soc 375 - exams - fall 2007

soc 375 - exams - fall 2007 - Sociology 375 Exam 1 Fall...

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Sociology 375 Exam 1 Fall 2007 Prof Montgomery Answer all questions. 250 points possible. 1) [30 points] The antisymmetry condition can be written as (xRy yRx) (x = y) . Using a truth table, determine whether the condition ¬ (x = y) ¬ (xRy) ¬ (yRx) is equivalent to the antisymmetry condition. [HINT: Applying some elementary rules of logic, you could potentially answer this question without a truth table, but you must construct the truth table to receive credit on this question.] 2) [20 points] a) State the (set-theoretic) conditions for symmetry and asymmetry. b) Consider the empty relation R = on a non-empty set S. Describe (in words) how this relation would appear if represented as a graph or as an adjacency matrix. Is this relation symmetric, asymmetric, both, or neither? Explain why. [HINT: Use the conditions that you stated in part (a).] 3) [90 points] Consider the relation R = {(1,2), (1,3), (3,2), (3,4), (4,2)} on the set S = {1, 2, 3, 4}. a) Show how the relation R could be represented i) as a (directed) graph ii) as an adjacency matrix iii) using infix notation b) Determine (by computation or inspection) i) the number of 2-paths between each pair of individuals ii) the number of 3-paths between each pair of individuals iii) the reachability matrix iv) the distance matrix c) State the matrix test for transitivity. Using this test, determine whether the relation R is transitive. If not, list any violations of transitivity. d) Give the transitive closure of R. Is the transitive closure of R an equivalence relation? If not, explain why. If so, give the equivalence classes. Is the transitive closure of R a strict partial order? If not, explain why. If so, draw the Hasse diagram. 1
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4) [15 points] Suppose you are given a symmetric adjacency matrix A that generates the reachability matrix R. Now consider the matrix U composed of the unique rows of the reachability matrix. (Using matlab, this matrix is computed as U = unique(R, ‘rows’).) For purposes of social network analysis, what do we learn by counting the number of rows of U (i.e., by computing size(U,1) in matlab)? What do we learn by computing the row sums of U (i.e., by computing sum(U ) in matlab)? What answer would we get if we computed the column sums of U (i.e., sum(U) in matlab)? Briefly explain. 5) [25 points] Briefly describe the main findings in the two papers by Duncan Watts ( American Journal of Sociology 1999, Science 2002) that we covered in lecture. [HINT: Two or three sentences about each paper should be sufficient. You might also want to use some diagrams to describe these findings. If so, make sure that they are properly labeled.] 6) [70 points] Consider the social relation A on the set of individuals {1,…, 8} which is represented by the following graph: 2 5 7 2 1 4 3 6 8 The A relation is also represented by the adjacency matrix A on the attached sheet, which also provides some relevant matlab computations. a) Briefly interpret the R1, R2, R3, and R4 matrices from the attached sheet.
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This note was uploaded on 08/08/2008 for the course SOC 375 taught by Professor Montgomery during the Fall '07 term at University of Wisconsin.

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soc 375 - exams - fall 2007 - Sociology 375 Exam 1 Fall...

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