$4\mathrm{.}2.$ Friendship Paradox

The degree distribution pk

expresses the probability that a randomly

selected node has k neighbors. However, if we randomly select a link, the

probability that a node at one of its ends has degree k is qk

$=$ Akpk

$,$ where A

is a normalization factor.

$($a$)$ Find the normalization factor A$,$ assuming that the network has

a power law degree distribution with $2<<3,$ with minimum

degree kmin and maximum degree kmax.

$($b$)$ In the configuration model qk

is also the probability that a ran$-$

domly chosen node has a neighbor with degree k$.$ What is the av$-$

erage degree of the neighbors of a randomly chosen node?

$($c$)$ Calculate the average degree of the neighbors of a randomly cho$-$

sen node in a network with N $=104,=2\mathrm{.}3,$ kmin$=1$ and kmax$=1,000.$

Compare the result with the average degree of the network, k$.$

$($d$)$ How can you explain the "paradox" of $($c$),$ that is a node's friends

have more friends than the node itself?