2023-212

Consider a growing network following the Barabási-Albert model. You don't know the initial number of links, but you do know that at time step \( t = 10,000 \), a node that joined at step \( t_i = 1,600 \) has degree \( k_i = 10 \). What would be the degree of a node that joined this same network at time step \( t_j = 6,400 \)?

1. 5
   
2. 14
3. 30
4. 120
5. None of the above

Original idea by: Yuri Souza Padua

2023-211

What is the volume generated by rotating the region bounded by the curves \( y=x^2 \) and \( y=4 \) around the \(x\)-axis?

1. 8π
2. 4π
3. 2π
4. π
5. None of the above

Original idea by: João Marcos

2023-210

Consider a random network with constant probability of connexion \( p = 0.1 \) at all times and a number of nodes that increases over time according to \( N(t) = 10t \). What would it be the function that estimates the variation of average links over time \( d \langle L \rangle/dt \)?

1. \( 10t - 1 \)
2. \( 0.1 * [10t - 1] \)
3. \( 10t - 1/2 \)
4. \( 0.1 * [10t - 1/2] \)
5. None of the above

Original idea by: Germán Darío Buitrago Salazar

2023-209

Consider a scale-free network with a fixed \( \gamma = 2.5 \) at all times and a number of nodes that increases over time according to the relation \( N(t) = e^{10^6t} \). Which alternative correctly estimates and interprets the variation of the average distance over time (\( d\langle d \rangle/dt \)):

1. \( e^{10^6t} \), as the network grows rapidly, \( d\langle d \rangle/dt \) also increases rapidly over time, causing the network to lose its ultra-small-world property.
2. \( 1/(te^{10^6} + \ln t) \), as the network grows rapidly, \( d\langle d \rangle/dt \) gets too small, a consequence of the ultra-small-world property of scale-free networks in this regime.
3. \( e^{10^6t}/(10^6t) \), as the network grows rapidly, \( d\langle d \rangle/dt \) also increases rapidly over time, but not too fast, which may cause the loss of its ultra-small-world property in some cases.
4. \( 1/(t(e^{10^6} + \ln t)) \), as the network grows rapidly, \( d\langle d \rangle/dt \) gets too small, a consequence of the ultra-small-world property of scale-free networks in this regime.
5. None of above

Original idea by: Anderson Nogueira Cotrim

2023-208

In a scale-free network with 8,000 nodes, 10% of the nodes have a degree of 10 or more. If the degree exponent γ is 2, what is the approximate percentage of nodes with a degree of 5 or more?

A. 20%

B. 30%

C. 40%

D. 50%

E. None of the above.

Original idea by: Olalekan Sadeeq Bello

2023-207

Consider an Erdős-Rényi random graph with \(N = 1200\) nodes. If the average degree of the graph is 50, what is the variance of the degree distribution (round to 1 decimal place)?

a) 49.9

b) 7.8

c) 47.9

d) 6.9

e) None of above

Original idea by: João Marcos

2023-206

 

Imagine a Cayley Tree, constructed starting from a central node, where each node has degree \(k\), except the nodes at distance P from the central node, that have degree one, as shown in the figure above. Considering that the number of nodes reachable in \( t \geq 1 \) steps from the central node is \( k(k-1)^{t-1} \), and the number of links is \( L = N-1 \), where \( N \) is the number of nodes, what is the probability of conexion between nodes for \(k=4\) and \(P=4\)?

1. 1/161
2. 2/161
3. 1/41
4. 2/41
5. None of above

Original idea by: Germán Darío Buitrago Salazar