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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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