The document discusses various methods for representing and measuring networks and their properties. It covers representing networks using adjacency matrices, edge lists, and adjacency lists. For measuring networks, it discusses degree distribution, connected components, distance metrics like average path length and diameter, clustering coefficients, and centrality measures like betweenness, closeness, and eigenvector centrality. The document provides examples and explanations of these network analysis concepts and techniques.

1. Social and Economic
Network Analysis
UNIT - I

2. Overview
Representing
Networks
Measuring
Networks
23-03-2021 VANI KANDHASAMY, PSGTECH 2

3. Summary
23-03-2021 VANI KANDHASAMY, PSGTECH 3

4. Representing Networks
SOCIAL AND ECONOMIC NETWORKS - CHAPTER 2
23-03-2021 VANI KANDHASAMY, PSGTECH 4

5. Network Vs Graph
23-03-2021 VANI KANDHASAMY, PSGTECH 5
nodes, vertices N
links, edges E
network, graph G(N,E)

6. Types of Networks
UNDIRECTED DIRECTED
23-03-2021 VANI KANDHASAMY, PSGTECH 6
Friendship on Facebook Following on Twitter/Instagram

7. 23-03-2021 VANI KANDHASAMY, PSGTECH 7
NETWORK NODES LINKS DIRECTED/
UNDIRECTED
Internet Device/Router Internet connections
WWW Webpages Links
Power Grid Power plants Cables
Phone Calls Subscribers Calls
Email Email address Emails
Actor Network Actors Co-acting
Guess Directedness!!!

8. 23-03-2021 VANI KANDHASAMY, PSGTECH 8
NETWORK NODES LINKS DIRECTED/
UNDIRECTED
Internet Device/Router Internet connections Undirected
WWW Webpages Links Directed
Power Grid Power plants Cables Undirected
Phone Calls Subscribers Calls Directed
Email Email address Emails Directed
Actor Network Actors Co-acting Undirected

9. Types of Networks (undirected)
UNWEIGHTED WEIGHTED
23-03-2021 VANI KANDHASAMY, PSGTECH 9
Friendship on Facebook Road/Airlines Network

10. Bipartite Graph
▪Examples
Authors-to-Papers (they authored)
Actors-to-Movies (they appeared in)
Users-to-Movies (they rated)
Recipes-to-Ingredients (they contain)
▪Folded / Projected Network
Author collaboration networks
Movie co-rating networks
23-03-2021 VANI KANDHASAMY, PSGTECH 10

11. 23-03-2021 VANI KANDHASAMY, PSGTECH 11

12. Representing Networks: Adjacency
Matrix
23-03-2021 VANI KANDHASAMY, PSGTECH 12

13. Representing Networks: Edge List
23-03-2021 VANI KANDHASAMY, PSGTECH 13

14. Representing Networks: Adjacency List
23-03-2021 VANI KANDHASAMY, PSGTECH 14
1:
2:3, 4
3: 2, 4
4: 5
5: 1, 2

15. Large & Sparse Graph
23-03-2021 VANI KANDHASAMY, PSGTECH 15
7.8 Billion
2.45
Billion
~400
7.8 Billion
1 Billion
~200
Facebook Instagram

16. Representing Networks
23-03-2021 VANI KANDHASAMY, PSGTECH 16
Representation Pros Cons
Adjacency Matrix Simple Memory utilization
Edge list Memory utilization High access time
Adjacency list Low access time
Memory utilization
_

17. Measuring Networks
SOCIAL AND ECONOMIC NETWORKS - CHAPTER 2
NETWORKS, CROWDS AND MARKETS – CHAPTER 2
23-03-2021 VANI KANDHASAMY, PSGTECH 17

18. Measuring Networks - Degree
▪Indegree
how many directed edges are incident on a node
▪Outdegree
how many directed edges originate at a node
▪Degree (in or out)
number of edges incident on a node
▪Average degree:
Undirected graph
Directed graph
outdegree=2
indegree=3
degree=4

19. Measuring Networks - Degree
23-03-2021 VANI KANDHASAMY, PSGTECH 19

20. Measuring Networks – Degree
Degree distribution: A frequency count of the occurrence of each degree
23-03-2021 VANI KANDHASAMY, PSGTECH 20
0
1
2
3
4
5
6
7
1 2 3 4
Degree
Frequency

21. Measuring Networks – Degree
Degree distribution P(k): Probability that a randomly chosen node has degree k
Nk = # nodes with degree k
P(k) = Nk /N
23-03-2021 VANI KANDHASAMY, PSGTECH 21
Normalized Histogram
Degree

22. Measuring Networks
CONNECTED COMPONENTS
23-03-2021 VANI KANDHASAMY, PSGTECH 22

23. 23-03-2021 VANI KANDHASAMY, PSGTECH 23
(a) (b)

24. Measuring Networks - Connectivity
23-03-2021 VANI KANDHASAMY, PSGTECH 24
(a) (b)

25. Measuring Networks - Connectivity
23-03-2021 VANI KANDHASAMY, PSGTECH 25

26. Measuring Networks - Connectivity
STRONGLY CONNECTED GRAPH
Has a directed path from each node to every
other node
WEEKLY CONNECTED GRAPH
Connected without considering the edge
directions
23-03-2021 VANI KANDHASAMY, PSGTECH 26
Strongly or Weekly?

27. Measuring Networks - Connectivity
Strongly connected component
23-03-2021 VANI KANDHASAMY, PSGTECH 27

28. Measuring Networks
DISTANCE METRICS
23-03-2021 VANI KANDHASAMY, PSGTECH 28

29. Measuring Networks
Node G wants to send a message
to node L.
What options does G have to
deliver the message?
23-03-2021 VANI KANDHASAMY, PSGTECH 29

30. Measuring Networks
Node G sends a message to node
L through
▪G-A-N-L,
▪G-A-N-O-L,
▪G-A-N-O-K-L,
▪G-J-O-L,
▪G-J-O-K-L,
▪G-J-F-G-A-N-L,
▪G-J-F-G-J-O-K-L
23-03-2021 VANI KANDHASAMY, PSGTECH 30

31. Measuring Networks - Terminology
Terminology Definition
Walk can have repeated links
Trails walk with no repeated links
Paths trail with no repeated nodes
Cycles path that starts and ends at the same node
23-03-2021 VANI KANDHASAMY, PSGTECH 31

32. 1) E-A-B-C-A-B-F
2) G-C-A-D
3) A-B-C-A-D
4) A-B-C-A
23-03-2021 VANI KANDHASAMY, PSGTECH 32
Identify walk, path, trial, cycle

33. Measuring Networks
1) E-A-B-C-A-B-F WALK
2) G-C-A-D PATH
3) A-B-C-A-D TRAIL
4) A-B-C-A CYCLE
23-03-2021 VANI KANDHASAMY, PSGTECH 33

34. Measuring Networks - Distance
Geodesic distance/Shortest path – Breadth First Search
23-03-2021 VANI KANDHASAMY, PSGTECH 34
Friends
Friends of friends
Friends of friends of friends
…

35. Measuring Networks - Distance
23-03-2021 VANI KANDHASAMY, PSGTECH 35
Avg path Length

36. Measuring Networks - Distance
A 5
B 4
C 3
D 4
E 3
F 3
G 4
H 4
I 4
J 5
K 5
23-03-2021 VANI KANDHASAMY, PSGTECH 36
Eccentricity
Largest distance
between all the
nodes

37. Measuring Networks - Distance
A 5
B 4
C 3
D 4
E 3
F 3
G 4
H 4
I 4
J 5
K 5
23-03-2021 VANI KANDHASAMY, PSGTECH 37
Diameter
Maximum
Eccentricity

38. Measuring Networks - Distance
A 5
B 4
C 3
D 4
E 3
F 3
G 4
H 4
I 4
J 5
K 5
23-03-2021 VANI KANDHASAMY, PSGTECH 38
Radius
Minimum
Eccentricity

39. Measuring Networks - Distance
A 5
B 4
C 3
D 4
E 3
F 3
G 4
H 4
I 4
J 5
K 5
23-03-2021 VANI KANDHASAMY, PSGTECH 39
Periphery
Eccentricity = Diameter

40. Measuring Networks - Distance
A 5
B 4
C 3
D 4
E 3
F 3
G 4
H 4
I 4
J 5
K 5
23-03-2021 VANI KANDHASAMY, PSGTECH 40
Center
Eccentricity = Radius

41. Measuring Networks - Distance
23-03-2021 VANI KANDHASAMY, PSGTECH 41
Metrics Value
Geodesic distance /
Shortest path (A-H)
4 (A-B-C-E-H)
Average path length (Network) 2.5272
Eccentricity (A) 5
Diameter (Network) 5
Radius (Network) 3
Periphery (Network) A, K, J
Center (Network) C, E, F

42. Measuring Networks - Distance
23-03-2021 VANI KANDHASAMY, PSGTECH 42
Walks of length 2
12-21, 13-31
13-34, 12-24

43. Measuring Networks - Distance
23-03-2021 VANI KANDHASAMY, PSGTECH 43
Walks of length 3
12-24-42
13-34-42
12-21-12
13-31-12

44. How similar are these networks?
23-03-2021 VANI KANDHASAMY, PSGTECH 44

45. How similar are these networks?
23-03-2021 VANI KANDHASAMY, PSGTECH 45
Average degree = 2
Degree Distribution =
all nodes have degree 2
Diameter = 7
Average degree = 1.9
Degree Distribution =
6 nodes have degree 3
1 node have degree 2
8 nodes have degree 1
Diameter = 6

46. Measuring Networks
CLUSTERING COEFFICIENTS
23-03-2021 VANI KANDHASAMY, PSGTECH 46

47. Measuring Networks – Network Density
23-03-2021 VANI KANDHASAMY, PSGTECH 47
Network Density =
Avg. Degree / N-1

48. Measuring Networks – Clustering
coefficients
23-03-2021 VANI KANDHASAMY, PSGTECH 48

49. Measuring Networks – Clustering
coefficients
23-03-2021 VANI KANDHASAMY, PSGTECH 49

50. Measuring Networks – Local Clustering
coefficients
How connected are C’s friends to each other?
23-03-2021 VANI KANDHASAMY, PSGTECH 50

51. Measuring Networks – Local Clustering
coefficients
How connected are J’s friends
to each other?
23-03-2021 VANI KANDHASAMY, PSGTECH 51

52. Measuring Networks – Local Clustering
coefficients
How connected are J’s friends
to each other?
Local clustering coefficient of J = 0
23-03-2021 VANI KANDHASAMY, PSGTECH 52

53. Calculate LCC
23-03-2021 VANI KANDHASAMY, PSGTECH 53
A B C

54. Calculate LCC
23-03-2021 VANI KANDHASAMY, PSGTECH 54
Ci = 1 Ci = 1/2 Ci = 0
A B C

55. Measuring Networks – Global Clustering
coefficients
AVERAGE CLUSTERING COEFFICIENT
Average CC =
TRANSITIVITY
Transitivity = 3 * # triangles
# triads
23-03-2021 VANI KANDHASAMY, PSGTECH 55

56. Measuring Networks – Global Clustering
coefficients
Average CC = (1 + 1 + 1/3 + 0) / 4
= 0.58
Transitivity = (3 * 1) / 5 = 0.6
23-03-2021 VANI KANDHASAMY, PSGTECH 56

57. Measuring Networks – Global Clustering
coefficients
Most nodes have high LCC
High degree node has less LCC
Average CC = 0.93
Transitivity = 0.23
23-03-2021 VANI KANDHASAMY, PSGTECH 57

58. Measuring Networks – Global Clustering
coefficients
23-03-2021 VANI KANDHASAMY, PSGTECH 58
Most nodes have low LCC
High degree node has high
LCC
Average CC = 0.25
Transitivity = 0.86

59. Measuring Networks
CENTRALITY
23-03-2021 VANI KANDHASAMY, PSGTECH 59

60. 23-03-2021 VANI KANDHASAMY, PSGTECH 60
(a) (b) (c) (d)

61. Measuring Networks - Centrality
Which nodes are most ‘central’?
Who’s important based on their network position?
❖ Local measure:
• degree centrality – based on degree of the node
❖ Relative to rest of network:
• Closeness centrality – based on average distances
• Betweenness centrality – based on shortest paths through the node
• Eigenvector centrality – based on how important the neighbors are
23-03-2021 VANI KANDHASAMY, PSGTECH 61

62. 23-03-2021 VANI KANDHASAMY, PSGTECH 62
indegree outdegree betweenness closeness

63. Degree Centrality
He who has many friends is most
important.
Normalize by N‐1 (most possible)
23-03-2021 VANI KANDHASAMY, PSGTECH 63

64. Degree Centrality
Degree isn’t everything
•Ease of reaching other nodes
•Ability to act as an broker/intermediary
23-03-2021 VANI KANDHASAMY, PSGTECH 64

65. Closeness Centrality
He who is closer to all other
nodes is most important
23-03-2021 VANI KANDHASAMY, PSGTECH 65
Normalizeby N‐1
A B C D E

66. Closeness Centrality
23-03-2021 VANI KANDHASAMY, PSGTECH 66

67. 23-03-2021 VANI KANDHASAMY, PSGTECH 67

68. Betweenness Centrality
He who connects all the nodes is
most important.
23-03-2021 VANI KANDHASAMY, PSGTECH 68
j≠k≠i Ɛ N
Undirected:
Normalize by (N‐1)(N-2) / 2
Directed:
Normalize by (N‐1)(N-2)

69. Betweenness Centrality
23-03-2021 VANI KANDHASAMY, PSGTECH 69
1-3 1-4 1-5 1-6 3-4 3-5 3-6 4-5 4-6 5-6
1 Shortest path
1-2-3
2 Shortest paths
3-2-5, 3-4-5

70. Betweenness Centrality
Node Betweenness Normalized
1 0 0
2 1.5 0.15
3 1 0.10
4 4 0.40
5 3 0.30
6 0 0
23-03-2021 VANI KANDHASAMY, PSGTECH 70

71. Eigen vector Centrality
He who is friends with important
nodes will also become important
Extended degree centrality
𝑪𝑬 𝒊 = ෍
𝒋:𝒇𝒓𝒊𝒆𝒏𝒅 𝒐𝒇 𝒊
𝑪𝑬 𝒋
𝑪𝑬 𝒊 =
𝟏
𝝀
෍
𝒋
𝑨𝒊𝒋𝑪𝑬 𝒋
𝝀𝑪 = 𝑨𝑪
23-03-2021 VANI KANDHASAMY, PSGTECH 71

72. Eigen vector Centrality
0 1 0 1
1 0 1 1
0 1 0 1
1 1 1 0
23-03-2021 VANI KANDHASAMY, PSGTECH 72
1 2 3
1
2
3
4
4
1 2 3 4
0.78 1 0.78 1
https://matrixcalc.org/en/vectors.html

73. 23-03-2021 VANI KANDHASAMY, PSGTECH 73

74. Overview of Network Measures
Network Level Node Level
• Number of nodes
• Number of edges
• Degree distribution
• Average path length
• Diameter
• Radius
• Network density
• Number of connected components
• Clustering coefficient
• Degree
• Degree centrality
• Betweenness centrality
• Closeness centrality
• Eigenvalue centrality
23-03-2021 VANI KANDHASAMY, PSGTECH 74