Chapter 4.8, Problem 4.145ACI

Social network densities. Social networking sites for business professionals are used to promote one’s business. Each social network involves interactions (connections) between members of the network. Researchers define network density as the ratio of actual network connections to the number of possible one-to-one connections. For example, a network with 10 members has $\left(\begin{array}{c}10\\ 2\end{array}\right)$ = 45 total possible connections. If that network has only 5 connections, the network density is $\frac{5}{45}$ = .111. Sociologists at the University of Michigan assumed that the density x of a social network would follow a uniform distribution between 0 and 1 (Social Networks, 2010).

1. a. On average, what is the density of a randomly selected social network?
2. b. What is the probability that the randomly selected network has a density higher than .7?
3. c. Consider a social network with only 2 members. Explain why the uniform model would not be a good approximation for the distribution of network density.