- 7 Graph Properties And Metrics Ii Bookmark This Page Exercises Due Jul 6 2022 07 59 Edt Small Graph 6 Points Possibl 1 (58.79 KiB) Viewed 12 times
7. Graph Properties and Metrics - II Bookmark this page Exercises due Jul 6, 2022 07:59 EDT Small Graph 6 points possible (graded) Download the file directed graph.txt. You may use any computational tool for this problem. The file is in the edge list format with each line ij" indicating a directed edge from vertex i to vertex j. For example, the edge list of the graph G = ({0, 1, 2}, {{0,1}, {0,2}}) is 01 02 The example above assumes an undirected graph, but the same edge list can represent the directed graph G({0, 1,2), ((0, 1), (0, 2)}). Answer the following questions. 1. How many nodes does the graph have? 2. How many edges does the graph possess? 3. Does the graph contain self loops? Yes No 4. Does the graph have directed cycles not involving self loops? Hint: Think along the lines of the presence or absence of nodes with no incoming edges other than self loops. Yes No 5. This graph was generated using the following model: We fixed the number of nodes and every possible directed edge was selected with a probability p independently of all other edges. Note that in the graph we also consider possible edges that construct self loops. What is the maximum likelihood estimate of p? Provide an answer accurate to at least five significant figures (graded to 0.01% tolerance), you can also enter your answer as an exact fraction. 6. Given the above generative model, what is the p-value of observing this graph for the null hypothesis that p = 0.1? Assume that the test statistic is where 5 is the sample mean and and are the mean and standard deviation of the sample mean under the null hypothesis. The sample mean is the fraction of number of edges observed to the total number of possible edges. Use the central limit theorem approximation to compute the required probability. Provide an answer accurate to at least three significant figures (graded to 1% tolerance). Submit You have used 0 of 6 attempts Save