The document discusses networks and graph concepts. It provides examples of social networks and the internet. It defines nodes, edges, and how networks can be represented using adjacency lists and edge lists. It discusses computing properties of networks like degrees, connected components, distances, and performing breadth-first search.

1. Networks
Part I
Sharad Goel
Columbia University
Computational Social Science: Lecture 5
February 22, 2013

2. High School Dating Network
[ Bearman, Moody, & Stovel, 2004 ]
Image by Mark Newman, via Easley & Kleinberg

3. Corporate E-mail Communication
[ Adamic & Adar, 2004 ]
via Easley & Kleinberg

4. “Internet map 2004” from Math Insight
http://mathinsight.org/image/internet_map_jurvetson_2004

5. Networks/Graphs
Nodes/vertices
people, organizations, webpages, computers
Edges
represent connections between pairs of nodes

9. 2 5
11
1
4 9
6 2

10. 1 2
4
6 4
2 1
2
3
7
13

11. 3
1 2
5
4
1 2 3 4 5 6 7
7 1 0 1 0 1 0 0 0
6
2 1 0 1 0 1 0 0
3 0 1 0 0 1 0 0
4 1 0 0 0 1 0 0
5 0 1 1 1 0 1 1
6 0 0 0 0 1 0 0
7 0 0 0 0 1 0 0

12. 3
1 2
5
4
7 1 {2, 4}
6
2 {1, 3, 5}
3 {2, 5}
4 {1, 5}
5 {2, 3, 4, 6, 7}
6 {5}
7 {5}

13. 3
1 2
5
4
(1, 2)
7 (1, 4)
6
(2, 3)
(2, 5)
(3, 5)
(4, 5)
(5, 6)
(5, 7)

14. 2 5 3
1 2
11
1
4 9 5
4 2
6
7
6 1 {2:2, 4:4}
2 {1:2, 3:5, 5:11}
3 {2:5, 5:1}
4 {1:4, 5:9}
5 {2:11, 3:1, 4:9, 6:6, 7:2}
6 {5:6}
7 {5:2}

15. 2 5 3
1 2
11
1
4 9 5
4 2
6
(1, 2, 2)
7 (1, 4, 4)
6
(2, 3, 5)
(2, 5, 11)
(3, 5, 1)
(4, 5, 9)
(5, 6, 6)
(5, 7, 2)

16. Adjacency list  edge list
(weighted) directed network
Input
Adjacency list
Output
Edge list

17. Adjacency list  edge list
(weighted) directed network
Map
input: u {w1, …, wk}
foreach wi:
output (u, wi)
Reduce
pass

18. Edge list  adjacency list
(weighted) undirected network
Input
Edge list
Output
Adjacency list

19. Edge list  adjacency list
(weighted) undirected network
Map
input: (u, w)
output: (u, w), key := u
output: (w, u), key := w
Reduce
input: u, {w1, …, wk}
identity

20. 3
1 2
5
4
7
6
Degree of node u
# of edges incident on u

21. Edge list  node degrees
undirected network
Input
Edge list
Output
Node degrees

22. Edge list  node degrees
undirected network
Map
input: (u, w)
output: (u, w), key := u
output: (w, u), key := w
Reduce
input: u, {w1, …, wk}
output: u, k

23. Edge list  degree distribution
undirected network
Input
Edge list
Output
Degree distribution

24. Edge list  degree distribution
undirected network
Map
input: (u, w)
output: (u, w), key := u
output: (w, u), key := w
Reduce
input: u, {w1, …, wk}
output: u, k

25. Edge list  degree distribution
undirected network
Map
input: u, k
identity, key := k
Reduce
input: k, {u1, …, um}
output: k, m

26. 3
1 2
5
4
7
6
Path
Sequence of nodes with each
consecutive pair connected by an edge

27. 3
1 2
5
4
7
6
Cycle
Path with at least three edges with first and last
nodes the same and all other nodes distinct

28. Connected Graph
There is a path between every pair of nodes

29. Connected Graph
There is a path between every pair of nodes

30. Connected Component
A connected subset of nodes that is not
contained in any larger connected subset

31. Distance
Length of the shortest path between two nodes

32. Distance
Length of the shortest path between two nodes

33. Breadth-first Search
iteratively explore nodes one layer at a time

34. # initialize distances
dist = {}
for u in G:
dist[u] = NA
dist [u1] = 0
d=0
periphery = { u1 }
while len(periphery) > 0:
# find nodes one step away from the periphery
next_level = {}
for u in periphery:
next_level += { w for w in neighbors[u] if dist[w] == NA }
# update distances
d += 1
for u in next_level:
dist[u] = d
# update periphery
periphery = next_level

35. BFS @ scale
undirected network
Input
edge list, starting node u0
Output
Distance to all nodes from u0

36. BFS @ scale
undirected network
Input: distances (u, d)
1. join distances with edge list
2. foreach (u, d, w) output (w, d+1)
[ also output (u0, 0) ]
3. group by w, and output min d