$$

$($k$)$ of this subgraph can be estimated as follows: consider a node in the giant component that, in the original network has degree k

$0$

$$

; it will have degree k in the giant component if k among its original k

$0$

$$

neighbors are also occupied, each with probability p$.$ Considering the binomial distribution, prove that P

p

$$

$($k$)=$

k

$0$

$$

$=$k

$\backslash$infty

$$

P$($k

$0$

$$

$)($

k

$0$

$$

k

$$

$)$p

k

$(1$p$)$

k

$0$

$$

$$k

$.$ Using the Molloy$-$Reed criterion, compute the percolation probability p

c

$$

$.$ To do so$,$ exchange the order of the summation variables.Problem $1$: Percolation on complex networks Consider a random uncorrelated complex network, completely defined by its degree distribution P$($k$).$ Le us define u as the probability that a randomly chosen edge does not point to the infinite percolation cluster, and P

$\backslash$infty

$$

$($p$)$ as the order parameter, that is$,$ the probability that a randomly chosen belongs to the percolating cluster. $(1)$ The existence of a giant component, scaling as the size of the network, can be shown to be given by the Molloy$-$Reed criterion,

$$k$$

$$k

$2$

$$

$$

$>=2.$ In a network percolation processes, the infinite percolating cluster can be identified as the giant component of the subgraph formed by the occupied nodes and their mutual connections. The degree distribution P

p

$$

$($k$)$ of this subgraph can be estimated as follows: consider a node in the giant component that, in the original network has degree k

$0$

$$

; it will have degree k in the giant component if k among its original k

$0$

$$

neighbors are also occupied, each with probability p$.$ Considering the binomial distribution, prove that P

p

$$

$($k$)=$

k

$0$

$$

$=$k

$\backslash$infty

$$

P$($k

$0$

$$

$)($

k

$0$

$$

k

$$

$)$p

k

$(1$p$)$

k

$0$

$$

$$k

$.$ Using the Molloy$-$Reed criterion, compute the percolation probability p

c

$$

$.$ To do so$,$ exchange the order of the summation variables.