Watts–Strogatz graph model. (See Exercise 4.5.27 and Exercise 4.5.28.) Watts and Strogatz...

Watts–Strogatz graph model. (See Exercise 4.5.27 and Exercise 4.5.28.) Watts and Strogatz proposed a hybrid model that contains typical links of vertices near each other (people know their geographic neighbors), plus some random long-range connection links. Plot the effect of adding random edges to an n-by-n grid graph on the average path length and on the cluster coefficient, for n = 100. Do the same for k-ring graphs on V vertices, for V = 10,000 and various values of k up to 10 log V.

EXERCISE 4.5.27

In a grid graph, vertices are arranged in an /7-by-n grid, with edges connecting each vertex to its neighbors above, below, to the left, and to the right in the grid. Compose a Smal 1 Worl d and Graph client that generates grid graphs and tests whether they exhibit the small-world phenomenon (first do Exercise 4.5.23).

6-by-6 grid graph

EXERCISE 4.5.23

Add to SmallWorld (Program 4.5.5) the function isSmallWorld() that takes a graph as an argument and returns true if the graph exhibits the small-world phenomenon (as defined by the specific thresholds given in the text) and false otherwise.

EXERCISE 4.5.28

Extend your solutions to the previous two exercises to also take a command-line argument m and to add m random edges to the graph. Experiment with your programs for graphs with approximately 1,000 vertices to find small-world graphs with relatively few edges.

previous two exercises

Exercise 1

Write a SmallWorld and Graph client that generates k-ring graphs and tests whether they exhibit the small-world phenomenon (first do Exercise 4.5.23).

3-ring graph

EXERCISE 4.5.23

Add to SmallWorld (Program 4.5.5) the function isSmallWorld() that takes a graph as an argument and returns true if the graph exhibits the small-world phenomenon (as defined by the specific thresholds given in the text) and false otherwise.

Exercise 2

In a grid graph, vertices are arranged in an /7-by-n grid, with edges connecting each vertex to its neighbors above, below, to the left, and to the right in the grid. Compose a Smal 1 Worl d and Graph client that generates grid graphs and tests whether they exhibit the small-world phenomenon (first do Exercise 4.5.23).

6-by-6 grid graph

EXERCISE 4.5.23

Add to SmallWorld (Program 4.5.5) the function isSmallWorld() that takes a graph as an argument and returns true if the graph exhibits the small-world phenomenon (as defined by the specific thresholds given in the text) and false otherwise.

Program 4.5.5 Small-world test