Social Network Analysis and Modelling I

A practical course

Davide Vega D’Aurelio <davide.vegadaurelio@unibo.it>

Nov. 2016

Attribution

Some parts of the lecture are taken from the MIT 6.042J course of “Mathematics for Computer Science”, though in 2011 by Tom Leighton.

It is therefore, used under the CC BY-NC-SA license.

Outline

• Background and fundamentals

• Applications / Tools

• Research problems

• Plan • Introduction

• Social Balance Theory / Connectivity / Centrality

• Network models

Outline

• Background and fundamentals

• Applications / Tools

• Research problems

• Plan • Introduction

• Social Balance Theory / Connectivity / Centrality

• Network models

Introduction / Graph Theory

What is Social Network Analysis?

5

Introduction / Graph Theory

What are (social) networks?

6

Introduction / Graph Theory

Difficult to answer? Here it is an example…

7

Example / Graph Theory

8

• An old “classical” debate about human behaviour:

On average, who has more opposite gender partners, men or women?

Example / Graph Theory

9

• In the popular literature it is much believed that men have more opposite-gender partners than woman.

• Examples • Literature: harem leader is always a man.

• Polygamist cultures: it’s men that has always multiple wives.

Example / Graph Theory

• Sociological study (2004) University Chicago

• 2,500 random interviews concludes

On average, men have 74% more opposite gender partners than women

(in USA)

Example / Graph Theory

11

• ABC News Poll (2004). The American Sex Survey: A Peek Beneath the Sheets • Promo: “a groundbreaking ABC News Primetime Live survey finds a range

of eye popping sexual activities, fantasies, and attitudes in this country, confirming some conventional wisdom, exploding some myths, and venturing where few scientific surveys have gone before.”

• Poll of 1,500 people concludes:

the average man has 20 partners, and the average woman has six (in USA - 2.5% margin error)

On average, men have 233% more opposite gender partners than women (in USA)

https://abcnews.go.com/images/Politics/959a1AmericanSexSurvey.pdf

Example / Graph Theory

Confused? Lets try a mathematical approach…

12

Example / Graph Theory

13

• Def: a Graph is a pair of sets , where: V is a nonempty set of items called vertices, and

E is a set of 2-item subsets of V called edges.

• Def: two nodes are adjacent if

• Def: an edge is incident to

vi, vj 2 V {vi, vj} 2 E

G = (V,E)

vi, vje = {vi, vj}

Example / Graph Theory

14

• Def: for any node , the set of nodes it is connected via. an edge is called its neighbourhood, and is represented as

• Def: the number of for edges incident of a node is called the degree of the node, and is represented as or

• Def: a graph is simple if it has no loops or multiple edges.

vi

N(vi)

di deg(i)

Introduction / Motivation

What is Social Network Analysis?

15

Katherine Faust, Social Network Analysis: Methods and Applications

“The fundamental difference between a social network explanation and a non-network

explanation of a process is the inclusion of concepts and information on relationships

among units in a study.”

16

Introduction / Motivation

17

The social network of friendships within a 34-person karate club

Wayne Zachary. An information flow model for conflict and fission in small groups. Journal of Anthropological Research, 33(4):452–473, 1977.

Introduction / Motivation

18

The network of loans among financial institutions

Morton L. Bech and Enghin Atalay. The topology of the federal funds market. Technical Report 354, Federal Reserve Bank of New York, November 2008.

Introduction / Motivation

19

The network structure of political blogs prior to the 2004 U.S. Presidential Election

Lada Adamic and Natalie Glance. The political blogosphere and the 2004 U.S. election: Divided they blog. In Proceedings of the 3rd International Workshop on Link Discovery, pages 36–43, 2005.

Introduction / Motivation

20

Air traffic connections from West African countries to the world

Gomes MFC, Pastore y Piontti A, Rossi L, Chao D, Longini I, Halloran ME, Vespignani A. Assessing the International Spreading Risk Associated with the 2014 West African Ebola Outbreak. PLOS Currents Outbreaks. 2014 Sep 2 . Edition 1

Introduction / Motivation

21

Real Madrid 1 : 2 Bayern Munchen Passing Distribution

https://scientometrics.wordpress.com/tag/social-network-analysis/

Katherine Faust, Social Network Analysis: Methods and Applications

“The ‘trick’ is to develop formal mathematical definitions that have known graph theoretic

properties, and also capture important intuitive and theoretical aspects of cohesive

subgroups.”

22

Introduction / Primary questions

23

• What we do know about networks?

• How do networks form? Do the “right networks form?

• How do networks influence behaviour?

Introduction / Areas for research

24

The network of Network Analysis Field

Outline

• Background and fundamentals

• Applications / Tools

• Research problems

• Plan • Introduction

• Social Balance Theory / Connectivity / Centrality

• Network models

Social Balance Theory

26

• Check these sentences

a) The friend of my friend is my friend b) The friend of my enemy is my enemy c) The enemy of my enemy is my enemy d) The enemy of my friend is my enemy

Social Balance Theory / Triadic closure

27

• Principle (transitive linking): If two people in a social network have a friend in common, then there is an increased likelihood that they will become friends themselves at some point in the future.

Network of referees for a job

Social Balance Theory / Triadic closure

28

• Principle (transitive linking): If two people in a social network have a friend in common, then there is an increased likelihood that they will become friends themselves at some point in the future.

Network of referees for a job

Social Balance Theory / Triadic closure

29

• Principle (transitive linking): If two people in a social network have a friend in common, then there is an increased likelihood that they will become friends themselves at some point in the future.

Network of referees for a job

Social Balance Theory / Triadic closure

30

• Clustering coefficient analyzes transitivity in undirected graphs

• Def: the clustering coefficient C is defined as:

or

C =(num. triangles)⇥ 6

|paths of length 2|

C =(num. triangles)⇥ 3

num. connected triples

Social Balance Theory / Triadic closure

31

• Example clustering coefficient

Social Balance Theory / Triadic closure

32

• Example clustering coefficient

C =2⇥ 3

2⇥ 3 + 12= 0.33

Social Balance Theory / Triadic closure

33

• Principle: If two people in a social network have an enemy in common, then …

Image source: Social Media Mining: an introduction. Cambridge Press, 2014

Homework

• Get gephi (https://gephi.org)

• Run the tutorial before class (https://gephi.org/tutorials/gephi-tutorial-quick_start.pdf)

• Read and try to understand the PageRank paper