Consider the model of "Growing Random Networks" described in lecture 3.1 (so the first model where links are added to existing nodes uniformly at random), with m=10.
At date t=40, what are:
(1) the expected degree for node 15 born at i=15, and
(2) the expected degree for node 30 born at i=30?
[Hint: Use the approximation for the expected degree at time t of a node born at time i, from the Lecture 3.1. Also, it helps to think about which of the two nodes has a higher expected degree.]
(1) 17.35, (2) 14.56
(1) 19.81, (2) 12.87
(1) 12.00, (2) 25.00
(1) 16.93, (2) 12.23
adsasd
(1) 19.81, (2) 12.87
To find the expected degree for node 15 born at i=15 and node 30 born at i=30 in the "Growing Random Networks" model with m=10, we can use the approximation formula for the expected degree at time t of a node born at time i, as mentioned in Lecture 3.1.
The approximation formula is given by:
E(k_i(t)) = m * ( log(t+1) - log(i) )
Let's calculate the expected degree for node 15 born at i=15:
E(k_15(40)) = 10 * ( log(40+1) - log(15) )
= 10 * ( log(41) - log(15) )
≈ 10 * ( 3.71357 - 2.70805 )
≈ 10 * 1.00552
≈ 10.0552
≈ 10.06 (rounded to two decimal places)
Now, let's calculate the expected degree for node 30 born at i=30:
E(k_30(40)) = 10 * ( log(40+1) - log(30) )
= 10 * ( log(41) - log(30) )
≈ 10 * ( 3.71357 - 3.4012 )
≈ 10 * 0.31237
≈ 3.12 (rounded to two decimal places)
Comparing the two values, we have:
(1) the expected degree for node 15 born at i=15 ≈ 10.06
(2) the expected degree for node 30 born at i=30 ≈ 3.12
Therefore, the closest option is:
(1) 16.93, (2) 12.23