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1. Compute the degree centrality, betweeness centrality, closeness centrality of nodes in the following graph.
2. A k-regular undirected network is a network in which every vertex has degree ??. Show that the vector ?? = (1, 1, 1, . . . , 1) is an eigenvector of the adjacency matrix with eigenvalue ??.
3. Consider an undirected tree of n vertices. A particular edge in the tree joins vertices 1 and 2 and divides the tree into two disjoint regions of n1 and n2 vertices as sketched here:
Show that the closeness centralities ??1 and ??2 of the two vertices are related by
4. Calculate the betweenness centrality of the ??^{th} vertex from the end of a "line graph" of n vertices. Here a line graph on n vertices 1, 2, 3,..., n has exactly n - 1 edges that connect vertices ?? and ?? + 1, for ?? = 1,..., n - 1.
5. Write a program in your preferred programming language to compute degree centrality of an undirected graph. The program will read the graph from a file called "graph.txt" and output the degree centrality of nodes to a file called "degree.txt".
The file "graph.txt" includes multiples lines in which the first line contains two integers n and m that correspond to the number of nodes and edges in the graph. Each of the following m lines contain two integers u and v, separated by one space, to denote an edge from u to v. Nodes are numbered from 1 to n.
The output file "degree.txt" contains exactly n lines in which the ????h line is the (unnormalized) degree centrality of node ??.
Your submission must include
- The source file(s)
- The sample input/output
- A README file that describes the compile and running instruction
6. Write a program in your preferred programming language to compute (normalized) closeness centrality of an undirected graph. The program will read the graph from a file called "graph.txt" and output the degree centrality of nodes to a file called "closeness.txt".
The file "graph.txt" includes multiples lines in which the first line contains two integers n and m that correspond to the number of nodes and edges in the graph. Each of the following m lines contain two integers u and v, separated by one space, to denote an edge from u to v. Nodes are numbered from 1 to n.
The output file "closeness.txt" contains exactly n lines in which the ????h line is the (normalized) closeness centrality of node ??.
Your submission must include
- The source file(s)
- The sample input/output
- A README file that describes the compile and running instruction