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Communities Question

 In the context of community detection, the generalized modularity \(M\) of a network with \(L\) links and \(n_c\) communities is given by: \[ M = \sum_{c=1}^{n_c}{\left[ \frac{L_c}{L} - \left( \frac{k_c}{2L} \right)^2 \right]}\] where \(L_c\) is the total number of links within the community \(C_c\) and \(k_c\) is the total degree of the nodes in this community. Considering the following statements: I. Modularity has a resolution limit, as it can't detect communities smaller than a factor proportional to \(\sqrt{2L}\). II. A community \(C_c\) contributes positively to \(M\) only if the fraction of internal links \(\frac{L_c}{L}\) exceeds the expected fraction \(\left(\frac{k_c}{2L}\right)^2\) under a null model that preserves the degree sequence. III. The modularity \(M\) is bounded in the interval \([0,1]\), where \(M = 1\) corresponds to a perfectly modular network. IV. The partition with the maximum modularity \(M\) offers an optimal community structure. It is correct  to ...
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Degree Correlations Question

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 of \(k_{\text{min}} = 2\), meaning every user follows at least two others, and an average degree of \(\langle k \rangle = 8\). 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? A) \(k_{\text{max}} = 120\), and the network presents structural assortativity because \(k_{s} < k_{\text{max}}\). B) \(k_{\text{max}} = 64\), and the network does not present structural disassortativity because \(\gamma > 2\). C) \(k_{\text{m...
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Question of Barabási-Albert Model

 Consider a network that follows the Barabási-Albert Model growth with \(m_0 = m = 2\). A node \(i\) joined the network at time \(t_i = 375\) and has degree \(k(t) = 40\) at current time step \(t\). What is the expected diameter of this network? Round to two decimal places. A) \( \langle D \rangle \approx 4.20 \) B) \( \langle D \rangle \approx 4.81 \) C) \( \langle D \rangle \approx 2.80 \) D) \( \langle D \rangle \approx 3.81 \) E) None of the above. Original idea by: Carlos Trindade
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Scale-free Networks question

 Consider a network with \( N = 10^4\) nodes. If the degree distribution \(p_k\) follows the power law \(p_k \sim k^{\gamma}\), with degree exponent \(\gamma = 3\) and a minimum degree \(k_{\text{min}} = 2\), which the following statements are true? I. The expected size of the largest hub is \(k_{\text{max}} = 200\). II. The first moment \(\langle k \rangle\) diverges, meaning the network has no scale. III. Doubling the number of nodes to \(N = 2 \times 10^4\) would double the expected size of the largest hub to \(k_{\text{max}} = 400\). IV. The second moment \(\langle k^2 \rangle\) diverges, meaning the network has no characteristic flutuation. a) I, II and IV b) I and IV c) Only I d) II and III e) None of the above. Original idea by: Carlos Trindade
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Random Graphs Question

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  Consider the T network below:   Which of the following statements are  true ?   I. T is in supercritical regime because its average degree \(\langle k \rangle = 4.5\) is higher than \(1\).   II. T can't be considered a random network because node G has degree \(k = 7\), representing a hub in the network and its degree is far from \(\langle k \rangle = 3.25\).   III. T is in connected regime, since its average degree is higher than \(\ln N\)   IV. T is a random network with \(N = 8\) nodes and \(p = 0.46\).     A) I and IV B) III and IV C) II and III D) I and II E) None of the above     Original idea by: Carlos Trindade  
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Graph Theory Question

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 A Distribution Center (DC) is a location where companies or industries store manufactured products with the goal of delivering them to other units or customers. The location of a DC is very strategic, since it must minimize the costs of delivery. Considering the graph below, what is the best location to place a DC, minimizing the expected distance to the other locations?     A) Node D, because it has the lowest average distance among nodes with 5 or more connections, indicating it is the most central high-degree node. B) Node E, because it connects the left cluster (C, D) to the right cluster (F, K) and has an average distance of approximately 1.73. C) Node D, because it has the highest number of direct connections, minimizing the maximum distance to any location. D) Node F, because it has an average distance of approximately 1.54. E) None of the above.   Original idea by: Carlos Trindade  
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