Midterm Review NETS 112
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1 Midterm Review NETS 112
2 Part 1: Network Structure Is this a network? A B C D
3 Part 1: Network Structure Is this a network? A B C D Yes -- G = (V, E), one or both can be empty
4 Part 1: Network Structure What is the distance between A and B? A B C D
5 Part 1: Network Structure What is the distance between A and B? A B C D Infinite or undefined
6 Part 1: Network Structure What is the diameter? A B C D
7 Part 1: Network Structure What is the diameter? A B C D Undefined
8
9 Part 1: Network Structure We say a network is complete if there is an edge between every pair of vertices.
10 Part 1: Network Structure We say a network is complete if there is an edge between every pair of vertices. a) What is the degree of a vertex in a complete network of N vertices?
11 Part 1: Network Structure We say a network is complete if there is an edge between every pair of vertices. a) What is the degree of a vertex in a complete network of N vertices? N-1
12 Part 1: Network Structure We say a network is complete if there is an edge between every pair of vertices. a) What is the degree of a vertex in a complete network of N vertices? N-1 b) What is the diameter of a complete network of N vertices?
13 Part 1: Network Structure We say a network is complete if there is an edge between every pair of vertices. a) What is the degree of a vertex in a complete network of N vertices? N-1 b) What is the diameter of a complete network of N vertices? 1
14 Part 1: Network Structure We say a network is a cycle if the vertices are connected in a ring, with each vertex having a clockwise and counterclockwise neighbor.
15 Part 1: Network Structure We say a network is a cycle if the vertices are connected in a ring, with each vertex having a clockwise and counterclockwise neighbor. a) What is the worst-case diameter in a cycle of N vertices?
16 Part 1: Network Structure We say a network is a cycle if the vertices are connected in a ring, with each vertex having a clockwise and counterclockwise neighbor. a) What is the worst-case diameter in a cycle of N vertices? N/2
17 Part 1: Network Structure We say a network is a cycle if the vertices are connected in a ring, with each vertex having a clockwise and counterclockwise neighbor. a) What is the worst-case diameter in a cycle of N vertices? N/2 b) What is the (average-case) diameter of a cycle of four vertices?
18 Part 1: Network Structure We say a network is a cycle if the vertices are connected in a ring, with each vertex having a clockwise and counterclockwise neighbor. a) What is the worst-case diameter in a cycle of N vertices? N/2 b) What is the (average-case) diameter of a cycle of four vertices? 4/3
19 Part 1: Network Structure In a directed network of N vertices, how many possible edges are there?
20 Part 1: Network Structure In a directed network of N vertices, how many possible edges are there? N(N-1)
21 Part 1: Network Structure In a directed network of N vertices, how many possible edges are there? N(N-1) Suppose you have two friends on Facebook who are not friends with each other. What is the distance between them in the Facebook network?
22 Part 1: Network Structure In a directed network of N vertices, how many possible edges are there? N(N-1) Suppose you have two friends on Facebook who are not friends with each other. What is the distance between them in the Facebook network? 2
23 Part 1: Network Structure We ve seen a lot of histograms that represent degree distributions. What are the x-axis and y-axis labels of a degree distribution?
24 Part 1: Network Structure We ve seen a lot of histograms that represent degree distributions. What are the x-axis and y-axis labels of a degree distribution? x-axis: degree y-axis: number of nodes with that degree
25 Part 1: Network Structure Professor Kearns often talks about how real-life networks have heavy-tailed degree distributions. What does this mean?
26 Part 1: Network Structure Professor Kearns often talks about how real-life networks have heavy-tailed degree distributions. What does this mean? Reliable number of vertices with degree much higher than average
27 Part 1: Network Structure Professor Kearns often talks about how real-life networks have heavy-tailed degree distributions. What does this mean? Reliable number of vertices with degree much higher than average What does the degree distribution of a complete network look like?
28 Part 1: Network Structure Professor Kearns often talks about how real-life networks have heavy-tailed degree distributions. What does this mean? Reliable number of vertices with degree much higher than average What does the degree distribution of a complete network look like? One bar of at degree N - 1 with height N
29 Part 1: Network Structure True/False: The sum of the degrees in a network must always equal the number of edges.
30 Part 1: Network Structure True/False: The sum of the degrees in a network must always equal the number of edges. False (sum of degrees = 2*edges)
31 Part 1: Network Structure True/False: The sum of the degrees in a network must always equal the number of edges. False (sum of degrees = 2*edges) True/False: Adding random long-distance edges to a grid network will reduce the diameter.
32 Part 1: Network Structure True/False: The sum of the degrees in a network must always equal the number of edges. False (sum of degrees = 2*edges) True/False: Adding random long-distance edges to a grid network will reduce the diameter. True
33 Part 2: Erdos Numbers In the Erdos collaboration network, each vertex is a mathematician or computer scientist, and there is an edge between two vertices if they have coauthored a paper together. An individual s Erdos number is the distance from their vertex to Paul Erdos vertex.
34 Part 2: Erdos Numbers In the Erdos collaboration network, each vertex is a mathematician or computer scientist, and there is an edge between two vertices if they have coauthored a paper together. An individual s Erdos number is the distance from their vertex to Paul Erdos vertex. How many individuals have an Erdos number of 0?
35 Part 2: Erdos Numbers In the Erdos collaboration network, each vertex is a mathematician or computer scientist, and there is an edge between two vertices if they have coauthored a paper together. An individual s Erdos number is the distance from their vertex to Paul Erdos vertex. How many individuals have an Erdos number of 0? 1 (Paul Erdos)
36 Part 2: Erdos Numbers Suppose Alex has an Erdos number of 5, and now writes a paper with Sarah and Mike. Sarah has an Erdos number of 3 and Mike has an Erdos number of 7. What is Alex s new Erdos number?
37 Part 2: Erdos Numbers Suppose Alex has an Erdos number of 5, and now writes a paper with Sarah and Mike. Sarah has an Erdos number of 3 and Mike has an Erdos number of 7. What is Alex s new Erdos number? 4
38 Part 2: Erdos Numbers Suppose Alex has an Erdos number of 5, and now writes a paper with Sarah and Mike. Sarah has an Erdos number of 3 and Mike has an Erdos number of 7. What is Alex s new Erdos number? 4 Suppose you write a new paper. Is it possible that your Erdos number increases? Decreases? Stays the same?
39 Part 2: Erdos Numbers Suppose Alex has an Erdos number of 5, and now writes a paper with Sarah and Mike. Sarah has an Erdos number of 3 and Mike has an Erdos number of 7. What is Alex s new Erdos number? 4 Suppose you write a new paper. Is it possible that your Erdos number increases? Decreases? Stays the same? Might decrease or stay the same, but cannot increase.
40 Part 2: Erdos Numbers What does the degree of an individual in the Erdos collaboration network represent?
41 Part 2: Erdos Numbers What does the degree of an individual in the Erdos collaboration network represent? Number of coauthors
42 Part 2: Erdos Numbers What does the degree of an individual in the Erdos collaboration network represent? Number of coauthors Is your degree related to your Erdos number?
43 Part 2: Erdos Numbers What does the degree of an individual in the Erdos collaboration network represent? Number of coauthors Is your degree related to your Erdos number? Not necessarily, but there is probably a correlation
44 Part 2: Erdos Numbers What do you think the degree distribution of this network looks like?
45 Part 2: Erdos Numbers What do you think the degree distribution of this network looks like? Heavy tailed (some mathematicians have many more coauthors than others)
46 Part 2: Erdos Numbers What do you think the degree distribution of this network looks like? Heavy tailed (some mathematicians have many more coauthors than others) Over time, what do you think will happen to the average Erdos number?
47 Part 2: Erdos Numbers What do you think the degree distribution of this network looks like? Heavy tailed (some mathematicians have many more coauthors than others) Over time, what do you think will happen to the average Erdos number? Probably increase
48
49 Part 2: Erdos Numbers Bacon number: Degrees of separation from Kevin Bacon in actor co-star network Erdos-Bacon number: Sum of Erdos and Bacon numbers Who has a finite Erdos-Bacon number? - Natalie Portman (7) - Colin Firth (7) - Carl Sagan (6) - Noam Chomsky (7)
50 Part 3: Contagion Consider the network below: In the forest fire model, if we start a fire at vertex D, how many vertices will burn?
51 Part 3: Contagion Consider the network below: In the forest fire model, if we start a fire at vertex D, how many vertices will burn? 5
52 Part 3: Contagion How did the forest fire demo work?
53 Part 3: Contagion How did the forest fire demo work? Deleted each vertex (and all its edges) with (independent) probability Fire spread to all adjacent cells
54 Part 3: Contagion How did the forest fire demo work? Deleted each vertex (and all its edges) with (independent) probability Fire spread to all adjacent cells How did the viral spread demo work?
55 Part 3: Contagion How did the forest fire demo work? Deleted each vertex (and all its edges) with (independent) probability Fire spread to all adjacent cells How did the viral spread demo work? Rewired local edges to random destination with certain probability Virus spread to adjacent cells with certain probability
56 Part 3: Contagion How can we design a network so as to maximize contagion?
57 Part 3: Contagion How can we design a network so as to maximize contagion? Mix of local / long distance edges
58 Part 3: Contagion How can we design a network so as to maximize contagion? Mix of local / long distance edges What was special about the equilibrium outcome of the altruistic contagion model?
59 Part 3: Contagion How can we design a network so as to maximize contagion? Mix of local / long distance edges What was special about the equilibrium outcome of the altruistic contagion model? Entirely determined by network structure (degrees)
60 Part 4: Navigation Recall the tennis ball experiment from the first day of class. a) How did we find paths between a source and a target?
61 Part 4: Navigation Recall the tennis ball experiment from the first day of class. a) How did we find paths between a source and a target? Find neighbors of source; choose one; find his neighbors; etc
62 Part 4: Navigation Recall the tennis ball experiment from the first day of class. a) How did we find paths between a source and a target? Find neighbors of source; choose one; find his neighbors; etc b) How did we modify this approach to find the shortest path?
63 Part 4: Navigation Recall the tennis ball experiment from the first day of class. a) How did we find paths between a source and a target? Find neighbors of source; choose one; find his neighbors; etc b) How did we modify this approach to find the shortest path? Find neighbors of source; find all neighbors of neighbors of source; etc
64 Part 4: Navigation Recall the tennis ball experiment from the first day of class. a) How did we find paths between a source and a target? Find neighbors of source; choose one; find his neighbors; etc b) How did we modify this approach to find the shortest path? Find neighbors of source; find all neighbors of neighbors of source; etc c) Would our approach in step b work in the real world?
65 Part 4: Navigation Recall the tennis ball experiment from the first day of class. a) How did we find paths between a source and a target? Find neighbors of source; choose one; find his neighbors; etc b) How did we modify this approach to find the shortest path? Find neighbors of source; find all neighbors of neighbors of source; etc c) Would our approach in step b work in the real world? No -- quickly blows up, not scalable
66 Part 4: Navigation What is the relationship between a heavy-tailed degree distribution and efficient navigation?
67 Part 4: Navigation What is the relationship between a heavy-tailed degree distribution and efficient navigation? Heavy-tailed degree distribution -> small diameter -> short paths exist
68 Part 4: Navigation What is the relationship between a heavy-tailed degree distribution and efficient navigation? Heavy-tailed degree distribution -> small diameter -> short paths exist Why can t a network have a large number of vertices, a small maximum degree, and a small diameter at the same time?
69 Part 4: Navigation What is the relationship between a heavy-tailed degree distribution and efficient navigation? Heavy-tailed degree distribution -> small diameter -> short paths exist Why can t a network have a large number of vertices, a small maximum degree, and a small diameter at the same time? Small max degree -> no connectors -> can t make big jumps across network
70 Part 4: Navigation Why is small diameter necessary but not sufficient for efficient navigation?
71 Part 4: Navigation Why is small diameter necessary but not sufficient for efficient navigation? Local / distributed algorithm may not be able to find the short paths
72
73 Part 4: Navigation Consider Kleinberg s model with r = 2. Say we are adding long distance edges to a vertex A. Suppose distance(a,b) = 1 and distance (A,C) = 2. The probability of adding an edge to B is proportional to:
74 Part 4: Navigation Consider Kleinberg s model with r = 2. Say we are adding long distance edges to a vertex A. Suppose distance(a,b) = 1 and distance (A,C) = 2. The probability of adding an edge to B is proportional to: 1
75 Part 4: Navigation Consider Kleinberg s model with r = 2. Say we are adding long distance edges to a vertex A. Suppose distance(a,b) = 1 and distance (A,C) = 2. The probability of adding an edge to B is proportional to: 1 The probability of adding an edge to C is proportional to:
76 Part 4: Navigation Consider Kleinberg s model with r = 2. Say we are adding long distance edges to a vertex A. Suppose distance(a,b) = 1 and distance (A,C) = 2. The probability of adding an edge to B is proportional to: 1 The probability of adding an edge to C is proportional to: 1/4
77 Part 4: Navigation In a grid network, there are 4 vertices at distance 1 away from A, and 8 vertices at distance 2 away from A.
78 Part 4: Navigation In a grid network, there are 4 vertices at distance 1 away from A, and 8 vertices at distance 2 away from A. The probability of adding an edge of distance 1 to A is:
79 Part 4: Navigation In a grid network, there are 4 vertices at distance 1 away from A, and 8 vertices at distance 2 away from A. The probability of adding an edge of distance 1 to A is: total weight = 4*1 + ¼*8 = 6 4/6 = 2/3
80 Part 4: Navigation In a grid network, there are 4 vertices at distance 1 away from A, and 8 vertices at distance 2 away from A. The probability of adding an edge of distance 1 to A is: total weight = 4*1 + ¼*8 = 6 4/6 = 2/3 The probability of adding an edge to of distance 2 to A is:
81 Part 4: Navigation In a grid network, there are 4 vertices at distance 1 away from A, and 8 vertices at distance 2 away from A. The probability of adding an edge of distance 1 to A is: total weight = 4*1 + ¼*8 = 6 4/6 = 2/3 The probability of adding an edge to of distance 2 to A is: 2/6 = 1/3
82 Part 4: Navigation What do we mean by r = 2 is a knife s edge result in the Kleinberg model?
83 Part 4: Navigation What do we mean by r = 2 is a knife s edge result in the Kleinberg model? Only r = 2 permits navigation in log(n) steps
84 Fourth Column
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