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 $t=10000$, a node that joined at step $t_i=1600$ has degree $k_i=10$. What would be the degree of a node that joined this same network at time step $t_j=6400$?

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 $d⟨L⟩/dt$?

1. $10t-1$
2. $0.1\ast [10t-1]$
3. $10t-1/2$
4. $0.1\ast [10t-1/2]$
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 $N(t)=e^{{10}^{6}t}$. Which alternative correctly estimates and interprets the variation of the average distance over time ($d⟨d⟩/dt$):

1. $e^{{10}^{6}t}$, as the network grows rapidly, $d⟨d⟩/dt$ also increases rapidly over time, causing the network to lose its ultra-small-world property.
2. $1/(t{e}^{{10}^6}+\ln t)$, as the network grows rapidly, $d⟨d⟩/dt$ gets too small, a consequence of the ultra-small-world property of scale-free networks in this regime.
3. $e^{{10}^{6}t}/({10}^{6}t)$, as the network grows rapidly, $d⟨d⟩/dt$ 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, $d⟨d⟩/dt$ 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 $t\ge 1$ 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