Consider the "Hybrid Model" described in lecture 3.4, with m=10. Let us compare the cases of a=0.8 and a=0.2.
At date t=10,000, consider a node born at date i=20 and a node born at date i=9999. Recall that the approximation of the expected degree at time t of a node born at time i is (m+2am/(1−a))⋅(t/i)(1−a)/2−2am/(1−a).
Which option makes the following statement correct?
At date t=10,000,
a node born at date i=9,999 has a --------- expected degree with a=0.8 than with a=0.2; and
a node born at date i=20 has a --------- expected degree with a=0.8 than with a=0.2.
higher; higher
higher; lower
lower; higher
lower; lower
To compare the expected degree of a node born at different dates with different values of "a" in the Hybrid Model, we will use the given formula:
Expected Degree = (m + 2am/(1−a))⋅(t/i)(1−a)/2−2am/(1−a)
For the first part of the question, we want to compare the expected degrees of a node born at date i=9,999 with a=0.8 and a=0.2 at date t=10,000.
1. For a=0.8:
Expected Degree, Node born at i=9,999 with a=0.8
= (m + 2am/(1−a))⋅(t/i)(1−a)/2−2am/(1−a)
= (10 + 2*10*0.8/(1−0.8))⋅(10,000/9,999)(1−0.8)/2−2*10*0.8/(1−0.8)
2. For a=0.2:
Expected Degree, Node born at i=9,999 with a=0.2
= (m + 2am/(1−a))⋅(t/i)(1−a)/2−2am/(1−a)
= (10 + 2*10*0.2/(1−0.2))⋅(10,000/9,999)(1−0.2)/2−2*10*0.2/(1−0.2)
Now we can compare these two values to determine which option makes the statement correct.
For the second part of the question, we want to compare the expected degrees of a node born at date i=20 with a=0.8 and a=0.2 at date t=10,000.
1. For a=0.8:
Expected Degree, Node born at i=20 with a=0.8
= (m + 2am/(1−a))⋅(t/i)(1−a)/2−2am/(1−a)
= (10 + 2*10*0.8/(1−0.8))⋅(10,000/20)(1−0.8)/2−2*10*0.8/(1−0.8)
2. For a=0.2:
Expected Degree, Node born at i=20 with a=0.2
= (m + 2am/(1−a))⋅(t/i)(1−a)/2−2am/(1−a)
= (10 + 2*10*0.2/(1−0.2))⋅(10,000/20)(1−0.2)/2−2*10*0.2/(1−0.2)
Again, we can compare these two values to determine which option makes the statement correct.