2026-351

 Consider the network below:

What is the Pearson degree correlation coefficient (r) for this network using Mark Newman's definition?

A. r = 0, since the average degree of every node’s neighbors equals ⟨k⟩.

B. r = -2/3, indicating the network is assortative.

C. r = +2/3, indicating the network is disassortative.

D. r = -1/2, indicating the network is assortative.

E. None of the above.

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

2026-350

Epidemiologists are studying the spreading of a disease through an interaction network. After collecting data, they find that the network's degree distribution follows a power law \(p_k \sim k^{-\gamma}\) with exponent \(\gamma = 2.5\), suggesting that a small number of highly connected individuals ("superspreaders") play a dominant role in transmission. The network has a minimum degree \( k_{min} = 2\), meaning every user follows at least two others, and an average degree \( \langle k \rangle = 8\). Structural analysis reveals a cutoff \(k_s = 64\), beyond which the simple-graph constraint begins to limit how hubs can connect to one another. Based on this information, which of the following alternatives is correct?

1. \(k_{max} = 128\), and the network presents structural assortativity because \(k_s < k_{max}\).
2. \(k_{max} = 64\), and the network does not present structural disassortativity because \(\gamma > 2\).
3. \(k_{max} = 128\), and the network presents structural disassortativity because \(k_s < k_{max}\).
4. \(k_{max} = 64\), and the network does not present structural disassortativity because \(k_s \geq k_{max}\).
5. None of the above

Original idea by: Carlos Trindade

2025-289

Considering the impact of degree correlations on the emergence of a giant component in random networks, which of the following statements is correct?

A. The phase transition point is the same for assortative and disassortative networks

B. The phase transition occurs at higher ⟨k⟩ in assortative networks and at lower ⟨k⟩ in disassortative networks

C. For large ⟨k⟩, the giant component is smaller in assortative networks than in disassortative networks

D. Regardless of ⟨k⟩, the giant component is always larger in neutral networks than in assortative or disassortative networks

E. None of the above

Original idea by: Mylena Roberta

2025-282

A large, undirected network, with no self-loops or multi-links, is analyzed. It exhibits a scale-free degree distribution described by the formula \(p(k) = k^\gamma\) with an exponent \(\gamma=2.5\). Empirical measurement yields a degree correlation coefficient \(r = -0.15\). A plot of the average nearest-neighbor degree \(k_{nn}(k)\) shows relatively independency of the degree \(k\) for small \(k\), but it begins to decay for nodes with \(k > \sqrt{\langle k\rangle N}\), where \(N\) is the network size. What is the most precise classification of this network's degree correlations?

A)  The network is intrinsically disassortative;

B) The network exhibits structural disassortativity, a phenomenon affecting scale-free networks with exponent \(\gamma < 3\) and the network's nature;

C) The network is neutral, as the average nearest-neighbor degree is independent of \(k\) for most nodes and \(r\) is close to 0;

D) The network is assortative, but the measurement is skewed by the high variance of high-degree nodes;

E)  None of the above.

Original idea by: Caio Rhoden

2025-280

Consider networks generated by the following models:

• Barabási-Albert model
• Erdős-Rényi model

With regard to the degree correlations, the generated networks are, with high probability, respectively:

A. Disassortative and Disassortative

B. Disassortative and Neutral

C. Neutral and Disassortative

D. Neutral and Neutral

E. None of the above

Original idea by: Jhessica Silva

2025-278

Consider a network with 4 nodes (1, 2, 3, 4) and the following 5 edges: (1, 2), (1, 3), (2, 3), (2, 4), (3, 4). What is the average nearest neighbor degree for nodes of degree \(k=3\), denoted by \(k_{nn}(3)\)?

1. 2.5
2. 3.0
3. 2.67
4. 2.33
5. None of the above

Original idea by: Gabriel Sato

2025-277

Consider the following example of a standard fully-connected Multilayer Perceptron (MLP), a type of artificial neural network. It has an input layer, a single hidden layer, and an output layer. In this architecture, neurons in one layer are connected to all neurons in the immediately following layer, but there are no connections within a layer or connections that skip layers. Here, the input and output layers have 5 neurons each, while the hidden layer has 10 neurons. Based on this description, would you expect the degree correlation coefficient, \(r\), to be:

a) undefined

b) \(r > 0\)

c) \(r < 0\)

d) \(r = 0\)

e) None of the above

Original idea by: Eduardo Bouhid

2025-276

In a network, for nodes of degree \(k=3\), the conditional degree distribution of their neighbors is given by:

\[p(k'=1|k=3) = 0.5\]

\[p(k'=2|k=3) = 0.3\]

\[p(k'=5|k=3) = 0.2\]

What is the value of \(k_{nn}(3)\), i.e., the average degree of the neighbors of nodes with degree 3?

1. 1.8
2. 1.9
3. 2.1
4. 3.0
5. None of the above.

Original idea by: Jonas Henrique Ribeiro Paula

2024-246

Given the networks below, which of the following options is most likely correct about them, where r is the degree correlation coefficient:

a) Network A is disassortative, as the hubs tend to connect with other hubs, while Network B is assortative, with hubs tending to connect to nodes of smaller degree.

b) Network A is assortative, with a positive r, and network B is disassortative, with a negative r.

c) Network A is a perfectly assortative network, where each node links only to nodes with the same degree, while network B is disassortative, with hubs preferring to link to low-degree nodes.

d) Network A is assortative with a negative r, and network B is disassortative with a positive r.

e) None of the above

Original idea by: Matheus C. Lindino

2025-274

Two undirected networks, Network A and Network B, have identical degree distributions \(p_k\)​ but distinct degree correlation patterns.

Regarding the average nearest-neighbor degree function $k_{nn}(k)$ and the degree correlation matrix $e_{jk}$ of each network, we have:

• In Network A, $k_{nn}(k)$ is approximately constant across $k$, and $e_{jk}$ roughly factorizes into $q_j{q}_k$.

• In Network B, $k_{nn}(k)$ decreases systematically with $k$, and the $e_{jk}$ matrix shows high values in the upper-left and lower-right corners rather than along the diagonal.

Consider the following statements:

I. Network A is neutral, while Network B is disassortative.
II. Network B is expected to have a negative Newman correlation coefficient $r$.
III. If a new Network C were highly assortative, its $e_{jk}$ would concentrate along the diagonal, and $k_{nn}(k)$ would increase with $k$.
IV. Because Networks A and B share the same $p_k$, their correlation coefficient $r$ must also be the same.

Which of the statements above are correct?

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

Original idea by: Mateus de Padua Vicente

2025-273

Consider an undirected network with no degree correlation. Let \(e_{jk}\) be the probability to find a node with degree \(j\) and degree \(k\) at the two ends of a randomly selected link and let \(q_k\) be the probability to have a degree \(k\) node at the end of a randomly selected link. You are given the average degree:

\(\langle k\rangle = 10\)

and the following degree-distribution probability masses (all other \(p_k\) can be anything consistent and are not needed):

\(p_{20} = 0.04\)

\(p_{30} = 0.03\)

\(p_{50} = 0.02\)

Using only the information above, compute the 3×3 submatrix of \(e_{jk}\) for degrees 20, 30, and 50. Which option matches this block? Consider the theoretical \(e_{jk}\) for the network with no degree correlation and that the matrix rows and columns correspond to degrees 20, 30, and 50 in that order.

A) \(\left[ \begin{array}{ccc} 0.0064 & 0.0072 & 0.0080\\ 0.0072 & 0.0081 & 0.0090\\ 0.0080 & 0.0090 & 0.0100\end{array}\right]\)

B) \(\left[ \begin{array}{ccc} 0.0032 & 0.0045 & 0.0060\\ 0.0045 & 0.0075 & 0.0080\\ 0.0060 & 0.0080 & 0.0090\end{array}\right]\)

C) \(\left[ \begin{array}{ccc} 0.0080 & 0.0090 & 0.0100\\ 0.0090 & 0.0100 & 0.0110\\ 0.0100 & 0.0110 & 0.0120\end{array}\right]\)

D) \(\left[ \begin{array}{ccc} 0.0064 & 0.0080 & 0.0096\\ 0.0080 & 0.0100 & 0.0120\\ 0.0096 & 0.0120 & 0.0144\end{array}\right]\)

E) None of the above.

Original idea by: Thiago Soares Laitz

2023-222

Consider a scale-free network with average degree 2.9, second moment of the degree distribution 32.3, and degree exponent 2.89. It is correct to state that:

1. The preferential attachment assumption is valid, with a probability Π(k) that a link of a new node links to another node of degree k proportional to the square of k.
2. This network is perfectly assortative, with a degree correlation coefficient equal to 1.
3. This network displays enhanced robustness, since its breakdown threshold is greater than the breakdown threshold of a random network with the same average degree.
4. This network is in a random network regime, since the degree exponent is close to 3.
5. None of the above.

Original idea by: Vitoria D. M. Pinho

2023-220

Consider a simple scale-free network with \(\gamma = 2.5\), \( k_\min = 1\), \( \langle k \rangle = 10 \), and \( N \) nodes. Choose the strongest condition of \( N \) for which we expect to observe the structural cutoff phenomenon. Tip: establish a relationship between \( k_S(N) \)  and \( k_\max \).

1. \( N > 2500 \)
2. \( N < 2500 \)
3. \( N > 1000 \)
4. \( N < 1000 \)
5. None of the above

Original idea by: Christian Konishi

2023-219

The degree correlation function depicted in the following figure (shown in purple) is derived from a Bianconi-Barabási model with parameters N = 10,000, m = 3, and a uniform fitness distribution:

(Extracted from Albert-László Barabási book "Network Science")

Please choose the accurate statement regarding this figure:

1. This network exhibits a neutral nature due to the presence of structural cutoff phenomena.
2. The network cannot be classified as disassortative since randomizing the network while maintaining the same degree distribution produces a similar behavior, as the orange dots (Randomized R-S).
3. While this network displays some disassortativity, it is not pronounced primarily due to structural cutoff.
4. This network does not exhibit any structural cutoff phenomena. 
5. none of above

Original idea by: Anderson Nogueira Cotrim

2022-182

Which of the following statements about degree correlations are true:

I. In assortative networks, nodes with small degree tend to connect with other nodes with small degree.

II. In a disassortative network, the probability of connecting a node with degree \( k \) to another of degree \( k' \) is \( k k' / (2L) \).

III. Perfectly assortative networks can only exist with the presence of cycles.

IV. The degree correlation coefficient is negative for assortative networks.

A. Only I

B. I and II

C. I and III

D. III and IV

E. None of the above
    

Original idea by: Victor Sotelo

2022-134

Not all networks display the expected behavior of hubs linking to other hubs. They do so on some networks, but not on others. These patterns manifest a common feature, degree correlations, which allow us to detect the presence or lack of correlations in a real network. Analyze the degree correlation function knn(k) which expresses the average degree of the neighbors of all degree-k-nodes, and, then, classify the sentences below as true or false.

                                                     

 Inspired by http://networksciencebook.com/chapter/7measuring-degree; purple: knn(k), horizontal line: prediction, green: fit to knn(k)=akμ

1) In a disassortative network hubs prefer to link to high-degree nodes and the best representation for this network is Image A.

2) In a neutral network, there is a lack of degree correlations. Plotting knn(k) in function of k results in a horizontal line as shown in Image C, which demonstrates that the average degree of a node's neighbors is independent of the node's degree k.

3) Image A is a good representation of an assortative network where hubs tend to connect to other hubs. Thus, the higher the degree k of a node, the higher the average degree of its nearest neighbors.

4) In a disassortative network, as shown in Image B, the degree correlation function decreases with k, indicating hubs prefer to link to low-degree nodes.

Now select the option that contains exactly the true statements:

    A. Only 1 and 3

    B. Only 1, 2 and 3

    C. Only 2, 3 and 4

    D. All the statements

    E. None of the above

Original idea by: Márcia Jacobina

2022-130

Consider the following statements about degree correlation in networks:

1. In neutral networks, nodes link to each other randomly, which in turn results in a lack of degree correlation for the linking pattern.
2. A perfectly assortative network is always a complete graph.
3. The correlation exponent can help determine the type of the network. When the correlation exponent is positive, we may say the network is assortative.
4. In assortative networks, nodes tend to link to nodes of similar degree. In other words, hubs tend to connect with hubs, and small-degree nodes tend to connect with small-degree nodes.
5. Degree correlations for directed networks are defined by two coefficients: \( r_{in,out} \) and \( r_{out,in} \).

Select the alternative that lists the correct statements:

1. I, II, and V are correct.
2. Only V is correct.
3. II, III, and IV are correct.
4. I, II, III, and IV are correct.
5. None of the above.

Original idea by: Heitor Mattosinho

2022-129

Which of the alternatives is true, regarding assortative networks?

1. Hubs tend to connect to small-degree nodes.
2. The average degree of the neighbors of nodes with degree \( k \) (denoted by \( k_{nn}(k) \)) increases with \( k \).
3. Technological networks are usually assortative.
4. For large \( \langle k \rangle \), the giant component is bigger in assortative networks than in neutral or disassortative networks.
5. None of the above.

Original idea by: Diogo Souza.

2022-125

Social networks display an assortative behavior. In the friendship network below, consider that John and Karen want to make new friends.

Which alternative lists very likely next friends for John and Karen, respectively:

1. Mike, John
2. Anne, Jane
3. Paul, Paul
4. Mark, Mike
5. None of the above

Original idea by: Iury Cleveston

2021-063

It is incorrect to say that:

1. The degree correlation function helps us capture the presence or absence of correlations in real networks.
2. In assortative networks, hubs tends to connect to other hubs, hence the higher is the degree of a node, the higher is the average degree of its nearest neighbors.
3. In a perfectly assortative network, each node links only to nodes with the same degree.
4. In disassortative network, hubs prefer to link to high-degree nodes.
5. None of the above

Original idea by: Adson N Alves