Question for MO412 - 17/04/2026

Consider the degree evolution in a network generated by the Barabási–Albert preferential attachment model with nodes following the same dynamical law and assume the continuum approximation when necessary. At time t=100, analyze the attachment probabilities.

Given parameters of the model:

• $m_0 = 2$
  
• $m = 2$ 

Consider the vertices:

• $v_a$ with degree $k_a = 10$
  
• $v_b$ with degree $k_b = 4$
  

Analyse the following statements:

I) The probability that a new vertex connects to $v_a$ is approximately $0.025$.

II) The probability of connecting to $v_a$ is $2.5$ times greater than connecting .

III) If both vertices had their degrees doubled, the probability of connecting to $v_a$ would be $1.25$ times greater than to $v_b$.

IV) If each vertex's degree were increased by $2$, a new vertex would be 2 times more likely to connect to $v_a$ than to $v$.

V) After $1{,}500$ more time steps, the expected degree of the vertex added at $t = 100$ is $16$.

Alternatives

A) II and IV only

B) I, II and IV only

C) I, II, IV and V only

D) I and II only

E) None of the above.

Original idea by: Gabriel Talasso