# CS 765 Complex Networks

## Due on Tuesday Apr 23, 2013 at 1:00 pm

Power-law network (7 points)

You may wish to refer to the Power-laws “Scale free” networks and the Generating and Fitting Power Law Distributions in Matlab to figure out how to complete the tasks.

Generate 100,000 random integers from a power law distribution with exponent alpha = 2.1

1. What is the largest value in your sample? Is it possible for a node in a network to have a degree this high (assuming you don't allow multiple edges between two nodes)?
2. Construct a histogram of the frequency of occurrence of each integer in your sample. Pajek will let you calculate the degree of each individual node (Net > Partitions > Degree > All). Then, export the partition as a '.clu' file by clicking on the save icon to the left of the partitions drop-down select menu. Now, you can import it into Excel or another program and histogram it. Try both a linear scale plot and a log-log scale plot.
3. What happens to the bins with zero count in the log-log plot?
4. Try a simple linear regression on the log transformation of both variables. In Matlab, you can plot two data sets together as follows: plot(x1,y1,'r-',x2,y2,'b:'). This will plot y1 vs. x1 as a red solid line, and y2 vs. x2 as a blue dotted line. (If you are using the fitlineonloglog.m Matlab script, you will feed it the binned data, and it will take the log of the x and y for you before doing a linear fit). What is your value of the power-law exponent alpha? Include a plot of the data with the fit superimposed.
5. Now exponentially bin the data and fit with a line. What is your value of alpha?
6. Finally, do a cumulative frequency plot of the original data sample. Fit, plot, and report on the fitted exponent and the corresponding value of alpha.
7. Which method was the most accurate? Which one, in your opinion, gave the best view of the data and the fit?
Growing networks: preferential and random growth (3 points)

Open the NetLogo applet: Click on 'setup' to start out with a cycle of 5 vertices. Click on 'Run' to add vertices one by one, each with m edges. Play with the m and gamma parameters.

1. Select m=1 and gamma=0. Add 300 vertices. Click on the 'resize nodes' button to size the vertices by their degree. Repeat the same, but with m = 1 and gamma = 1. What differences do you observe between the two networks, e.g. in terms of appearance, the number of vertices with degree 1, and the maximum degree of any vertex?
2. Generate two networks with 1000 vertices and m = 4. (You can run this faster by adjusting the speed slider at the top.) For one network select gamma = 0, and for the other gamma = 1. Which degree distribution looks more like a power law?