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Question 6. Modularity

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  Imagine a Cayley Tree, constructed starting from a central node of degree k. Each node at distance d from the central node has degree k, until the nodes reachable at distance P that have degree one, as shown in figure above. Consider that there are k+1 communities, where each community corresponds to a leaf of the network and the central node is other community. Considering that the number of nodes reachable in t steps from the central node is computed as k(k-1) t-1 , and  number of links L = N-1, being N the number of nodes, what is the modularity if k=3 and t =4? Round it up to three decimals places. Tip: Try to model the total number of links within the community , L c, and the total degree of the nodes in the community k c as a function of k and t. A. 0.621 B. 0.523 C. 0.785 D. 0.591 E. None of the above. Original idea by: Germán Darío Buitrago Salazar
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Question 5. Degree correlation

 Which of the following statements are true about degree correlations: I. The tendency for nodes to connect to other nodes with similar properties within a network is a characteristic of an assortative network. II. When we a degree correlation analyse, we can find networks with correlation exponent (μ) and Pearson correlation coefficient (r) with opposite sign. III. For directed networks, positive degree correlation means nodes with high in-degree tend to have high out-degree. IV. The point where the degree distribution decays rapidly is considered structural cutoffs in a network. Select the alternative that lists the correct statements: I, II, III and IV are correct. I, II and IV are correct. II, III, and IV are correct. I and III are correct. None of the above. Original idea by: Germán Darío Buitrago Salazar  
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Question 4. Strongly Connected Components

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  Based on figure above, we can assert: I. If we apply Kosaraju-Sharir's algorithm, it will return 5 strongly connected components (SCCs), when we assume the node's list are sorted in descending order. II. The component {G, J, I} is not a SCCs because it is inside the component {G, H, K, L, J, I} III. If there is a direct link from node I to node A, we would just have 4 SCCs. IV. If there is a direct link from node D to node C, the SCC {C, F} would still exist. Which statements from the options above are correct ?  A. I, II, III and IV  B. II and IV  C. I, II, IV  D. II and III  E. none of above Original idea by: Germán Darío Buitrago Salazar
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Question 3. Calculus

Consider a random network with constant probability of conexion 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) a . 10t - 1 b. 0.1 * [10t - 1] c. 10t - 1/2 d. 0.1 * [10t - 1/2] e. None of above Original idea by: Germán Darío Buitrago Salazar
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Question 2. Random Networks

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Imagine a Cayley Tree, constructed starting from a central node of degree k. Each node at distance d from the central node has degree k, until the nodes reachable at distance P that have degree one, as shown in figure above. Considering that the number of nodes reachable in t steps from the central node is computed as k(k-1) t-1 , and its average number of links <L> = N-1, being N the number of nodes, what is the probability of conexion between nodes if k=4 and t=4?   A. 1/161 B. 2/161 C. 1/41 D. 2/41 E. None of above Original idea by: Germán Darío Buitrago Salazar
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