# CS 790g Seminar: Complex Networks

## Due on Monday Nov 16, 2009 at 2:30 pm

Diffusion in a random graph

In the simple SI model, each node is either susceptible (S) or infected (I). That is, the population starts out with all individuals susceptible to infection, and one individual infected, and thereafter, each noninfected individual who has an infected neighbor is at risk of infection. We'll first look at the Erdos-Renyi random graph.

Access ERDiffusion

1. Vary the probability that an infected node infects a particular neighbor at each time step. How does this influence the rate at which the infection is spreading?
2. How does varying the density (average degree) of the network influence the speed of diffusion?
3. Explain in terms of the network structure.
Diffusion in a network resulting from a growth process

Access RAndPrefAttachment

1. How does varying gamma (which influences whether the growth is preferential or not) influence the speed of diffusion?
2. How does the degree of each node correlate with the how early it gets infected?
Diffusion in a small world In this model, access SmallWorldDiffusionEB
1. Start by setting the rewiring probability to 0. Observe the speed of diffusion in terms of the cumulative number of individuals infected.
2. Increase the rewiring probability just a bit, so you have only a few shortcut edges. How is the rate of diffusion affected?
Diffusion in a small world

Next we'll be looking at a SIS model. Individuals have a constant probability of returning to a susceptible state if they are in an infected state at every time step. This means that they may recover before getting to infect all of their neighbors.

Access SmallWorldDiffusionSIS

1. Try plotting the values for different rewiring probabilities and observe how long the infection survives in the network, and how far it spreads. What happens as the probability of recovery increases?
2. What happens as the probability of infection increases?
3. Can you find a critical threshold in the infection and recovery probabilities such that for a given rewiring probability, below these threshold values the disease always dies out, and above the threshold value, it tends to persist in the network?
Diffusion of an innovation in math teaching (bonus: 4 points)

We'll be using Pajek for this, and following the Ch 8 of the Pajek book. (questions)

Submitting your files

Submission of your homework is via WebCT. You must submit all the required files in a single tar or zip file containing all the files for your submission.

Acknowledgement: The assignment is modified from Lada Adamic.