Representing & Measuring networks

Social and Economic
Network Analysis
UNIT - I

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

Summary

Representing Networks
SOCIAL AND ECONOMIC NETWORKS - CHAPTER 2

Network Vs Graph
nodes, vertices N
links, edges E
network, graph G(N,E)

Types of Networks
UNDIRECTED DIRECTED
Friendship on Facebook Following on Twitter/Instagram

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!!!

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

Types of Networks (undirected)
UNWEIGHTED WEIGHTED
Friendship on Facebook Road/Airlines Network

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

Representing Networks: Adjacency
Matrix

Representing Networks: Edge List

Representing Networks: Adjacency List
1:
2:3, 4
3: 2, 4
4: 5
5: 1, 2

Large & Sparse Graph
7.8 Billion
2.45
Billion
~400
7.8 Billion
1 Billion
~200
Facebook Instagram

Representing Networks
Representation Pros Cons
Adjacency Matrix Simple Memory utilization
Edge list Memory utilization High access time
Adjacency list Low access time
Memory utilization
_

Measuring Networks
SOCIAL AND ECONOMIC NETWORKS - CHAPTER 2
NETWORKS, CROWDS AND MARKETS – CHAPTER 2

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

Measuring Networks - Degree

Measuring Networks – Degree
Degree distribution: A frequency count of the occurrence of each degree
0
1
2
3
4
5
6
7
1 2 3 4
Degree
Frequency

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
Normalized Histogram
Degree

Measuring Networks
CONNECTED COMPONENTS

(a) (b)

Measuring Networks - Connectivity
(a) (b)

STRONGLY CONNECTED GRAPH
Has a directed path from each node to every
other node
WEEKLY CONNECTED GRAPH
Connected without considering the edge
directions
Strongly or Weekly?

Strongly connected component

Measuring Networks
DISTANCE METRICS

Measuring Networks
Node G wants to send a message
to node L.
What options does G have to
deliver the message?

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

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

1) E-A-B-C-A-B-F
2) G-C-A-D
3) A-B-C-A-D
4) A-B-C-A
Identify walk, path, trial, cycle

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

Measuring Networks - Distance
Geodesic distance/Shortest path – Breadth First Search
Friends
Friends of friends
Friends of friends of friends
…

Avg path Length

A 5
B 4
C 3
D 4
E 3
F 3
G 4
H 4
I 4
J 5
K 5
Eccentricity
Largest distance
between all the
nodes

A 5
B 4
C 3
D 4
E 3
F 3
G 4
H 4
I 4
J 5
K 5
Diameter
Maximum
Eccentricity

A 5
B 4
C 3
D 4
E 3
F 3
G 4
H 4
I 4
J 5
K 5
Radius
Minimum
Eccentricity

A 5
B 4
C 3
D 4
E 3
F 3
G 4
H 4
I 4
J 5
K 5
Periphery
Eccentricity = Diameter

A 5
B 4
C 3
D 4
E 3
F 3
G 4
H 4
I 4
J 5
K 5
Center
Eccentricity = Radius

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

Walks of length 2
12-21, 13-31
13-34, 12-24

Walks of length 3
12-24-42
13-34-42
12-21-12
13-31-12

How similar are these networks?

How similar are these networks?
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

Measuring Networks
CLUSTERING COEFFICIENTS

Measuring Networks – Network Density
Network Density =
Avg. Degree / N-1

Measuring Networks – Clustering
coefficients

Measuring Networks – Local Clustering
coefficients
How connected are C’s friends to each other?

coefficients
How connected are J’s friends
to each other?

coefficients
How connected are J’s friends
to each other?
Local clustering coefficient of J = 0

Calculate LCC
A B C

Calculate LCC
Ci = 1 Ci = 1/2 Ci = 0
A B C

Measuring Networks – Global Clustering
coefficients
AVERAGE CLUSTERING COEFFICIENT
Average CC =
TRANSITIVITY
Transitivity = 3 * # triangles
# triads

coefficients
Average CC = (1 + 1 + 1/3 + 0) / 4
= 0.58
Transitivity = (3 * 1) / 5 = 0.6

coefficients
Most nodes have high LCC
High degree node has less LCC
Average CC = 0.93
Transitivity = 0.23

coefficients
Most nodes have low LCC
High degree node has high
LCC
Average CC = 0.25
Transitivity = 0.86

Measuring Networks
CENTRALITY

(a) (b) (c) (d)

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

indegree outdegree betweenness closeness

Degree Centrality
He who has many friends is most
important.
Normalize by N‐1 (most possible)

Degree Centrality
Degree isn’t everything
•Ease of reaching other nodes
•Ability to act as an broker/intermediary

Closeness Centrality
He who is closer to all other
nodes is most important
Normalizeby N‐1
A B C D E

Closeness Centrality

Betweenness Centrality
He who connects all the nodes is
most important.
j≠k≠i Ɛ N
Undirected:
Normalize by (N‐1)(N-2) / 2
Directed:
Normalize by (N‐1)(N-2)

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

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

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

Eigen vector Centrality
0 1 0 1
1 0 1 1
0 1 0 1
1 1 1 0
1 2 3
1
2
3
4
4
1 2 3 4
0.78 1 0.78 1
https://matrixcalc.org/en/vectors.html

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

Representing & Measuring networks

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