2026-344

In a Barabási-Albert model with m = 2, node A is added at time tA= 1 and node B at time tB = 4, as illustrated in the Figure. Here tA and tB are the birth times ti of each node. What is the ratio kA(100) / kB(100)?

1. 1/2
2. 1
3. 4
4. 10
5. None of the above

Original idea by: Gustavo P. C. P. da Luz

2026-343

Consider a network that follows the Barabási-Albert model with \(m_0 = m = 2\). A node \(i\) joined the network at time \(t_i = 375\) and has degree \(k_i(t) = 40\) at the current time step. What is the expected diameter of this network now? Round to two decimal places.

1. \(\langle D \rangle \approx 4.20\)
2. \(\langle D \rangle \approx 4.81\)
3. \(\langle D \rangle \approx 2.80\)
4. \(\langle D \rangle \approx 3.81\)
5. None of the above

Original idea by: Carlos Trindade

2025-279

Consider that the following network obeys the Barabási–Albert model, assess each statement from I to V regarding probabilities of receiving a link, and then mark the correct alternative. Round probability values to two decimal places.

I) Node M is the node with the highest probability of receiving a link from a new node, of 0.13.

II) Node D is the node with the second highest probability of receiving a link from a new node, of 0.10.

III) Nodes A, C, L, P, and I have the same probability of receiving a new link, of 0.02.

IV) Nodes N and E have the same probability of receiving a new link, of 0.08.

V) Nodes H and D have different probabilities of receiving a new link, of 0.08 and 0.09 respectively.

Which alternative contains the truth values of the above statements in the correct order?

A) F F T T F

B) T F T T F

C) T T F F T

D) T F T F T

E) None of the above

Original idea by: Carolina Albuquerque

2025-272

A friend of yours has published a pioneering research paper in a niche area, and it has gathered several citations from other researchers. The directed graph below shows the citation relationships among papers in this area.

Your friend is extremely happy with the results, given that his Paper A is already the most influent paper in the network, with an incoming degree of 4. Curious, with your knowledge on network science, you want to understand what the probability is of a new paper citing paper A. 

You are happy with rough approximations, so you frame this problem using a copying model for a directed network. With probability \(p = 0.4\) of linking uniformly at random to an existing paper, and probability \(1 - p\) of copying a link from a random paper, what is the probability of a new paper connecting to paper A? Round to two decimal places.

a) 0.10

b) 0.20

c) 0.21

d) 0.29

e) None of the above

Original idea by: Alexandre Petrachini

2025-271

Consider a network generated using the Barabasi-Albert model, whose sequence of node addition is A,B,C,D,E. Consider m=2 and that new nodes always connect to nodes with higher degrees. Which of the alternatives below represents a valid network in this context?

A. 

B.

C.

D.

E. None of the above

Original idea by: Jhessica Silva

2025-270

A research group is creating a new social media platform and wants to model its growth using the Barabási–Albert model. They start with a small group of well-connected users. According to the principles of this model, which of the following strategies will most likely lead to the formation of a scale-free network, characteristic of real-world social platforms?

A) Each new user connects to existing users at random, with equal probability, regardless of how many followers those users already have.

B) Each new user connects to existing users with a probability proportional to their current number of followers (popular users are more likely to attract new followers).

C) Each new user connects only to the newest users who joined in the previous step, ensuring recent users stay active and visible.

D) Restrict new users to connecting only with users in their immediate geographical surroundings to create localized clusters.

E) None of the above.

Original idea by: João Medrado Gondim

2025-269

Consider a Barabási-Albert network with parameter \(m = 2\). A node was introduced at time \(t_i = 4\). At a later time, this node has degree \(k_i(t) = 100\). What is the expected average clustering coefficient \(\langle C \rangle\) of the network at time \(t\)?

1. \(\langle C \rangle \approx 0.030\)
2. \(\langle C \rangle \approx 0.025\)
3. \(\langle C \rangle \approx 0.010\)
4. \(\langle C \rangle \approx 0.008\)
5. None of the above

Original idea by: Yan Prada.

2025-268

Bitu is an explorer who sets out to visit the mysterious Bara-Al islands — a chain of 90 islands. Each island is connected by a few bridges. Whenever a new island was settled, its bridges were built with a preference for linking to islands that already had many connections. In this way, the Bara-Al islands grew according to Barabási–Albert Model of networks.  Still, as in any graph, there are pairs of islands that maximize the pairwise distance between islands.  These are called extreme islands.

So, he wonders: what is the expected number of bridges he would need to cross, starting from an extreme island and traveling to the farthest one by means of a shortest path?

Note 1: \(90 \approx e^{4.5}\)
Note 2: \(4.5 \approx e^{1.5}\) 

A. 8

B. 6

C. 5

D. 3

E. None of the above

Original idea by: Rodolfo Bitu

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

2022-175

The sequence of images below shows some steps for generating the network based on the Barabási-Albert model. Consider that starting with \( m_0 \) nodes, the empty circles mark the newly added node to the network in each timestep \( t \), and the new node's number of links is \( m = 2 \).

Analyze the following statements about the network:

1. Choosing which node will be connected uses the preferential attachment criteria.

2. After \( t \) timesteps, this model generates a network with \( t + m_0 \) nodes.

3. The degree distribution of this network follows the Poisson distribution.

4. This network has the scale-free property.

5. The diameter of this network with N nodes can be estimated by the equation \( D \sim \frac{\ln N}{\ln \ln N} \)

Select the correct alternative:

1. All statements are correct. 
2. Only statements 1, 2, and 3 are correct.  
3. Only statements 3, 4, and 5 are correct.
4. Only statements 1, 2, 4, and 5 are correct. 
5. None of the above.

Original idea by: Rubens de Castro Pereira

2022-174

In the Non-linear preferential attachment, the preferential attachment probability becomes \( \Pi(k) \sim k^{\alpha} \). For a starting network with 1 node, approximately how many nodes we have to add on the network for it to achieve \( k_{max} = 100 \), considering the cases where \( \alpha \) is 0.5, 1, and 1.5, respectively?

1. 22026 nodes; 10000 nodes; 100 nodes, respectively
2. 1024 nodes; 1000 nodes; 100 nodes respectively
3. 22026 nodes; 10000 nodes; 200 nodes, respectively
4. 1024 nodes; 1000 nodes; 200 nodes, respectively
5. None of the above.

Original idea by: Pedro Henrique Di Francia Rosso

2022-173

Consider a scale-free network that has evolved according to the BA model and choose the best possible option given below. Let P and Q be two nodes such that the rate at which node P acquires new links is twice the rate at which node Q acquires new links. If node P joined the network at time 100 units, find the time at which node Q joined the network.

1. 800
2. 600
3. 400
4. 200
5. None of the above

Original idea by: Muhammad Idrees

2022-172

By considering the Barabási–Albert model for a network, the degree of a node is k=50, and it joined the network at time t=10, what will be the degree of another node which joined the network at time t=100?

1. 10.53
2. 15.81
3. 21.09
4. 26.37
5. None of the above

Original idea by: Meer Muhammad Khan

2022-171

Suppose you are simulating a network that follows the Barabási-Albert model. At each step, you add 5 links to this network. One day, you were feeling sleepy, and you decided to stop for coffee. When you returned to your work, you noted that your simulation was at time step 100 thousand, and there was a node whose degree was 500. Puzzled, you wonder when this node was introduced to the network. After some calculations, you figured it out! Which step did you find?

1. 100.
2. 10.
3. 50.
4. 5.
5. None of the above.

Original idea by: Gabriel Oliveira

2022-170

Consider an undirected network that follows the Barabási-Albert model with \( m = 4\).  What is the network's expected diameter at time \( t = 1000 \) (round to 1 decimal place)? 

1. 3.6
2. 12.1
3. 503.5
4. 999.7
5. None of the above.

Original idea by: André Nóbrega

2022-127

 Analyze the following statements:

      I.  The Barabási–Albert model describes a process to generate scale-free networks, and tries to capture the idea that, in real networks, the number of nodes continually grows, thanks to the addition of new nodes;

     II.  Scale-free networks don't follow a power-law degree distribution, because there are almost no hubs in this type of network, which means the degree distribution can't follow an exponential curve (be proportional to an exponential term);

    III.  The number of links in a Barabási-Albert-generated network can decrease over time, since there is a significant chance the algorithm will remove links from hubs, to keep the network from having a linear degree distribution;

    IV.  The Barabási-Albert model uses a preferential attachment mechanism. The network grows by adding new nodes over time, and a new node links to existing nodes in the network with probability proportional to their degree.

Now, mark the correct alternative:

    A. I and IV are correct
    B. I and III are correct
    C. II and III are correct
    D. All statements are correct
    E. None of the above

Original idea by: Victória Pedrazzoli

2022-114

Analyze the following statements and select the correct alternative.

1. The Barabási-Albert model describes a network where a new node is twice as likely to connect to a degree-four node than a degree-two node;
2. When the dynamical exponent is β=1/2 in a BA network, the first-mover advantage might emerge over time, making hubs larger because they arrived earlier. But in real networks, that is not always the case. BA is a minimum model and does not account for nodes' intrinsic properties;
3. The Barabási-Albert model predicts networks that result in rich-gets-richer dynamics which can be replicated in a real network regardless of the intrinsic properties of its nodes.

1. Statements I and III are true.
2. Statements I and II are true.
3. Statements II and III are true.
4. Only statement III is true.
5. None of the above.

Original idea by: André Portela

2022-113

 

Non-Linear Barabási-Albert Model

  Analyze the scaling regimes based on the non-linear Barabási-Albert model. To aid with this analysis, use the image below.

Image extracted from http://networksciencebook.com/chapter/5#non-linear

1. In the sublinear regime, the degrees follow the stretched exponential distribution
2. The Barabási-Albert Model corresponds to the linear regime and follows a power law distribution
3. In the superlinear regime, one can observe the formation of a hub-and-spoke network in which most nodes connect directly to a few central nodes
4. The rich-gets-richer process takes place in the linear regime where all nodes connect to super-hubs

Now select the true statements:

    A. 1, 2, and 3

    B. 2, 3, and 4

    C. 2 and 4

    D. All the statements

    E. None of the above

Original idea by: Márcia Jacobina

2022-112

The network below was generated using the Barabasi-Albert model:

Which alternative represents a possible order of node addition:

1. ECABD
2. DACBE
3. CABDE
4. BACDE
5. None of the above

Original idea by: Iury Cleveston

2022-110

Consider the following statements about the Barabási-Albert model.

1. The degree exponent is dependent of m.
2. Growth and preferential attachment are simultaneously needed for the emergence of the scale-free property.
3. When deriving the degree distribution, we assume that, at any moment, the total number of nodes equals the number of timesteps, that is, \( N(t) = t \).
4. If a new node has a choice between a degree-two and a degree-four node, it is twice as likely that it connects to the degree-four node.

Select the correct alternative.

1. Only statements I and IV are true.
2. Only statements I, II and III are true.
3. Only statements II, III and IV are true.
4. Only statements II and IV are true.
5. None of the above.

Original idea by: Diogo Souza