MO412 - Mateus de Padua Vicente

Two undirected networks, Network A and Network B , have identical degree distributions p k p_k ​ but distinct mixing patterns. For each, it is measured the average nearest-neighbor degree function k n n ( k ) k_{nn}(k)  and the degree correlation matrix e j k e_{jk} . So: In Network A , k n n ( k ) k_{nn}(k)  is approximately constant across k k , and e j k e_{jk} ​ roughly factorizes into q j q k q_j q_k ​ . In Network B , k n n ( k ) k_{nn}(k)  decreases systematically with k k , and the e j k 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 value of Newman’s correlation coefficient r r . III. If a new Network C were highly assortative, its e j k e_{jk} ​ would concentrate along the diagonal, and k n n ( k ) k_{nn}(k)  would increase with k k . IV. Be...

Consider the flow network below, with vertices  V = { s , a , b , t } , source  s , and sink  t . The edge capacities are given as: ( s , a ) = 10 ,  ( s , b ) = 15 ,  ( a , b ) = 15 ,  ( a , t ) = 10 ,  ( b , t ) = 10 A student applies the Ford-Fulkerson algorithm, choosing augmenting paths in the order they are found by DFS, without using BFS or any other fixed rule. After a few iterations, the student notices that the total flow found is 15, but there still exists an augmenting path in the residual graph. What does this example illustrate about the Ford-Fulkerson method? A)  This behavior only occurs if there are cycles in the graph, which is not the case here . B) The Ford-Fulkerson algorithm may stop before the optimum if the choice of augmenting paths is not made carefully. C) The choice of augmenting paths may affect the number of iterations and efficiency, but the algorithm still finds the maximum flow in the end. D)  The Ford-Fulkers...

Consider a directed graph  G = ( V , E )  and the Kosaraju-Sharir's algorithm for finding strongly connected components (SCCs). Analyze the following statements: I.   The SCC decomposition of a directed graph is unique, but the particular order in which SCCs are discovered by Kosaraju’s algorithm may depend on the order of adjacency lists. II.   If a directed graph has no back edges in a DFS traversal, then each vertex forms a distinct SCC. III.  If  G  is strongly connected, then the second DFS in Kosaraju’s algorithm produces a single tree covering all vertices, regardless of the order of exploration. IV.   The order of vertices given by decreasing finishing times of the first DFS guarantees that the second DFS on the transposed graph  G T  discovers one SCC at a time. Select the correct alternative: A) Only I and II are correct. B) Only I and IV are correct. C) Only I, II and IV are correct. D) Only I, III and IV are correct. E) ...

Consider a graph G = ( V , E ) G = (V, E) , and the Depth-First Search (DFS) and Breadth-First Search (BFS) algorithms. Analyze the following statements: I. In a DFS execution on a directed graph, the classification of edges as tree, forward, backward, and cross edges depends on the order in which vertices are explored. II. In a BFS executed from a source vertex s s  in an unweighted graph, the order in which vertices are discovered at the same level may vary depending on the order of edges in the adjacency list of the graph, but the minimum distance from each vertex to s s  remains unchanged. III. In directed graphs, a back edge detected during DFS indicates the presence of at least one cycle that includes the origin vertex of the edge. IV. In undirected graphs, BFS can detect all cycles of length 3 (triangles), but it does not guarantee detection of larger cycles. V. DFS guarantees that in a connected undirected graph, the tree generated contains exactly ∣ V ∣ − 1 |V| -...