Download the file Dining-table_partners.net from the cTools website. This data set is provided with the book 'Exploratory Social Network Analysis with Pajek'. It represents the first and second choices in dining table partners by the residents of a girls' school dormitory.

1 Basic Network Properties

1A (20 points) Strongly connected components and bowtie structure
Suppose the girls are all sitting together at a very large table and they each ordered a different dish from a chinese restaurant (with an extensive menu). Each girl will share her dish only with her 1st or 2nd choice of dining partner. But once she has shared her dish, her 1st and 2nd choice partners can share her dish with their 1st or 2nd choice, etc.

• Find groups of girls who can all sample each other's dishes (hint: try Net>Components>Strong and Draw>Draw-partition)
• Which girls will get to sample no other dishes but their own?
• Display the network of strongly connected components (Operations>Shrink Network>Partition and Draw>Draw-partition) (*I*)
• Identify the bowtie structure (Net>Partitions>Bow-Tie).
  What part of the bowtie is missing? What does this imply for the circulation of the different dishes?

1B (20 points) Clustering

• Create an undirected network (Net>Transform> Arcs->Edges). Compute the local clustering coefficient for each vertex (Net>Vector> Clustering Coefficients>CC1). Draw the network (Draw > Draw Vector). Label each vertex with its clustering coefficient (Options > Mark Vertices Using > Vector Values). (*I*)
• What is the average clustering coefficient for the network? (hint: select the clustering coefficients in the vector drop-down menu, and run Info> Vector. In the report window, the arithmetic mean will give you the value of the average clustering coefficient).
• Sketch (by hand or computer) two of the configurations that produced different clustering coefficients (i.e. take a vertex, draw its neighborhood, and explain why the clustering coefficient of the vertex is what it is).

2 (10 points) Snowball Sampling
Go back to the directed network. You are a prince who just met an enchanting young lady at a ball, but she left at the stroke of midnight and left a shoe behind. Now you'd like to find the shoe's owner. All you know about her is that she lives in this particular girls' dorm. The headmistress won't let you talk to the girls, so the only way you can find your princess is to covertly ask the one girl you know, Ella, to introduce you to her two favorite friends. Once you know her friends, you can ask them to introduce you to their two favorite friends, etc. This is the snowball sampling technique we covered in class.

Highlight the vertices that you will reach using snowball sampling (Net > K-Neighbors > ...) (*I*).
Which girls will you not find using snowball sampling starting with Ella?

3 (20 points) Network Density

Create two random graphs in Pajek (Net>Random Network > Total No. of Arcs) of two different sizes (number of vertices), but the same density (you will need to select the number of arcs accordingly)..
Verify that the density is the same (Info>Network>General) for the two graphs.
Draw both graphs. (*I*)
What do you observe about the appearance of the networks?
What is the average degree for each network?
What does this tell you about density comparisons across differently sized networks?

4 (30 points) Shortest Paths

The inhabitants of circle-world are evenly spaced on a ring. Most are poor and do not own a telephone. They are pretty lazy, so instead of walking over to deliver a message, they shout the message to their neighbor, who shouts it to their neighbor, until the message reaches the intended recipient. There are n = k*i inhabitants in all. Every k^th inhabitant is well off and has a phone and can call any of the i-1 other telephone owners. The inhabitants are called ‘one’, ‘two’, ‘three’, .., according to where they are on the ring. Each is aware where the closest telephone is, so given an intended recipient, say “ninety-six”, they know which way to route a message so that it reaches the recipient in the minimum number of hops (a shout or a phone call count as one hop). The network looks like this:


a) What is the maximum number of hops a message has to make to go between any two individuals (assume k is even).
b) The inhabitants are sometimes forgetful, and if they are not directly the sender or intended recipient of a message, there is a 10% chance that they’ll forget to pass the message on. If two inhabitants are 7 hops removed, what is the probability that a message from one reaches the other?

Extra Credit (10 points) Random sampling - You are a researcher on a budget. You only have time to interview some of the girls in the dormintory. You decide only to interview girls whose names start with the letters M through Z. So what you have is just the 1st and 2nd choices for those girls.

Create a new network that only includes these girls and their choices. You can either create a new .net file (note that you can't just delete lines from the file because vertices need to be consecutively numbered) or try removing nodes and edges within Pajek. (*I*)
What is the average node in-degree? Compare it to the average in-degree for the complete network. What do you observe?
Find the strongly connected components. How do they compare in size and number to the complete network?
Find the largest weakly connected component. How many of the vertices of the original network are in it? What fraction of the girls that were not interviewed are now in the network because of being named as a first or second choice?