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Communities

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Consider the undirected graph shown below. The network contains two dense regions: nodes (1-4) and nodes (5-7), connected through node 3. Node 3 also connects to node 8 through the dashed edge (3,8). We wish to apply the Girvan–Newman divisive community detection algorithm, that is based on the betweenness of nodes. Assume that in the first iteration, the dashed edge (3,8) has the highest betweenness and is therefore removed. After removing the edge (3,8), which of the following statements best describes the next step of the algorithm and its structural implications? A) Since the graph becomes disconnected after removing (3,8), the algorithm terminates immediately and each connected component is taken as a final community. B) The algorithm recomputes the betweenness of all remaining edges. Because node 3 still acts as the bridge between the two dense regions (1–4 and 5–7), the edges incident to node 3 are expected to gain higher betweenness, making edges such as (3,5) or (3,7) the next...
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Traveling Salesperson Problem

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A post office must plan a route to visit five cities (A, B, C, D, E) and return to the start. Distances (cost units) are given in the graph below. The graph satisfies the triangle inequality and is undirected. Apply the cheapest insertion heuristic (from the traveling salesperson problem), starting with triangle ABC (initial cycle: A → B → C → A). At each step, insert the vertex that yields the smallest cost increase according to the cheapest insertion heuristic. Which final cycle is obtained and what is its total cost? A) A → B → D → C → E → A, total cost 25 B) A → B → E → D → C → A, total cost 27 C) A → C → B → D → E → A, total cost 29 D) A → D → B → E → C → A, total cost 27 E) None of the above Original idea by: Thiago Soares Laitz
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Degree Correlations

Considering an undirected graph 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 edge and let (q_k) be the probability to have a degree k node at the end of a link. You are given the average degree: <k> = 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) {0.0064  0.0072  0.0080 } {0.0072  0.0081  0.0090 } {0.0080  0.0090  0.0100 } B) {0.0032   0.0045   0.0060 } {0.0045   0.0075   0.0080 } {0.0060   0.0080...
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Flow network

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Given the flow network below with labels (capacity, flow), let (S) be the source and (T) be the sink. Build the residual network from the given (capacity, flow) labels. Consider this as the   initial residual network and evaluate the statements: I. In the initial residual network, the path S -> B -> E -> F -> T is an augmenting path with bottleneck 3 . II. In the initial residual network, there is no augmenting path. The current flow is already maximum. III. Starting from the initial residual network, after performing any best single-path augmentation (i.e., any   augmenting path   with maximum bottleneck possible) , there still exists an augmenting path with bottleneck 1 . IV. The value of the current flow in the given network (before any new augmentation) is 13 . Choose the correct alternative: A) I and III only B) I and IV only C) II and III only D) I, III, and IV E) None of the above Original idea by: Thiago Soares Laitz
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Graph algorithms

A software engineering team is modernizing a legacy system with hundreds of code modules. The goal is to refactor the system by splitting it into independent microservices. A major challenge is the complex network of “imports” between modules. They found that many modules are interdependent; that is, Module A depends on Module B, which in turn, directly or indirectly, depends on Module A. To carry out the microservice separation, the team first needs to identify these groups of mutually dependent modules, since they must be treated as a single unit. Which algorithmic approach would be most effective to identify these groups of modules, considering this problem can be modeled as a directed graph? Analyze the statements below and select the correct alternative: I. Topological Sorting can be applied because it arranges modules in a linear order of dependencies and, when it fails, it indicates the existence of cycles, making it useful as a diagnostic tool and making it easy to identify th...
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DFS

 Consider the execution of Depth-First Search (DFS) on a graph G. Analyze the following statements: I. The existence of at least one backward edge during the DFS traversal guarantees that G contains at least one cycle. II. In an undirected graph, DFS can generate forward and cross edges if the graph is disconnected. III. After running DFS on a graph, if a backward edge is found, it is still possible to produce a valid topological sort of its vertices. IV. Cross edges alone are sufficient to guarantee that G contains a cycle. Which of the following combinations of statements is true? A) I and III only. B) I and II only. C) Only I. D) II and IV only. E) None of the above. Original idea by: Thiago Laitz
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