1. DFS and strongly connected components
Please complete this part by drawing directly on the image of the network.

• Do a depth first traversal of the graph starting with the vertex "A".
• For each vertex, write the discovery and finish time inside the circle (e.g. "1/15" means discovered at time 1, finished at time 15).
• Mark the edges in your depth first forest (or tree) with a "t"
• Mark back edges with "b"s, forward edges with "f"s and cross edges with "c".
• Now do a depth first traversal of the transpose graph below in reverse order of finishing times.

• Write the discovery/finishing times on each vertex as you go.
• As you identify the strongly connected components, draw a boundary around them.

2. Growing networks in NetLogo
Download and install NetLogo. Download the RandAndPrefNet.nlogo.

• Open the model. 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. For each of the m-edges:
  • With probability gamma the endpoint is chosen randomly
  • With probability (1-gamma) the endpoint is chosen preferrentially according to degree
  Play with the parameters. Then 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. Comment on the differences between the two networks, .e.g. in terms of appearance, the number of vertices with degree 1, and the maximum degree of any vertex.
• Next generate two networks with 1000 vertices and m=4. (you can run this faster by turning off the display, layout, and even the degree distribution plots and then turn them back on when you've added about the desired number of vertices). For one network select gamma = 0, and for the other gamma = 1. Comment on the degree distributions. Which one looks more like a power-law?
  

Extra credit (up to 10 points)
Modify the NetLogo model such that vertices are added without any edges automatically attached to them. Instead, every time you add a vertex, you add m edges, but choose the starting points of the edges randomly and the other preferentially with probability (1-gamma) in proportion to k+1, where k is the degree of each vertex.

• Submit a screenshot of your network after 500 vertices have been added (along with your m and gamma).
• Make an observation or two about how your new network differs from the original network with the same m and gamma.