introduction to social network analysis in rstatnet.org/workshops/introtosnainr.pdfintroduction to...

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

1-133

Introduction to Social Network Analysis inR

Lorien Jasny1

1Q-Step Centre, Exeter Universityl.jasny@exeter.ac.uk

Sunbelt XXXVIII, Utrecht University26 June 2018

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

2-133

Think Formally

A network is not just ametaphor: it is aprecise, mathematicalconstruct

of nodes(vertices, actors) N andedges (ties, relations) Ethat can be directed orundirected. We caninclude information(attributes) on thenodes as well as theedges.

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

2-133

Think Formally

A network is not just ametaphor: it is aprecise, mathematicalconstruct of nodes(vertices, actors) N

andedges (ties, relations) Ethat can be directed orundirected. We caninclude information(attributes) on thenodes as well as theedges.

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

2-133

Think Formally

A network is not just ametaphor: it is aprecise, mathematicalconstruct of nodes(vertices, actors) N andedges (ties, relations) E

that can be directed orundirected. We caninclude information(attributes) on thenodes as well as theedges.

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

2-133

Think Formally

A network is not just ametaphor: it is aprecise, mathematicalconstruct of nodes(vertices, actors) N andedges (ties, relations) Ethat can be directed

orundirected. We caninclude information(attributes) on thenodes as well as theedges.

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

2-133

Think Formally

A network is not just ametaphor: it is aprecise, mathematicalconstruct of nodes(vertices, actors) N andedges (ties, relations) Ethat can be directed orundirected.

We caninclude information(attributes) on thenodes as well as theedges.

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

2-133

Think Formally

A network is not just ametaphor: it is aprecise, mathematicalconstruct of nodes(vertices, actors) N andedges (ties, relations) Ethat can be directed orundirected. We caninclude information(attributes) on thenodes

as well as theedges.

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

2-133

Think Formally

A network is not just ametaphor: it is aprecise, mathematicalconstruct of nodes(vertices, actors) N andedges (ties, relations) Ethat can be directed orundirected. We caninclude information(attributes) on thenodes as well as theedges.

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

3-133

Network Intuition

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

3-133

Network Intuition

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

4-133

Why network methods

• We need a new language to describe what’s going on

• Cannot simply use existing statistical methods

• The whole point is that observations are interdependent

• Want to explicitly model these interdependencies

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

4-133

Why network methods

• We need a new language to describe what’s going on

• Cannot simply use existing statistical methods

• The whole point is that observations are interdependent

• Want to explicitly model these interdependencies

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

4-133

Why network methods

• We need a new language to describe what’s going on

• Cannot simply use existing statistical methods

• The whole point is that observations are interdependent

• Want to explicitly model these interdependencies

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

4-133

Why network methods

• We need a new language to describe what’s going on

• Cannot simply use existing statistical methods

• The whole point is that observations are interdependent

• Want to explicitly model these interdependencies

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

4-133

Why network methods

• We need a new language to describe what’s going on

• Cannot simply use existing statistical methods

• The whole point is that observations are interdependent

• Want to explicitly model these interdependencies

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

5-133

Data Structures

n1 n2 n3 n4 n5 n6n1

- 1 0 0 1 0

n2

0 - 1 1 0 0

n3

0 1 - 0 0 0

n4

0 0 0 - 1 0

n5

0 0 0 0 - 1

n6

0 1 0 0 0 -

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

5-133

Data Structures

n1 n2 n3 n4 n5 n6n1

- 1 0 0 1 0

n2

0 - 1 1 0 0

n3

0 1 - 0 0 0

n4

0 0 0 - 1 0

n5

0 0 0 0 - 1

n6

0 1 0 0 0 -

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

5-133

Data Structures

n1 n2 n3 n4 n5 n6n1

-

1

0 0 1 0

n2

0 - 1 1 0 0

n3

0 1 - 0 0 0

n4

0 0 0 - 1 0

n5

0 0 0 0 - 1

n6

0 1 0 0 0 -

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

5-133

Data Structures

n1 n2 n3 n4 n5 n6n1

-

1

0 0

1

0

n2

0 - 1 1 0 0

n3

0 1 - 0 0 0

n4

0 0 0 - 1 0

n5

0 0 0 0 - 1

n6

0 1 0 0 0 -

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

5-133

Data Structures

n1 n2 n3 n4 n5 n6n1

-

1

0 0

1

0

n2

0 -

1 1

0 0

n3

0

1

- 0 0 0

n4

0 0 0 -

1

0

n5

0 0 0 0 -

1n6

0

1

0 0 0 -

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

5-133

Data Structures

n1 n2 n3 n4 n5 n6n1 - 1

0 0

1

0

n2

0

- 1 1

0 0

n3

0

1 -

0 0 0

n4

0 0 0

- 1

0

n5

0 0 0 0

- 1n6

0

1

0 0 0

-

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

5-133

Data Structures

n1 n2 n3 n4 n5 n6n1 - 1 0 0 1 0n2 0 - 1 1 0 0n3 0 1 - 0 0 0n4 0 0 0 - 1 0n5 0 0 0 0 - 1n6 0 1 0 0 0 -

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

6-133

Data Structures

Sender Receiver Weight

n1 n2 1n1 n5 1n2 n3 1n2 n6 1n3 n2 1n4 n5 1n5 n6 1n6 n2 1

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

6-133

Data Structures

Sender Receiver Weight

n1 n2 1n1 n5 1n2 n3 1n2 n6 1n3 n2 1n4 n5 1n5 n6 1n6 n2 1

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

6-133

Data Structures

Sender Receiver Weightn1 n2 1

n1 n5 1n2 n3 1n2 n6 1n3 n2 1n4 n5 1n5 n6 1n6 n2 1

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

6-133

Data Structures

Sender Receiver Weightn1 n2 1n1 n5 1n2 n3 1n2 n6 1n3 n2 1n4 n5 1n5 n6 1n6 n2 1

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

7-133

The R Environment

• R is both a language and a software platform

• R software is open-source, cross-platform, and free

• Its home on the web: http://www.r-project.org

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

7-133

The R Environment

• R is both a language and a software platform

• R software is open-source, cross-platform, and free

• Its home on the web: http://www.r-project.org

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

7-133

The R Environment

• R is both a language and a software platform

• R software is open-source, cross-platform, and free

• Its home on the web: http://www.r-project.org

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

7-133

The R Environment

• R is both a language and a software platform

• R software is open-source, cross-platform, and free

• Its home on the web: http://www.r-project.org

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

8-133

Basics

• R uses a command-like environment, like stata or sas

• R is highly extendable• You can write your own custom functions• There are 6000 free add-on packages

• Generally good at reading in/writing out other fileformats

• Everything in R is an object – data, functions,everything

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

8-133

Basics

• R uses a command-like environment, like stata or sas

• R is highly extendable• You can write your own custom functions• There are 6000 free add-on packages

• Generally good at reading in/writing out other fileformats

• Everything in R is an object – data, functions,everything

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

8-133

Basics

• R uses a command-like environment, like stata or sas

• R is highly extendable

• You can write your own custom functions• There are 6000 free add-on packages

• Generally good at reading in/writing out other fileformats

• Everything in R is an object – data, functions,everything

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

8-133

Basics

• R uses a command-like environment, like stata or sas

• R is highly extendable• You can write your own custom functions

• There are 6000 free add-on packages

• Generally good at reading in/writing out other fileformats

• Everything in R is an object – data, functions,everything

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

8-133

Basics

• R uses a command-like environment, like stata or sas

• R is highly extendable• You can write your own custom functions• There are 6000 free add-on packages

• Generally good at reading in/writing out other fileformats

• Everything in R is an object – data, functions,everything

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

8-133

Basics

• R uses a command-like environment, like stata or sas

• R is highly extendable• You can write your own custom functions• There are 6000 free add-on packages

• Generally good at reading in/writing out other fileformats

• Everything in R is an object – data, functions,everything

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

8-133

Basics

• R uses a command-like environment, like stata or sas

• R is highly extendable• You can write your own custom functions• There are 6000 free add-on packages

• Generally good at reading in/writing out other fileformats

• Everything in R is an object – data, functions,everything

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

9-133

Fundamentals

• When you type commands at the prompt ‘>’ and hitENTER

• R tries to interpret what you’ve asked it to do(evaluation)

• If it understands what you’ve written, it does it(execution)

• If it doesn’t, it will likely give you an error or a warning

• Some commands trigger R to print to the screen, othersdon’t

• If you type an incomplete command, R will usuallyrespond by changing the command prompt to the ‘+’character

• Hit ESC on a Mac to cancel• Type in Ctrl + C on Windows and Linux to cancel

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

9-133

Fundamentals

• When you type commands at the prompt ‘>’ and hitENTER

• R tries to interpret what you’ve asked it to do(evaluation)

• If it understands what you’ve written, it does it(execution)

• If it doesn’t, it will likely give you an error or a warning

• Some commands trigger R to print to the screen, othersdon’t

• If you type an incomplete command, R will usuallyrespond by changing the command prompt to the ‘+’character

• Hit ESC on a Mac to cancel• Type in Ctrl + C on Windows and Linux to cancel

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

9-133

Fundamentals

• When you type commands at the prompt ‘>’ and hitENTER

• R tries to interpret what you’ve asked it to do(evaluation)

• If it understands what you’ve written, it does it(execution)

• If it doesn’t, it will likely give you an error or a warning

• Some commands trigger R to print to the screen, othersdon’t

• If you type an incomplete command, R will usuallyrespond by changing the command prompt to the ‘+’character

• Hit ESC on a Mac to cancel• Type in Ctrl + C on Windows and Linux to cancel

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

9-133

Fundamentals

• When you type commands at the prompt ‘>’ and hitENTER

• R tries to interpret what you’ve asked it to do(evaluation)

• If it understands what you’ve written, it does it(execution)

• If it doesn’t, it will likely give you an error or a warning

• Some commands trigger R to print to the screen, othersdon’t

• If you type an incomplete command, R will usuallyrespond by changing the command prompt to the ‘+’character

• Hit ESC on a Mac to cancel• Type in Ctrl + C on Windows and Linux to cancel

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

9-133

Fundamentals

• When you type commands at the prompt ‘>’ and hitENTER

• R tries to interpret what you’ve asked it to do(evaluation)

• If it understands what you’ve written, it does it(execution)

• If it doesn’t, it will likely give you an error or a warning

• Some commands trigger R to print to the screen, othersdon’t

• If you type an incomplete command, R will usuallyrespond by changing the command prompt to the ‘+’character

• Hit ESC on a Mac to cancel• Type in Ctrl + C on Windows and Linux to cancel

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

9-133

Fundamentals

• When you type commands at the prompt ‘>’ and hitENTER

• R tries to interpret what you’ve asked it to do(evaluation)

• If it understands what you’ve written, it does it(execution)

• If it doesn’t, it will likely give you an error or a warning

• Some commands trigger R to print to the screen, othersdon’t

• If you type an incomplete command, R will usuallyrespond by changing the command prompt to the ‘+’character

• Hit ESC on a Mac to cancel• Type in Ctrl + C on Windows and Linux to cancel

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

9-133

Fundamentals

• When you type commands at the prompt ‘>’ and hitENTER

• R tries to interpret what you’ve asked it to do(evaluation)

• If it understands what you’ve written, it does it(execution)

• If it doesn’t, it will likely give you an error or a warning

• Some commands trigger R to print to the screen, othersdon’t

• If you type an incomplete command, R will usuallyrespond by changing the command prompt to the ‘+’character

• Hit ESC on a Mac to cancel• Type in Ctrl + C on Windows and Linux to cancel

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

9-133

Fundamentals

• When you type commands at the prompt ‘>’ and hitENTER

• R tries to interpret what you’ve asked it to do(evaluation)

• If it understands what you’ve written, it does it(execution)

• If it doesn’t, it will likely give you an error or a warning

• Some commands trigger R to print to the screen, othersdon’t

• If you type an incomplete command, R will usuallyrespond by changing the command prompt to the ‘+’character

• Hit ESC on a Mac to cancel• Type in Ctrl + C on Windows and Linux to cancel

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

10-133

Data Structures in R

• R has several built-in data types and structures

• Common data types:• Numeric (integers, numbers, etc.)

• 12• 3.14

• Strings (alphanumeric characters in quotation marks)

• “hello”• “3.14”

• Common data structures• Vectors• Matrices• Data Frames• Network objects

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

10-133

Data Structures in R

• R has several built-in data types and structures

• Common data types:• Numeric (integers, numbers, etc.)

• 12• 3.14

• Strings (alphanumeric characters in quotation marks)

• “hello”• “3.14”

• Common data structures• Vectors• Matrices• Data Frames• Network objects

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

10-133

Data Structures in R

• R has several built-in data types and structures

• Common data types:

• Numeric (integers, numbers, etc.)• 12• 3.14

• Strings (alphanumeric characters in quotation marks)

• “hello”• “3.14”

• Common data structures• Vectors• Matrices• Data Frames• Network objects

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

10-133

Data Structures in R

• R has several built-in data types and structures

• Common data types:• Numeric (integers, numbers, etc.)

• 12• 3.14

• Strings (alphanumeric characters in quotation marks)

• “hello”• “3.14”

• Common data structures• Vectors• Matrices• Data Frames• Network objects

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

10-133

Data Structures in R

• R has several built-in data types and structures

• Common data types:• Numeric (integers, numbers, etc.)

• 12• 3.14

• Strings (alphanumeric characters in quotation marks)

• “hello”• “3.14”

• Common data structures• Vectors• Matrices• Data Frames• Network objects

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

10-133

Data Structures in R

• R has several built-in data types and structures

• Common data types:• Numeric (integers, numbers, etc.)

• 12• 3.14

• Strings (alphanumeric characters in quotation marks)

• “hello”• “3.14”

• Common data structures• Vectors• Matrices• Data Frames• Network objects

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

10-133

Data Structures in R

• R has several built-in data types and structures

• Common data types:• Numeric (integers, numbers, etc.)

• 12• 3.14

• Strings (alphanumeric characters in quotation marks)• “hello”• “3.14”

• Common data structures• Vectors• Matrices• Data Frames• Network objects

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

10-133

Data Structures in R

• R has several built-in data types and structures

• Common data types:• Numeric (integers, numbers, etc.)

• 12• 3.14

• Strings (alphanumeric characters in quotation marks)• “hello”• “3.14”

• Common data structures• Vectors• Matrices• Data Frames• Network objects

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

10-133

Data Structures in R

• R has several built-in data types and structures

• Common data types:• Numeric (integers, numbers, etc.)

• 12• 3.14

• Strings (alphanumeric characters in quotation marks)• “hello”• “3.14”

• Common data structures

• Vectors• Matrices• Data Frames• Network objects

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

10-133

Data Structures in R

• R has several built-in data types and structures

• Common data types:• Numeric (integers, numbers, etc.)

• 12• 3.14

• Strings (alphanumeric characters in quotation marks)• “hello”• “3.14”

• Common data structures• Vectors

• Matrices• Data Frames• Network objects

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

10-133

Data Structures in R

• R has several built-in data types and structures

• Common data types:• Numeric (integers, numbers, etc.)

• 12• 3.14

• Strings (alphanumeric characters in quotation marks)• “hello”• “3.14”

• Common data structures• Vectors• Matrices

• Data Frames• Network objects

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

10-133

Data Structures in R

• R has several built-in data types and structures

• Common data types:• Numeric (integers, numbers, etc.)

• 12• 3.14

• Strings (alphanumeric characters in quotation marks)• “hello”• “3.14”

• Common data structures• Vectors• Matrices• Data Frames

• Network objects

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

10-133

Data Structures in R

• R has several built-in data types and structures

• Common data types:• Numeric (integers, numbers, etc.)

• 12• 3.14

• Strings (alphanumeric characters in quotation marks)• “hello”• “3.14”

• Common data structures• Vectors• Matrices• Data Frames• Network objects

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

11-133

Vectors

A vector is a one-dimensional data structure

n1 2 3 4

• Vectors are indexed starting at 1

• A vector of length n has n cells

• Each cell can hold a single value

• Vectors can only store data of the same type – either allstrings or all numerical but not both

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

11-133

Vectors

A vector is a one-dimensional data structure

n

1 2 3 4

• Vectors are indexed starting at 1

• A vector of length n has n cells

• Each cell can hold a single value

• Vectors can only store data of the same type – either allstrings or all numerical but not both

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

11-133

Vectors

A vector is a one-dimensional data structure

n1 2 3 4

• Vectors are indexed starting at 1

• A vector of length n has n cells

• Each cell can hold a single value

• Vectors can only store data of the same type – either allstrings or all numerical but not both

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

11-133

Vectors

A vector is a one-dimensional data structure

n1 2 3 4

• Vectors are indexed starting at 1

• A vector of length n has n cells

• Each cell can hold a single value

• Vectors can only store data of the same type – either allstrings or all numerical but not both

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

11-133

Vectors

A vector is a one-dimensional data structure

n1 2 3 4

• Vectors are indexed starting at 1

• A vector of length n has n cells

• Each cell can hold a single value

• Vectors can only store data of the same type – either allstrings or all numerical but not both

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

12-133

Working with Vectors in R

• We index vectors in R using “square bracket notation”

• Example:• you have a vector of numeric values called testScores• To retrieve the value in the third cell, typetestScores[3]

• To retrieve all BUT the third value, typetestScoress[-3]

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

12-133

Working with Vectors in R

• We index vectors in R using “square bracket notation”

• Example:• you have a vector of numeric values called testScores• To retrieve the value in the third cell, typetestScores[3]

• To retrieve all BUT the third value, typetestScoress[-3]

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

12-133

Working with Vectors in R

• We index vectors in R using “square bracket notation”

• Example:• you have a vector of numeric values called testScores

• To retrieve the value in the third cell, typetestScores[3]

• To retrieve all BUT the third value, typetestScoress[-3]

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

12-133

Working with Vectors in R

• We index vectors in R using “square bracket notation”

• Example:• you have a vector of numeric values called testScores• To retrieve the value in the third cell, typetestScores[3]

• To retrieve all BUT the third value, typetestScoress[-3]

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

12-133

Working with Vectors in R

• We index vectors in R using “square bracket notation”

• Example:• you have a vector of numeric values called testScores• To retrieve the value in the third cell, typetestScores[3]

• To retrieve all BUT the third value, typetestScoress[-3]

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

13-133

Two-dimensional data in R

• Most (all?) of us are familiar with two-dimensionaldata like that in spreadsheets

• R has two built-in data structures for storingtwo-dimensional data

• Matrices• Data Frames

• In most instances, they behave the same

• Most functions will accept either a matrix or a dataframe

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

13-133

Two-dimensional data in R

• Most (all?) of us are familiar with two-dimensionaldata like that in spreadsheets

• R has two built-in data structures for storingtwo-dimensional data

• Matrices• Data Frames

• In most instances, they behave the same

• Most functions will accept either a matrix or a dataframe

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

13-133

Two-dimensional data in R

• Most (all?) of us are familiar with two-dimensionaldata like that in spreadsheets

• R has two built-in data structures for storingtwo-dimensional data

• Matrices

• Data Frames

• In most instances, they behave the same

• Most functions will accept either a matrix or a dataframe

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

13-133

Two-dimensional data in R

• Most (all?) of us are familiar with two-dimensionaldata like that in spreadsheets

• R has two built-in data structures for storingtwo-dimensional data

• Matrices• Data Frames

• In most instances, they behave the same

• Most functions will accept either a matrix or a dataframe

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

13-133

Two-dimensional data in R

• Most (all?) of us are familiar with two-dimensionaldata like that in spreadsheets

• R has two built-in data structures for storingtwo-dimensional data

• Matrices• Data Frames

• In most instances, they behave the same

• Most functions will accept either a matrix or a dataframe

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

13-133

Two-dimensional data in R

• Most (all?) of us are familiar with two-dimensionaldata like that in spreadsheets

• R has two built-in data structures for storingtwo-dimensional data

• Matrices• Data Frames

• In most instances, they behave the same

• Most functions will accept either a matrix or a dataframe

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

14-133

Matrices versus data frames in R

• Matrices can only store data of one type• Either all strings or all numbers, but not both• If you try to give it multiple types, R converts

everything to string format

• Data frames can store data of multiple types• Ideal for classical data analysis where you might have a

mix of numerical and string data

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

14-133

Matrices versus data frames in R

• Matrices can only store data of one type

• Either all strings or all numbers, but not both• If you try to give it multiple types, R converts

everything to string format

• Data frames can store data of multiple types• Ideal for classical data analysis where you might have a

mix of numerical and string data

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

14-133

Matrices versus data frames in R

• Matrices can only store data of one type• Either all strings or all numbers, but not both

• If you try to give it multiple types, R convertseverything to string format

• Data frames can store data of multiple types• Ideal for classical data analysis where you might have a

mix of numerical and string data

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

14-133

Matrices versus data frames in R

• Matrices can only store data of one type• Either all strings or all numbers, but not both• If you try to give it multiple types, R converts

everything to string format

• Data frames can store data of multiple types• Ideal for classical data analysis where you might have a

mix of numerical and string data

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

14-133

Matrices versus data frames in R

• Matrices can only store data of one type• Either all strings or all numbers, but not both• If you try to give it multiple types, R converts

everything to string format

• Data frames can store data of multiple types

• Ideal for classical data analysis where you might have amix of numerical and string data

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

14-133

Matrices versus data frames in R

• Matrices can only store data of one type• Either all strings or all numbers, but not both• If you try to give it multiple types, R converts

everything to string format

• Data frames can store data of multiple types• Ideal for classical data analysis where you might have a

mix of numerical and string data

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

15-133

Working with matrices

id name age sex handed lastDocVisit

5012 Danielle 44 F R 20122331 Josh 44 M R 20081989 Mark 40 M R 20102217 Emma 32 F L 20122912 Sarah 33 F R 2011

• Can also use “square bracket notation”

• Inside the square brackets, the first position refers tothe row(s) and the second to the column(s)

• If this matrix is called friendSurvey, the command toretrieve Josh’s age is friendSurvey[2,3]

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

15-133

Working with matrices

id name age sex handed lastDocVisit

5012 Danielle 44 F R 20122331 Josh 44 M R 20081989 Mark 40 M R 20102217 Emma 32 F L 20122912 Sarah 33 F R 2011

• Can also use “square bracket notation”

• Inside the square brackets, the first position refers tothe row(s) and the second to the column(s)

• If this matrix is called friendSurvey, the command toretrieve Josh’s age is friendSurvey[2,3]

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

15-133

Working with matrices

id name age sex handed lastDocVisit

5012 Danielle 44 F R 20122331 Josh 44 M R 20081989 Mark 40 M R 20102217 Emma 32 F L 20122912 Sarah 33 F R 2011

• Can also use “square bracket notation”

• Inside the square brackets, the first position refers tothe row(s) and the second to the column(s)

• If this matrix is called friendSurvey, the command toretrieve Josh’s age is friendSurvey[2,3]

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

15-133

Working with matrices

id name age sex handed lastDocVisit

5012 Danielle 44 F R 20122331 Josh 44 M R 20081989 Mark 40 M R 20102217 Emma 32 F L 20122912 Sarah 33 F R 2011

• Can also use “square bracket notation”

• Inside the square brackets, the first position refers tothe row(s) and the second to the column(s)

• If this matrix is called friendSurvey, the command toretrieve Josh’s age is friendSurvey[2,3]

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

16-133

Working with data frames

• Square bracket notation works for data frames as well

• Data frames provide another option: dollar signnotation

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

16-133

Working with data frames

• Square bracket notation works for data frames as well

• Data frames provide another option: dollar signnotation

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

16-133

Working with data frames

• Square bracket notation works for data frames as well

• Data frames provide another option: dollar signnotation

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

17-133

Working with data frames

id name age sex handed lastDocVisit

5012 Danielle 44 F R 20122331 Josh 44 M R 20081989 Mark 40 M R 20102217 Emma 32 F L 20122912 Sarah 33 F R 2011

• To retrieve the ‘sex’ column as a vector, usefriendSurvey$sex

• To retrieve Josh’s age, use friendSurvey$age[2]

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

17-133

Working with data frames

id name age sex handed lastDocVisit

5012 Danielle 44 F R 20122331 Josh 44 M R 20081989 Mark 40 M R 20102217 Emma 32 F L 20122912 Sarah 33 F R 2011

• To retrieve the ‘sex’ column as a vector, usefriendSurvey$sex

• To retrieve Josh’s age, use friendSurvey$age[2]

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

17-133

Working with data frames

id name age sex handed lastDocVisit

5012 Danielle 44 F R 20122331 Josh 44 M R 20081989 Mark 40 M R 20102217 Emma 32 F L 20122912 Sarah 33 F R 2011

• To retrieve the ‘sex’ column as a vector, usefriendSurvey$sex

• To retrieve Josh’s age, use friendSurvey$age[2]

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

18-133

Network Objects

• stores an adjacency matrix or an edgelist as well asmetadata

• vertex, edge, and network attributes

• can use square-bracket notation just like a matrix

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

18-133

Network Objects

• stores an adjacency matrix or an edgelist as well asmetadata

• vertex, edge, and network attributes

• can use square-bracket notation just like a matrix

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

18-133

Network Objects

• stores an adjacency matrix or an edgelist as well asmetadata

• vertex, edge, and network attributes

• can use square-bracket notation just like a matrix

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

18-133

Network Objects

• stores an adjacency matrix or an edgelist as well asmetadata

• vertex, edge, and network attributes

• can use square-bracket notation just like a matrix

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

18-133

Network Objects

• stores an adjacency matrix or an edgelist as well asmetadata

• vertex, edge, and network attributes

• can use square-bracket notation just like a matrix

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

19-133

Code Time!

• Sections 1-3

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

20-133

Descriptives

• One isolate

• Two components

• Diameter is 5

• Medici is most popular

• Three triads

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

20-133

Descriptives

• One isolate

• Two components

• Diameter is 5

• Medici is most popular

• Three triads

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

20-133

Descriptives

• One isolate

• Two components

• Diameter is 5

• Medici is most popular

• Three triads

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

20-133

Descriptives

• One isolate

• Two components

• Diameter is 5

• Medici is most popular

• Three triads

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

20-133

Descriptives

• One isolate

• Two components

• Diameter is 5

• Medici is most popular

• Three triads

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

20-133

Descriptives

• One isolate

• Two components

• Diameter is 5

• Medici is most popular

• Three triads

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

21-133

Degree

For each node, its degree is

• the number of nodes adjacent to it

• or, the number of lines incident with it

• Pucci has degree 0

• Lamberteschi has degree 1

• Guadagni has degree 4

• Medici has degree 6

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

21-133

Degree

For each node, its degree is

• the number of nodes adjacent to it

• or, the number of lines incident with it

• Pucci has degree 0

• Lamberteschi has degree 1

• Guadagni has degree 4

• Medici has degree 6

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

21-133

Degree

For each node, its degree is

• the number of nodes adjacent to it

• or, the number of lines incident with it

• Pucci has degree 0

• Lamberteschi has degree 1

• Guadagni has degree 4

• Medici has degree 6

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

21-133

Degree

For each node, its degree is

• the number of nodes adjacent to it

• or, the number of lines incident with it

• Pucci has degree 0

• Lamberteschi has degree 1

• Guadagni has degree 4

• Medici has degree 6

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

21-133

Degree

For each node, its degree is

• the number of nodes adjacent to it

• or, the number of lines incident with it

• Pucci has degree 0

• Lamberteschi has degree 1

• Guadagni has degree 4

• Medici has degree 6

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

21-133

Degree

For each node, its degree is

• the number of nodes adjacent to it

• or, the number of lines incident with it

• Pucci has degree 0

• Lamberteschi has degree 1

• Guadagni has degree 4

• Medici has degree 6

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

21-133

Degree

For each node, its degree is

• the number of nodes adjacent to it

• or, the number of lines incident with it

• Pucci has degree 0

• Lamberteschi has degree 1

• Guadagni has degree 4

• Medici has degree 6

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

21-133

Degree

For each node, its degree is

• the number of nodes adjacent to it

• or, the number of lines incident with it

• Pucci has degree 0

• Lamberteschi has degree 1

• Guadagni has degree 4

• Medici has degree 6

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

22-133

Directed Degree

In directed graphs,

• Indegree indicates the number of received ties

• Outdegree indicates the number of sent ties

A

B

C

D

• A has 1 outdegree

• B has 1 outdegree and 3indegree

• C has 2 outdegree and 1indegree

• D has 1 outdegreeand 1indegree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

22-133

Directed Degree

In directed graphs,

• Indegree indicates the number of received ties

• Outdegree indicates the number of sent ties

A

B

C

D

• A has 1 outdegree

• B has 1 outdegree and 3indegree

• C has 2 outdegree and 1indegree

• D has 1 outdegreeand 1indegree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

22-133

Directed Degree

In directed graphs,

• Indegree indicates the number of received ties

• Outdegree indicates the number of sent ties

A

B

C

D

• A has 1 outdegree

• B has 1 outdegree and 3indegree

• C has 2 outdegree and 1indegree

• D has 1 outdegreeand 1indegree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

22-133

Directed Degree

In directed graphs,

• Indegree indicates the number of received ties

• Outdegree indicates the number of sent ties

A

B

C

D

• A has 1 outdegree

• B has 1 outdegree and 3indegree

• C has 2 outdegree and 1indegree

• D has 1 outdegreeand 1indegree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

22-133

Directed Degree

In directed graphs,

• Indegree indicates the number of received ties

• Outdegree indicates the number of sent ties

A

B

C

D

• A has 1 outdegree

• B has 1 outdegree

and 3indegree

• C has 2 outdegree and 1indegree

• D has 1 outdegreeand 1indegree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

22-133

Directed Degree

In directed graphs,

• Indegree indicates the number of received ties

• Outdegree indicates the number of sent ties

A

B

C

D

• A has 1 outdegree

• B has 1 outdegree and 3indegree

• C has 2 outdegree and 1indegree

• D has 1 outdegreeand 1indegree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

22-133

Directed Degree

In directed graphs,

• Indegree indicates the number of received ties

• Outdegree indicates the number of sent ties

A

B

C

D

• A has 1 outdegree

• B has 1 outdegree and 3indegree

• C has 2 outdegree

and 1indegree

• D has 1 outdegreeand 1indegree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

22-133

Directed Degree

In directed graphs,

• Indegree indicates the number of received ties

• Outdegree indicates the number of sent ties

A

B

C

D

• A has 1 outdegree

• B has 1 outdegree and 3indegree

• C has 2 outdegree and 1indegree

• D has 1 outdegreeand 1indegree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

22-133

Directed Degree

In directed graphs,

• Indegree indicates the number of received ties

• Outdegree indicates the number of sent ties

A

B

C

D

• A has 1 outdegree

• B has 1 outdegree and 3indegree

• C has 2 outdegree and 1indegree

• D has 1 outdegree

and 1indegree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

22-133

Directed Degree

In directed graphs,

• Indegree indicates the number of received ties

• Outdegree indicates the number of sent ties

A

B

C

D

• A has 1 outdegree

• B has 1 outdegree and 3indegree

• C has 2 outdegree and 1indegree

• D has 1 outdegreeand 1indegree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

23-133

Calculate degree centrality?

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

23-133

Calculate degree centrality?

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

24-133

Pseudo Code

for each vertex{Count the number of 1’s in the row

}

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

24-133

Pseudo Code

for each vertex{Count the number of 1’s in the row

}

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

25-133

R Code

network rs← vector(length=nrow(network))

for(i in 1:nrow(network)){network rs← sum(network[i,])}

Or, rowSums(network)

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

25-133

R Code

network rs← vector(length=nrow(network))

for(i in 1:nrow(network)){network rs← sum(network[i,])}

Or, rowSums(network)

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

25-133

R Code

network rs← vector(length=nrow(network))

for(i in 1:nrow(network)){network rs← sum(network[i,])}

Or, rowSums(network)

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C

= 1

• A→ D

= 1

• C→ D

= 0

• D→ C

= 1

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C

= 1

• A→ D

= 1

• C→ D

= 0

• D→ C

= 1

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C

= 1

• A→ D

= 1

• C→ D

= 0

• D→ C

= 1

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C

= 1

• A→ D

= 1

• C→ D

= 0

• D→ C

= 1

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:

• A→ C

= 1

• A→ D

= 1

• C→ D

= 0

• D→ C

= 1

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C

= 1

• A→ D

= 1

• C→ D

= 0

• D→ C

= 1

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C

= 1

• A→ D

= 1

• C→ D

= 0

• D→ C

= 1

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C = 1• A→ D

= 1

• C→ D

= 0

• D→ C

= 1

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C = 1• A→ D

= 1

• C→ D

= 0

• D→ C

= 1

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C = 1• A→ D = 1• C→ D

= 0

• D→ C

= 1

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C = 1• A→ D = 1• C→ D

= 0

• D→ C

= 1

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C = 1• A→ D = 1• C→ D = 0• D→ C

= 1

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C = 1• A→ D = 1• C→ D = 0• D→ C

= 1

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C = 1• A→ D = 1• C→ D = 0• D→ C = 1• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C = 1• A→ D = 1• C→ D = 0• D→ C = 1• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C = 1• A→ D = 1• C→ D = 0• D→ C = 1• C→ A = 0• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C = 1• A→ D = 1• C→ D = 0• D→ C = 1• C→ A = 0• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

26-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• A sits on no pathsbetween others

• B sits on some paths:• A→ C = 1• A→ D = 1• C→ D = 0• D→ C = 1• C→ A = 0• D→ A = 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

27-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• B sits on some paths:• A→ C

= 0

• A→ D

= 1/2

• C→ D

= 0

• D→ C

= 0

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

27-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• B sits on some paths:• A→ C

= 0

• A→ D

= 1/2

• C→ D

= 0

• D→ C

= 0

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

27-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• B sits on some paths:• A→ C = 0• A→ D

= 1/2

• C→ D

= 0

• D→ C

= 0

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

27-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• B sits on some paths:• A→ C = 0• A→ D

= 1/2

• C→ D

= 0

• D→ C

= 0

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

27-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• B sits on some paths:• A→ C = 0• A→ D

= 1/2

• C→ D

= 0

• D→ C

= 0

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

27-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• B sits on some paths:• A→ C = 0• A→ D = 1/2• C→ D

= 0

• D→ C

= 0

• C→ A

= 0

• D→ A

= 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

27-133

Betweenness Centrality

Proportion of shortest paths between all other pairs ofnodes that the given node lies on

A

B

C

D

• B sits on some paths:• A→ C = 0• A→ D = 1/2• C→ D = 0• D→ C = 0• C→ A = 0• D→ A = 0

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

28-133

Betweenness Centrality

Forrest Pitts 1978 “The River Trade Network of Russia,Revisited”

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

28-133

Betweenness Centrality

Forrest Pitts 1978 “The River Trade Network of Russia,Revisited”

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

29-133

Closeness Centrality

How far is each node from the other nodes in the graph

A

B

C

D

• C(v) = |V |−1∑d(v,i)

• A’s Distance to• B = 1• C = 2• D = 3

• A’s Closeness Centrality= 3

1+2+3 = .5

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

29-133

Closeness Centrality

How far is each node from the other nodes in the graph

A

B

C

D

• C(v) = |V |−1∑d(v,i)

• A’s Distance to• B = 1• C = 2• D = 3

• A’s Closeness Centrality= 3

1+2+3 = .5

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

29-133

Closeness Centrality

How far is each node from the other nodes in the graph

A

B

C

D

• C(v) = |V |−1∑d(v,i)

• A’s Distance to• B = 1• C = 2• D = 3

• A’s Closeness Centrality= 3

1+2+3 = .5

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

29-133

Closeness Centrality

How far is each node from the other nodes in the graph

A

B

C

D

• C(v) = |V |−1∑d(v,i)

• A’s Distance to

• B = 1• C = 2• D = 3

• A’s Closeness Centrality= 3

1+2+3 = .5

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

29-133

Closeness Centrality

How far is each node from the other nodes in the graph

A

B

C

D

• C(v) = |V |−1∑d(v,i)

• A’s Distance to• B = 1

• C = 2• D = 3

• A’s Closeness Centrality= 3

1+2+3 = .5

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

29-133

Closeness Centrality

How far is each node from the other nodes in the graph

A

B

C

D

• C(v) = |V |−1∑d(v,i)

• A’s Distance to• B = 1• C = 2

• D = 3

• A’s Closeness Centrality= 3

1+2+3 = .5

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

29-133

Closeness Centrality

How far is each node from the other nodes in the graph

A

B

C

D

• C(v) = |V |−1∑d(v,i)

• A’s Distance to• B = 1• C = 2• D = 3

• A’s Closeness Centrality= 3

1+2+3 = .5

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

29-133

Closeness Centrality

How far is each node from the other nodes in the graph

A

B

C

D

• C(v) = |V |−1∑d(v,i)

• A’s Distance to• B = 1• C = 2• D = 3

• A’s Closeness Centrality= 3

1+2+3 = .5

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

30-133

Closeness Centrality

How far is each node from the other nodes in the graph

A

B

C

D

• C(v) = |V |−1∑d(v,i)

• B’s Distance to

• C = 1• D = 2• B = inf

• B’s Closeness Centrality= 3

inf+1+2 = inf

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

30-133

Closeness Centrality

How far is each node from the other nodes in the graph

A

B

C

D

• C(v) = |V |−1∑d(v,i)

• B’s Distance to• C = 1

• D = 2• B = inf

• B’s Closeness Centrality= 3

inf+1+2 = inf

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

30-133

Closeness Centrality

How far is each node from the other nodes in the graph

A

B

C

D

• C(v) = |V |−1∑d(v,i)

• B’s Distance to• C = 1• D = 2

• B = inf

• B’s Closeness Centrality= 3

inf+1+2 = inf

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

30-133

Closeness Centrality

How far is each node from the other nodes in the graph

A

B

C

D

• C(v) = |V |−1∑d(v,i)

• B’s Distance to• C = 1• D = 2• B = inf

• B’s Closeness Centrality= 3

inf+1+2 = inf

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

30-133

Closeness Centrality

How far is each node from the other nodes in the graph

A

B

C

D

• C(v) = |V |−1∑d(v,i)

• B’s Distance to• C = 1• D = 2• B = inf

• B’s Closeness Centrality= 3

inf+1+2 = inf

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

31-133

Closeness Centrality – AlternateMeasure

How far is each node from the other nodes in the graph

A

B

C

D

• C2(v) =

∑ 1d(v,i)

|V |−1• B’s Distance to

• C = 1• D = 2• B = inf

• B’s Closeness Centrality

=1

inf+ 1

1+ 1

2

3 = 12

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

31-133

Closeness Centrality – AlternateMeasure

How far is each node from the other nodes in the graph

A

B

C

D

• C2(v) =

∑ 1d(v,i)

|V |−1• B’s Distance to

• C = 1• D = 2• B = inf

• B’s Closeness Centrality

=1

inf+ 1

1+ 1

2

3 = 12

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

32-133

Page Rank and EigenvectorCentrality

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

33-133

Page Rank

Erica

Amy Bryan

Carter

David

1

1 1

1

1

1

0 1+.5

1+.5

1

1

0 1+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

0+.5

1+1

1+1

0 0+.5

0+.5

1+1

1+1

0+1 0+.5

0+.5

0+1

1+1

0+1 0+.5

0+.5

0+1

2

1 .5

.5

1

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

33-133

Page Rank

Erica

Amy Bryan

Carter

David1

1 1

1

1

1

0 1+.5

1+.5

1

1

0 1+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

0+.5

1+1

1+1

0 0+.5

0+.5

1+1

1+1

0+1 0+.5

0+.5

0+1

1+1

0+1 0+.5

0+.5

0+1

2

1 .5

.5

1

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

33-133

Page Rank

Erica

Amy Bryan

Carter

David1

1 1

1

1

1

0 1+.5

1+.5

1

1

0 1+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

0+.5

1+1

1+1

0 0+.5

0+.5

1+1

1+1

0+1 0+.5

0+.5

0+1

1+1

0+1 0+.5

0+.5

0+1

2

1 .5

.5

1

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

33-133

Page Rank

Erica

Amy Bryan

Carter

David

1

1 1

1

1

1

0 1+.5

1+.5

1

1

0 1+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

0+.5

1+1

1+1

0 0+.5

0+.5

1+1

1+1

0+1 0+.5

0+.5

0+1

1+1

0+1 0+.5

0+.5

0+1

2

1 .5

.5

1

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

33-133

Page Rank

Erica

Amy Bryan

Carter

David

1

1 1

1

1

1

0 1+.5

1+.5

1

1

0 1+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

0+.5

1+1

1+1

0 0+.5

0+.5

1+1

1+1

0+1 0+.5

0+.5

0+1

1+1

0+1 0+.5

0+.5

0+1

2

1 .5

.5

1

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

33-133

Page Rank

Erica

Amy Bryan

Carter

David

1

1 1

1

1

1

0 1+.5

1+.5

1

1

0 1+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

0+.5

1+1

1+1

0 0+.5

0+.5

1+1

1+1

0+1 0+.5

0+.5

0+1

1+1

0+1 0+.5

0+.5

0+1

2

1 .5

.5

1

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

33-133

Page Rank

Erica

Amy Bryan

Carter

David

1

1 1

1

1

1

0 1+.5

1+.5

1

1

0 1+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

0+.5

1+1

1+1

0 0+.5

0+.5

1+1

1+1

0+1 0+.5

0+.5

0+1

1+1

0+1 0+.5

0+.5

0+1

2

1 .5

.5

1

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

33-133

Page Rank

Erica

Amy Bryan

Carter

David

1

1 1

1

1

1

0 1+.5

1+.5

1

1

0 1+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

0+.5

1+1

1+1

0 0+.5

0+.5

1+1

1+1

0+1 0+.5

0+.5

0+1

1+1

0+1 0+.5

0+.5

0+1

2

1 .5

.5

1

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

33-133

Page Rank

Erica

Amy Bryan

Carter

David

1

1 1

1

1

1

0 1+.5

1+.5

1

1

0 1+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

0+.5

1+1

1+1

0 0+.5

0+.5

1+1

1+1

0+1 0+.5

0+.5

0+1

1+1

0+1 0+.5

0+.5

0+1

2

1 .5

.5

1

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

33-133

Page Rank

Erica

Amy Bryan

Carter

David

1

1 1

1

1

1

0 1+.5

1+.5

1

1

0 1+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

0+.5

1+1

1+1

0 0+.5

0+.5

1+1

1+1

0+1 0+.5

0+.5

0+1

1+1

0+1 0+.5

0+.5

0+1

2

1 .5

.5

1

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

33-133

Page Rank

Erica

Amy Bryan

Carter

David

1

1 1

1

1

1

0 1+.5

1+.5

1

1

0 1+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

0+.5

1+1

1+1

0 0+.5

0+.5

1+1

1+1

0+1 0+.5

0+.5

0+1

1+1

0+1 0+.5

0+.5

0+1

2

1 .5

.5

1

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

33-133

Page Rank

Erica

Amy Bryan

Carter

David

1

1 1

1

1

1

0 1+.5

1+.5

1

1

0 1+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

1+.5

1

1+1

0 0+.5

0+.5

1+1

1+1

0 0+.5

0+.5

1+1

1+1

0+1 0+.5

0+.5

0+1

1+1

0+1 0+.5

0+.5

0+1

2

1 .5

.5

1

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

34-133

Graph Level Indices

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

35-133

Density

• Number of ties, expressed as a percentage of thenumber of possible ties

• For directed graphs:E

N(N−1)

• For undirected graphs:

EN(N−1)

2

1

2

3

=2

3(3−1)

2 = 46

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

35-133

Density

• Number of ties, expressed as a percentage of thenumber of possible ties

• For directed graphs:E

N(N−1)

• For undirected graphs:

EN(N−1)

2

1

2

3

=2

3(3−1)

2 = 46

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

35-133

Density

• Number of ties, expressed as a percentage of thenumber of possible ties

• For directed graphs:E

N(N−1)

• For undirected graphs:

EN(N−1)

2

1

2

3

=2

3(3−1)

2 = 46

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

35-133

Density

• Number of ties, expressed as a percentage of thenumber of possible ties

• For directed graphs:E

N(N−1)

• For undirected graphs:

EN(N−1)

2

1

2

3

=2

3(3−1)

2 = 46

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

35-133

Density

• Number of ties, expressed as a percentage of thenumber of possible ties

• For directed graphs:E

N(N−1)

• For undirected graphs:

EN(N−1)

2

1

2

3

=2

3(3−1)

2 = 46

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

35-133

Density

• Number of ties, expressed as a percentage of thenumber of possible ties

• For directed graphs:E

N(N−1)

• For undirected graphs:

EN(N−1)

2

1

2

3

=2

3(3−1)

2 = 46

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

36-133

Graph Level Indices

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

37-133

Density

• Number of ties, expressed as a percentage of thenumber of possible ties

• For directed graphs:E

N(N−1)

• For undirected graphs:

EN(N−1)

2

1

2

3

=2

3(3−1)

2 = 46

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

37-133

Density

• Number of ties, expressed as a percentage of thenumber of possible ties

• For directed graphs:E

N(N−1)

• For undirected graphs:

EN(N−1)

2

1

2

3

=2

3(3−1)

2 = 46

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

37-133

Density

• Number of ties, expressed as a percentage of thenumber of possible ties

• For directed graphs:E

N(N−1)

• For undirected graphs:

EN(N−1)

2

1

2

3

=2

3(3−1)

2 = 46

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

37-133

Density

• Number of ties, expressed as a percentage of thenumber of possible ties

• For directed graphs:E

N(N−1)

• For undirected graphs:

EN(N−1)

2

1

2

3

=2

3(3−1)

2 = 46

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

37-133

Density

• Number of ties, expressed as a percentage of thenumber of possible ties

• For directed graphs:E

N(N−1)

• For undirected graphs:

EN(N−1)

2

1

2

3

=2

3(3−1)

2 = 46

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

37-133

Density

• Number of ties, expressed as a percentage of thenumber of possible ties

• For directed graphs:E

N(N−1)

• For undirected graphs:

EN(N−1)

2

1

2

3

=2

3(3−1)

2 = 46

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

38-133

Mean Degree

The mean degree, d, of all nodes in the graph is

d(n) =∑Ni=1 d(ni)N = 2E

N

= 1+2+13 = 4

3

1

2

3

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

38-133

Mean Degree

The mean degree, d, of all nodes in the graph is

d(n) =∑Ni=1 d(ni)N = 2E

N

= 1+2+13 = 4

3

1

2

3

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

38-133

Mean Degree

The mean degree, d, of all nodes in the graph is

d(n) =∑Ni=1 d(ni)N = 2E

N

= 1+2+13 = 4

3

1

2

3

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

39-133

Size, Density, and Mean Degree

If we hold the meandegree constant, butvary size, whathappens to density?

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

39-133

Size, Density, and Mean Degree

If we hold the meandegree constant, butvary size, whathappens to density?

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

40-133

Centralization

• Extent to which centrality is concentrated on a singlevertex

• Calculated as the sum of the differences between eachnode’s centrality score and the maximum score

• Most centralized structure is usually a star network

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

40-133

Centralization

• Extent to which centrality is concentrated on a singlevertex

• Calculated as the sum of the differences between eachnode’s centrality score and the maximum score

• Most centralized structure is usually a star network

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

40-133

Centralization

• Extent to which centrality is concentrated on a singlevertex

• Calculated as the sum of the differences between eachnode’s centrality score and the maximum score

• Most centralized structure is usually a star network

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

40-133

Centralization

• Extent to which centrality is concentrated on a singlevertex

• Calculated as the sum of the differences between eachnode’s centrality score and the maximum score

• Most centralized structure is usually a star network

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

41-133

Bavelas Experiments

• People in positionspassed messages toone another tosolve a problem

• Studied the effectof structure on

• Efficiency• Leadership• Satisfaction

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

41-133

Bavelas Experiments

• People in positionspassed messages toone another tosolve a problem

• Studied the effectof structure on

• Efficiency• Leadership• Satisfaction

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

41-133

Bavelas Experiments

• People in positionspassed messages toone another tosolve a problem

• Studied the effectof structure on

• Efficiency• Leadership• Satisfaction

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

42-133

Bavelas Experiments

Circle Line

Star Y

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

43-133

Bavelas Experiments

Circle

Star

Slowest to completion

Most errors

Most satisfied

FastestFewest errors

Most dissatisfied

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

43-133

Bavelas Experiments

Circle

Star

Slowest to completion

Most errors

Most satisfied

FastestFewest errors

Most dissatisfied

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

43-133

Bavelas Experiments

Circle

Star

Slowest to completion

Most errors

Most satisfied

FastestFewest errors

Most dissatisfied

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

43-133

Bavelas Experiments

Circle

Star

Slowest to completion

Most errors

Most satisfied

FastestFewest errors

Most dissatisfied

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

43-133

Bavelas Experiments

Circle

Star

Slowest to completion

Most errors

Most satisfied

Fastest

Fewest errors

Most dissatisfied

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

43-133

Bavelas Experiments

Circle

Star

Slowest to completion

Most errors

Most satisfied

FastestFewest errors

Most dissatisfied

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

43-133

Bavelas Experiments

Circle

Star

Slowest to completion

Most errors

Most satisfied

FastestFewest errors

Most dissatisfied

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

44-133

Dyad Census

Mutual (M)

Assymetric (A)

Null (N)

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

44-133

Dyad Census

Mutual (M)

Assymetric (A)

Null (N)

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

44-133

Dyad Census

Mutual (M)

Assymetric (A)

Null (N)

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

44-133

Dyad Census

Mutual (M)

Assymetric (A)

Null (N)

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

45-133

Reciprocity

• Dyadic: the proportion of dyads that are symmetric

M+NM+A+N

• Dyadic non-null: the proportion of non-null dyads that

are reciprocalM

M+A

• Edgewise:2∗M

2∗M+A

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

45-133

Reciprocity

• Dyadic: the proportion of dyads that are symmetric

M+NM+A+N

• Dyadic non-null: the proportion of non-null dyads that

are reciprocalM

M+A

• Edgewise:2∗M

2∗M+A

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

45-133

Reciprocity

• Dyadic: the proportion of dyads that are symmetric

M+NM+A+N

• Dyadic non-null: the proportion of non-null dyads that

are reciprocalM

M+A

• Edgewise:2∗M

2∗M+A

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

45-133

Reciprocity

• Dyadic: the proportion of dyads that are symmetric

M+NM+A+N

• Dyadic non-null: the proportion of non-null dyads that

are reciprocalM

M+A

• Edgewise:2∗M

2∗M+A

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

46-133

Triad Census

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

46-133

Triad Census

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

47-133

Triad Census

16 different triad types

On

ero

wp

ern

etw

ork

i,j cell is the numberof triad type jin network i

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

47-133

Triad Census

16 different triad types

On

ero

wp

ern

etw

ork i,j cell is the number

of triad type jin network i

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

48-133

Triad Census

Dominance Relationships

Friendship, Assistance

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

48-133

Triad Census

Dominance Relationships

Friendship, Assistance

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

48-133

Triad Census

Dominance Relationships

Friendship, Assistance

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

49-133

Transitivity

1

2

3

• Usually calculated as thefraction of completedtwo-paths

• Related to Grannovetter’s‘forbidden triad’

• Can be directed or undirected

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

49-133

Transitivity

1

2

3

• Usually calculated as thefraction of completedtwo-paths

• Related to Grannovetter’s‘forbidden triad’

• Can be directed or undirected

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

49-133

Transitivity

1

2

3

• Usually calculated as thefraction of completedtwo-paths

• Related to Grannovetter’s‘forbidden triad’

• Can be directed or undirected

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

49-133

Transitivity

1

2

3

• Usually calculated as thefraction of completedtwo-paths

• Related to Grannovetter’s‘forbidden triad’

• Can be directed or undirected

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

49-133

Transitivity

1

2

3

• Usually calculated as thefraction of completedtwo-paths

• Related to Grannovetter’s‘forbidden triad’

• Can be directed or undirected

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

49-133

Transitivity

1

2

3

• Usually calculated as thefraction of completedtwo-paths

• Related to Grannovetter’s‘forbidden triad’

• Can be directed or undirected

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

49-133

Transitivity

1

2

3

• Usually calculated as thefraction of completedtwo-paths

• Related to Grannovetter’s‘forbidden triad’

• Can be directed or undirected

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

49-133

Transitivity

1

2

3

• Usually calculated as thefraction of completedtwo-paths

• Related to Grannovetter’s‘forbidden triad’

• Can be directed or undirected

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

50-133

Forbidden Triad orStructural Hole?

• Granovetter, Mark S. 1973.“The Strength of Weak Ties”

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

50-133

Forbidden Triad orStructural Hole?

• Granovetter, Mark S. 1973.“The Strength of Weak Ties”

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

51-133

Forbidden Triad orStructural Hole?

• Granovetter, Mark S. 1973.“The Strength of Weak Ties”

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

52-133

Forbidden Triad orStructural Hole?

• Burt, Ronald S. 2004.“Structural Holes: The SocialStructure of Competition”

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

53-133

Extensions

Attributes!

• Properties of nodes, edges, or even networks

• Pretty much anything you can measure could be anattribute

• Extension based on node attributes: Brokerage

• Extension based on edge attributes: Structural Balance

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

53-133

Extensions

Attributes!

• Properties of nodes, edges, or even networks

• Pretty much anything you can measure could be anattribute

• Extension based on node attributes: Brokerage

• Extension based on edge attributes: Structural Balance

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

53-133

Extensions

Attributes!

• Properties of nodes, edges, or even networks

• Pretty much anything you can measure could be anattribute

• Extension based on node attributes: Brokerage

• Extension based on edge attributes: Structural Balance

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

53-133

Extensions

Attributes!

• Properties of nodes, edges, or even networks

• Pretty much anything you can measure could be anattribute

• Extension based on node attributes: Brokerage

• Extension based on edge attributes: Structural Balance

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

53-133

Extensions

Attributes!

• Properties of nodes, edges, or even networks

• Pretty much anything you can measure could be anattribute

• Extension based on node attributes: Brokerage

• Extension based on edge attributes: Structural Balance

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

54-133

Brokerage

• Brokerage is a process “by which intermediary actorsfacilitate transactions between other actors lackingaccess to or trust in one another” (Marsden 1982)

• Brokers play a crucial role in knitting together diversegroups of people, organizations, parties

• Brokers can gain a lot – early access to information,prestige

• But can also be distrusted by everyone

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

54-133

Brokerage

• Brokerage is a process “by which intermediary actorsfacilitate transactions between other actors lackingaccess to or trust in one another” (Marsden 1982)

• Brokers play a crucial role in knitting together diversegroups of people, organizations, parties

• Brokers can gain a lot – early access to information,prestige

• But can also be distrusted by everyone

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

54-133

Brokerage

• Brokerage is a process “by which intermediary actorsfacilitate transactions between other actors lackingaccess to or trust in one another” (Marsden 1982)

• Brokers play a crucial role in knitting together diversegroups of people, organizations, parties

• Brokers can gain a lot – early access to information,prestige

• But can also be distrusted by everyone

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

54-133

Brokerage

• Brokerage is a process “by which intermediary actorsfacilitate transactions between other actors lackingaccess to or trust in one another” (Marsden 1982)

• Brokers play a crucial role in knitting together diversegroups of people, organizations, parties

• Brokers can gain a lot – early access to information,prestige

• But can also be distrusted by everyone

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

55-133

Brokerage: Formal Concept

In a network N withedges E,

node j brokersnodes i and kif eij ∈ Eand ejk ∈ Ebut eik /∈ E

ji

k

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

55-133

Brokerage: Formal Concept

In a network N withedges E,node j brokers

nodes i and kif eij ∈ Eand ejk ∈ Ebut eik /∈ E

j

i

k

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

55-133

Brokerage: Formal Concept

In a network N withedges E,node j brokersnodes i and k

if eij ∈ Eand ejk ∈ Ebut eik /∈ E

ji

k

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

55-133

Brokerage: Formal Concept

In a network N withedges E,node j brokersnodes i and kif eij ∈ E

and ejk ∈ Ebut eik /∈ E

ji

k

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

55-133

Brokerage: Formal Concept

In a network N withedges E,node j brokersnodes i and kif eij ∈ Eand ejk ∈ E

but eik /∈ E

ji

k

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

55-133

Brokerage: Formal Concept

In a network N withedges E,node j brokersnodes i and kif eij ∈ Eand ejk ∈ Ebut eik /∈ E

ji

k

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

56-133

Brokerage: Formal Concept

Gould and Fernandez (1989, 1994)

• formalized the concept

• added a vertex attribute component

• compared empirical brokerage counts to counts fromrandom graphs conditioned on the number of edges

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

56-133

Brokerage: Formal Concept

Gould and Fernandez (1989, 1994)

• formalized the concept

• added a vertex attribute component

• compared empirical brokerage counts to counts fromrandom graphs conditioned on the number of edges

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

56-133

Brokerage: Formal Concept

Gould and Fernandez (1989, 1994)

• formalized the concept

• added a vertex attribute component

• compared empirical brokerage counts to counts fromrandom graphs conditioned on the number of edges

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

56-133

Brokerage: Formal Concept

Gould and Fernandez (1989, 1994)

• formalized the concept

• added a vertex attribute component

• compared empirical brokerage counts to counts fromrandom graphs conditioned on the number of edges

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

57-133

Brokerage: One Mode

Coordinator

Representative Gatekeeper Itinerant Liaison

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

57-133

Brokerage: One Mode

Coordinator Representative

Gatekeeper Itinerant Liaison

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

57-133

Brokerage: One Mode

Coordinator Representative Gatekeeper

Itinerant Liaison

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

57-133

Brokerage: One Mode

Coordinator Representative Gatekeeper Itinerant

Liaison

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

57-133

Brokerage: One Mode

Coordinator Representative Gatekeeper Itinerant Liaison

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

58-133

Gould and Fernandez’ Findings

• the benefits of brokerage are mediated both by the typeof organization (the node sets) and the type ofbrokerage chain

• non-governmental organizations were found to havemore influence when they held any type of brokerageposition

• governmental organizations gained influence only whenthey held “outsider” brokerage roles in itinerant andliaison chains

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

58-133

Gould and Fernandez’ Findings

• the benefits of brokerage are mediated both by the typeof organization (the node sets) and the type ofbrokerage chain

• non-governmental organizations were found to havemore influence when they held any type of brokerageposition

• governmental organizations gained influence only whenthey held “outsider” brokerage roles in itinerant andliaison chains

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

58-133

Gould and Fernandez’ Findings

• the benefits of brokerage are mediated both by the typeof organization (the node sets) and the type ofbrokerage chain

• non-governmental organizations were found to havemore influence when they held any type of brokerageposition

• governmental organizations gained influence only whenthey held “outsider” brokerage roles in itinerant andliaison chains

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

58-133

Gould and Fernandez’ Findings

• the benefits of brokerage are mediated both by the typeof organization (the node sets) and the type ofbrokerage chain

• non-governmental organizations were found to havemore influence when they held any type of brokerageposition

• governmental organizations gained influence only whenthey held “outsider” brokerage roles in itinerant andliaison chains

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

59-133

Structural Balance

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

60-133

Bernoulli ‘Random’ Graphs

• A probabilitydistribution for asuccess/failure

• Best known example isthe coin flip

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

60-133

Bernoulli ‘Random’ Graphs

• A probabilitydistribution for asuccess/failure

• Best known example isthe coin flip

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

60-133

Bernoulli ‘Random’ Graphs

• A probabilitydistribution for asuccess/failure

• Best known example isthe coin flip

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

61-133

Bernoulli ‘Random’ Graphs

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

61-133

Bernoulli ‘Random’ Graphs

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

61-133

Bernoulli ‘Random’ Graphs

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

61-133

Bernoulli ‘Random’ Graphs

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

61-133

Bernoulli ‘Random’ Graphs

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

61-133

Bernoulli ‘Random’ Graphs

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

61-133

Bernoulli ‘Random’ Graphs

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

61-133

Bernoulli ‘Random’ Graphs

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

61-133

Bernoulli ‘Random’ Graphs

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

61-133

Bernoulli ‘Random’ Graphs

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

62-133

Small Worlds

What are the characteristics of real world?

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

63-133

Small Worlds

Stanley Milgram “The Small World Problem”, PsychologyToday, vol. 1, no. 1, May 1967, pp61-67

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

63-133

Small Worlds

Stanley Milgram “The Small World Problem”, PsychologyToday, vol. 1, no. 1, May 1967, pp61-67

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

64-133

Watts-Strogatz Model

• high clustering coefficient

• low diameter

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

64-133

Watts-Strogatz Model

• high clustering coefficient

• low diameter

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

64-133

Watts-Strogatz Model

• high clustering coefficient

• low diameter

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

65-133

Clustering Coefficient andDiameter

• The diameter of a graph is the longest shortest pathbetween any pair nodes

• The clustering coefficient of a graph is the numberof triangles divided by the number of two-paths

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

65-133

Clustering Coefficient andDiameter

• The diameter of a graph is the longest shortest pathbetween any pair nodes

• The clustering coefficient of a graph is the numberof triangles divided by the number of two-paths

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

65-133

Clustering Coefficient andDiameter

• The diameter of a graph is the longest shortest pathbetween any pair nodes

• The clustering coefficient of a graph is the numberof triangles divided by the number of two-paths

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

66-133

Clustering Coefficient Tangent

• remember the triad census...

• your numerator is...the number of triads with 3ties

• your denominator is...the number of triads with 2ties PLUS 3*the number oftriads with 3 ties

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

66-133

Clustering Coefficient Tangent

• remember the triad census...

• your numerator is...the number of triads with 3ties

• your denominator is...the number of triads with 2ties PLUS 3*the number oftriads with 3 ties

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

66-133

Clustering Coefficient Tangent

• remember the triad census...

• your numerator is...

the number of triads with 3ties

• your denominator is...the number of triads with 2ties PLUS 3*the number oftriads with 3 ties

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

66-133

Clustering Coefficient Tangent

• remember the triad census...

• your numerator is...the number of triads with 3ties

• your denominator is...the number of triads with 2ties PLUS 3*the number oftriads with 3 ties

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

66-133

Clustering Coefficient Tangent

• remember the triad census...

• your numerator is...the number of triads with 3ties

• your denominator is...

the number of triads with 2ties PLUS 3*the number oftriads with 3 ties

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

66-133

Clustering Coefficient Tangent

• remember the triad census...

• your numerator is...the number of triads with 3ties

• your denominator is...the number of triads with 2ties PLUS 3*the number oftriads with 3 ties

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

67-133

Watts-Strogatz Model

• high clusteringcoefficient

• low diameter

• starts with alattice structure

• randomlyre-wires ties

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

67-133

Watts-Strogatz Model

• high clusteringcoefficient

• low diameter

• starts with alattice structure

• randomlyre-wires ties

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

67-133

Watts-Strogatz Model

• high clusteringcoefficient

• low diameter

• starts with alattice structure

• randomlyre-wires ties

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

68-133

Watts-Strogatz Model

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

69-133

Watts-Strogatz Model

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

70-133

Watts-Strogatz Model

Lactual L1 Cactual C1

Film Actors

3.65 2.99 0.79 0.00027

Power Grid

18.7 12.4 0.08 0.005

C. Elegans

2.65 2.25 0.28 0.05

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

70-133

Watts-Strogatz Model

Lactual

L1 Cactual C1

Film Actors

3.65 2.99 0.79 0.00027

Power Grid

18.7 12.4 0.08 0.005

C. Elegans

2.65 2.25 0.28 0.05

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

70-133

Watts-Strogatz Model

Lactual L1

Cactual C1

Film Actors

3.65 2.99 0.79 0.00027

Power Grid

18.7 12.4 0.08 0.005

C. Elegans

2.65 2.25 0.28 0.05

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

70-133

Watts-Strogatz Model

Lactual L1 Cactual

C1

Film Actors

3.65 2.99 0.79 0.00027

Power Grid

18.7 12.4 0.08 0.005

C. Elegans

2.65 2.25 0.28 0.05

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

70-133

Watts-Strogatz Model

Lactual L1 Cactual C1

Film Actors

3.65 2.99 0.79 0.00027

Power Grid

18.7 12.4 0.08 0.005

C. Elegans

2.65 2.25 0.28 0.05

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

70-133

Watts-Strogatz Model

Lactual L1 Cactual C1

Film Actors 3.65

2.99 0.79 0.00027

Power Grid 18.7

12.4 0.08 0.005

C. Elegans 2.65

2.25 0.28 0.05

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

70-133

Watts-Strogatz Model

Lactual L1 Cactual C1

Film Actors 3.65 2.99

0.79 0.00027

Power Grid 18.7 12.4

0.08 0.005

C. Elegans 2.65 2.25

0.28 0.05

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

70-133

Watts-Strogatz Model

Lactual L1 Cactual C1

Film Actors 3.65 2.99 0.79

0.00027

Power Grid 18.7 12.4 0.08

0.005

C. Elegans 2.65 2.25 0.28

0.05

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

70-133

Watts-Strogatz Model

Lactual L1 Cactual C1

Film Actors 3.65 2.99 0.79 0.00027Power Grid 18.7 12.4 0.08 0.005C. Elegans 2.65 2.25 0.28 0.05

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

71-133

Watts-Strogatz Model

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

71-133

Watts-Strogatz Model

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

71-133

Watts-Strogatz Model

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

72-133

Barabasi-Albert Model

• Alternative network generating model

• Where Watts and Strogatz’s model results in a worldwhere everyone has approximately the same number ofties,

• Barabasi and Albert thought about a skeweddistribution of ties

• Based on the idea of ‘preferential attachment’ aka ’richget richer’

• Unlike Watts-Strogatz, this model starts with one node,add additional nodes one at a time

• Nodes ‘preferentially’ attach to those with higher degree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

72-133

Barabasi-Albert Model

• Alternative network generating model

• Where Watts and Strogatz’s model results in a worldwhere everyone has approximately the same number ofties,

• Barabasi and Albert thought about a skeweddistribution of ties

• Based on the idea of ‘preferential attachment’ aka ’richget richer’

• Unlike Watts-Strogatz, this model starts with one node,add additional nodes one at a time

• Nodes ‘preferentially’ attach to those with higher degree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

72-133

Barabasi-Albert Model

• Alternative network generating model

• Where Watts and Strogatz’s model results in a worldwhere everyone has approximately the same number ofties,

• Barabasi and Albert thought about a skeweddistribution of ties

• Based on the idea of ‘preferential attachment’ aka ’richget richer’

• Unlike Watts-Strogatz, this model starts with one node,add additional nodes one at a time

• Nodes ‘preferentially’ attach to those with higher degree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

72-133

Barabasi-Albert Model

• Alternative network generating model

• Where Watts and Strogatz’s model results in a worldwhere everyone has approximately the same number ofties,

• Barabasi and Albert thought about a skeweddistribution of ties

• Based on the idea of ‘preferential attachment’ aka ’richget richer’

• Unlike Watts-Strogatz, this model starts with one node,add additional nodes one at a time

• Nodes ‘preferentially’ attach to those with higher degree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

72-133

Barabasi-Albert Model

• Alternative network generating model

• Where Watts and Strogatz’s model results in a worldwhere everyone has approximately the same number ofties,

• Barabasi and Albert thought about a skeweddistribution of ties

• Based on the idea of ‘preferential attachment’ aka ’richget richer’

• Unlike Watts-Strogatz, this model starts with one node,add additional nodes one at a time

• Nodes ‘preferentially’ attach to those with higher degree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

72-133

Barabasi-Albert Model

• Alternative network generating model

• Where Watts and Strogatz’s model results in a worldwhere everyone has approximately the same number ofties,

• Barabasi and Albert thought about a skeweddistribution of ties

• Based on the idea of ‘preferential attachment’ aka ’richget richer’

• Unlike Watts-Strogatz, this model starts with one node,add additional nodes one at a time

• Nodes ‘preferentially’ attach to those with higher degree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

72-133

Barabasi-Albert Model

• Alternative network generating model

• Where Watts and Strogatz’s model results in a worldwhere everyone has approximately the same number ofties,

• Barabasi and Albert thought about a skeweddistribution of ties

• Based on the idea of ‘preferential attachment’ aka ’richget richer’

• Unlike Watts-Strogatz, this model starts with one node,add additional nodes one at a time

• Nodes ‘preferentially’ attach to those with higher degree

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

73-133

Barabasi-Albert Model

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

73-133

Barabasi-Albert Model

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

73-133

Barabasi-Albert Model

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

73-133

Barabasi-Albert Model

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

73-133

Barabasi-Albert Model

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

73-133

Barabasi-Albert Model

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

74-133

Barabasi-Albert Model

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

75-133

Barabasi-Albert Model

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

76-133

Barabasi-Albert Model

Barabsi, A.-L.; R. Albert (1999). “Emergence of scaling inrandom networks”. Science

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

77-133

Code Time!

• Section 4

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

78-133

Hypothesis Testing

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

79-133

Relating Node level indices tocovariates

• Node Level Indices: centrality measures, brokerage,constraint

• Node Covariates: measures of power, careeradvancement, gender – really anything you want tostudy that varies at the node level

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

79-133

Relating Node level indices tocovariates

• Node Level Indices: centrality measures, brokerage,constraint

• Node Covariates: measures of power, careeradvancement, gender – really anything you want tostudy that varies at the node level

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

80-133

Emergent Multi-OrganizationalNetworks (EMON) Dataset

• 7 case studies of EMONs in the context of search andrescue activities from Drabek et. al. (1981)

• Ties between organizations are self-reported levels ofcommunication coded from 1 to 4 with 1 as mostfrequent

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

80-133

Emergent Multi-OrganizationalNetworks (EMON) Dataset

• 7 case studies of EMONs in the context of search andrescue activities from Drabek et. al. (1981)

• Ties between organizations are self-reported levels ofcommunication coded from 1 to 4 with 1 as mostfrequent

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

81-133

Emergent Multi-OrganizationalNetworks (EMON) Dataset

Attribute Data

• Command Rank Score (CRS): mean rank (reversed) forprominence in the command structure

• Decision Rank Score (DRS): mean rank (reversed) forprominence in decision making process

• Paid Staff: number of paid employees

• Volunteer Staff: number of volunteer staff

• Sponsorship: organization type (City, County, State,Federal, or Private)

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

82-133

Correlation between DRS andDegree?

• Subsample of MutuallyReported “ContinuousCommunication” inTexas EMON

• Degree is shown in color(darker is bigger)

• DRS in size

• Empirical corelationρ = 0.86

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

82-133

Correlation between DRS andDegree?

• Subsample of MutuallyReported “ContinuousCommunication” inTexas EMON

• Degree is shown in color(darker is bigger)

• DRS in size

• Empirical corelationρ = 0.86

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

82-133

Correlation between DRS andDegree?

• Subsample of MutuallyReported “ContinuousCommunication” inTexas EMON

• Degree is shown in color(darker is bigger)

• DRS in size

• Empirical corelationρ = 0.86

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

82-133

Correlation between DRS andDegree?

• Subsample of MutuallyReported “ContinuousCommunication” inTexas EMON

• Degree is shown in color(darker is bigger)

• DRS in size

• Empirical corelationρ = 0.86

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

82-133

Correlation between DRS andDegree?

• Subsample of MutuallyReported “ContinuousCommunication” inTexas EMON

• Degree is shown in color(darker is bigger)

• DRS in size

• Empirical corelationρ = 0.86

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

83-133

Correlation between DRS andDegree?

ρ = 0.86

ρ = −0.07

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

83-133

Correlation between DRS andDegree?

ρ = 0.86

ρ = −0.07

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

83-133

Correlation between DRS andDegree?

ρ = 0.86 ρ = −0.07

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

84-133

Correlation between DRS andDegree?

ρ = 0.86

ρ = −0.07

ρ = −0.12

ρ = −0.39

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

84-133

Correlation between DRS andDegree?

ρ = 0.86

ρ = −0.07

ρ = −0.12

ρ = −0.39

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

84-133

Correlation between DRS andDegree?

ρ = 0.86

ρ = −0.07

ρ = −0.12

ρ = −0.39

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

85-133

Correlation between DRS andDegree?

ρobs = 0.86

Pr(ρ ≥ ρobs)= 3e− 5

Pr(ρ < ρobs) = 0.9999

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

85-133

Correlation between DRS andDegree?

ρobs = 0.86

Pr(ρ ≥ ρobs)= 3e− 5

Pr(ρ < ρobs) = 0.9999

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

85-133

Correlation between DRS andDegree?

ρobs = 0.86

Pr(ρ ≥ ρobs)= 3e− 5

Pr(ρ < ρobs) = 0.9999

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

85-133

Correlation between DRS andDegree?

ρobs = 0.86

Pr(ρ ≥ ρobs)= 3e− 5

Pr(ρ < ρobs) = 0.9999

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

86-133

Regression?

• Can use Node Level Indices as independent variables ina regression

• Big assumption: position predicts the properties ofthose who hold them

• Conditioning on NLI values, so dependence inaccounted for assuming no error in the network

• NLIs as dependent variables more problematic due toautocorrelation

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

86-133

Regression?

• Can use Node Level Indices as independent variables ina regression

• Big assumption: position predicts the properties ofthose who hold them

• Conditioning on NLI values, so dependence inaccounted for assuming no error in the network

• NLIs as dependent variables more problematic due toautocorrelation

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

86-133

Regression?

• Can use Node Level Indices as independent variables ina regression

• Big assumption: position predicts the properties ofthose who hold them

• Conditioning on NLI values, so dependence inaccounted for assuming no error in the network

• NLIs as dependent variables more problematic due toautocorrelation

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

86-133

Regression?

• Can use Node Level Indices as independent variables ina regression

• Big assumption: position predicts the properties ofthose who hold them

• Conditioning on NLI values, so dependence inaccounted for assuming no error in the network

• NLIs as dependent variables more problematic due toautocorrelation

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

86-133

Regression?

• Can use Node Level Indices as independent variables ina regression

• Big assumption: position predicts the properties ofthose who hold them

• Conditioning on NLI values, so dependence inaccounted for assuming no error in the network

• NLIs as dependent variables more problematic due toautocorrelation

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

87-133

Code Time!

• 5.1-5.3

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

88-133

Quadratic Assignment Procedure

Marriage Business

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

89-133

Quadratic AssignmentProceedure

Marriage Business

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

90-133

Graph Correlation

• Simple way of comparing graphs on the same vertex setby element

• gcor([

1 11 0

],

[1 12 2

])= cor([1, 1, 1, 0], [1, 1, 2, 2])

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

90-133

Graph Correlation

• Simple way of comparing graphs on the same vertex setby element

• gcor([

1 11 0

],

[1 12 2

])= cor([1, 1, 1, 0], [1, 1, 2, 2])

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

90-133

Graph Correlation

• Simple way of comparing graphs on the same vertex setby element

• gcor([

1 11 0

],

[1 12 2

])= cor([1, 1, 1, 0], [1, 1, 2, 2])

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

91-133

Do business ties coincide withmarriages?

Marriage Business

ρ = 0.372

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

92-133

Do business ties coincide withmarriages?

Marriage Business

ρ = 0.169

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

92-133

Do business ties coincide withmarriages?

Marriage Business

ρ = 0.169

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

93-133

Do business ties coincide withmarriages?

Marriage Business

ρ = −0.034

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

94-133

Do business ties coincide withmarriages?

Marriage Business

ρ = −0.101

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

95-133

QAP Test

ρobs = 0.372

Pr(ρ ≥ ρobs)= .001

Pr(ρ < ρobs) = 0.999

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

95-133

QAP Test

ρobs = 0.372

Pr(ρ ≥ ρobs)= .001

Pr(ρ < ρobs) = 0.999

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

95-133

QAP Test

ρobs = 0.372

Pr(ρ ≥ ρobs)= .001

Pr(ρ < ρobs) = 0.999

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

95-133

QAP Test

ρobs = 0.372

Pr(ρ ≥ ρobs)= .001

Pr(ρ < ρobs) = 0.999

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

96-133

Network Regression

• Family of models predicting social ties• Special case of standard OLS regression• Dependent variable is a network adjacency matrix

• EYij = β0 + β1X1ij + β2X2ij + · · ·+ βρXρij

• Where E is the expectation operator (analagous to“mean” or “average”)

• Yij is the value from i to j on the dependent relationwith adjacency matrix Y

• Xkij is the value of the kth predictor for the (i, j)ordered pair, and β0, . . . βρ are coefficients

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

96-133

Network Regression

• Family of models predicting social ties

• Special case of standard OLS regression• Dependent variable is a network adjacency matrix

• EYij = β0 + β1X1ij + β2X2ij + · · ·+ βρXρij

• Where E is the expectation operator (analagous to“mean” or “average”)

• Yij is the value from i to j on the dependent relationwith adjacency matrix Y

• Xkij is the value of the kth predictor for the (i, j)ordered pair, and β0, . . . βρ are coefficients

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

96-133

Network Regression

• Family of models predicting social ties• Special case of standard OLS regression

• Dependent variable is a network adjacency matrix

• EYij = β0 + β1X1ij + β2X2ij + · · ·+ βρXρij

• Where E is the expectation operator (analagous to“mean” or “average”)

• Yij is the value from i to j on the dependent relationwith adjacency matrix Y

• Xkij is the value of the kth predictor for the (i, j)ordered pair, and β0, . . . βρ are coefficients

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

96-133

Network Regression

• Family of models predicting social ties• Special case of standard OLS regression• Dependent variable is a network adjacency matrix

• EYij = β0 + β1X1ij + β2X2ij + · · ·+ βρXρij

• Where E is the expectation operator (analagous to“mean” or “average”)

• Yij is the value from i to j on the dependent relationwith adjacency matrix Y

• Xkij is the value of the kth predictor for the (i, j)ordered pair, and β0, . . . βρ are coefficients

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

96-133

Network Regression

• Family of models predicting social ties• Special case of standard OLS regression• Dependent variable is a network adjacency matrix

• EYij = β0 + β1X1ij + β2X2ij + · · ·+ βρXρij

• Where E is the expectation operator (analagous to“mean” or “average”)

• Yij is the value from i to j on the dependent relationwith adjacency matrix Y

• Xkij is the value of the kth predictor for the (i, j)ordered pair, and β0, . . . βρ are coefficients

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

96-133

Network Regression

• Family of models predicting social ties• Special case of standard OLS regression• Dependent variable is a network adjacency matrix

• EYij = β0 + β1X1ij + β2X2ij + · · ·+ βρXρij

• Where E is the expectation operator (analagous to“mean” or “average”)

• Yij is the value from i to j on the dependent relationwith adjacency matrix Y

• Xkij is the value of the kth predictor for the (i, j)ordered pair, and β0, . . . βρ are coefficients

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

96-133

Network Regression

• Family of models predicting social ties• Special case of standard OLS regression• Dependent variable is a network adjacency matrix

• EYij = β0 + β1X1ij + β2X2ij + · · ·+ βρXρij

• Where E is the expectation operator (analagous to“mean” or “average”)

• Yij is the value from i to j on the dependent relationwith adjacency matrix Y

• Xkij is the value of the kth predictor for the (i, j)ordered pair, and β0, . . . βρ are coefficients

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

96-133

Network Regression

• Family of models predicting social ties• Special case of standard OLS regression• Dependent variable is a network adjacency matrix

• EYij = β0 + β1X1ij + β2X2ij + · · ·+ βρXρij

• Where E is the expectation operator (analagous to“mean” or “average”)

• Yij is the value from i to j on the dependent relationwith adjacency matrix Y

• Xkij is the value of the kth predictor for the (i, j)ordered pair, and β0, . . . βρ are coefficients

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

97-133

Data Prep

• Dependent variable is an adjacency matrix• Standard case: dichotomous data• Valued case

• Independent variables also in adjacency matrix form• Always takes matrix form, but may be based on vector

data• eg. simple adjacency matrix, sender/receiver effects,

attribute differences, elements held in common

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

97-133

Data Prep

• Dependent variable is an adjacency matrix

• Standard case: dichotomous data• Valued case

• Independent variables also in adjacency matrix form• Always takes matrix form, but may be based on vector

data• eg. simple adjacency matrix, sender/receiver effects,

attribute differences, elements held in common

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

97-133

Data Prep

• Dependent variable is an adjacency matrix• Standard case: dichotomous data

• Valued case

• Independent variables also in adjacency matrix form• Always takes matrix form, but may be based on vector

data• eg. simple adjacency matrix, sender/receiver effects,

attribute differences, elements held in common

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

97-133

Data Prep

• Dependent variable is an adjacency matrix• Standard case: dichotomous data• Valued case

• Independent variables also in adjacency matrix form• Always takes matrix form, but may be based on vector

data• eg. simple adjacency matrix, sender/receiver effects,

attribute differences, elements held in common

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

97-133

Data Prep

• Dependent variable is an adjacency matrix• Standard case: dichotomous data• Valued case

• Independent variables also in adjacency matrix form

• Always takes matrix form, but may be based on vectordata

• eg. simple adjacency matrix, sender/receiver effects,attribute differences, elements held in common

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

97-133

Data Prep

• Dependent variable is an adjacency matrix• Standard case: dichotomous data• Valued case

• Independent variables also in adjacency matrix form• Always takes matrix form, but may be based on vector

data

• eg. simple adjacency matrix, sender/receiver effects,attribute differences, elements held in common

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

97-133

Data Prep

• Dependent variable is an adjacency matrix• Standard case: dichotomous data• Valued case

• Independent variables also in adjacency matrix form• Always takes matrix form, but may be based on vector

data• eg. simple adjacency matrix, sender/receiver effects,

attribute differences, elements held in common

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

98-133

Code Time!

• Sections 5.4 and 5.5

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

99-133

Network Autocorrelation Models

• Family of models for estimating how covariates relate toeach other through ties

• Special case of standard OLS regression• Dependent variable is a vertex attribute

• y = (I −ΘW )−1(Xβ + (I − ψZ)−1v)• where Θ is the matrix for the Auto-Regressive weights• and ψ is the matrix for the Moving Average weights

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

99-133

Network Autocorrelation Models

• Family of models for estimating how covariates relate toeach other through ties

• Special case of standard OLS regression• Dependent variable is a vertex attribute

• y = (I −ΘW )−1(Xβ + (I − ψZ)−1v)• where Θ is the matrix for the Auto-Regressive weights• and ψ is the matrix for the Moving Average weights

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

99-133

Network Autocorrelation Models

• Family of models for estimating how covariates relate toeach other through ties

• Special case of standard OLS regression

• Dependent variable is a vertex attribute

• y = (I −ΘW )−1(Xβ + (I − ψZ)−1v)• where Θ is the matrix for the Auto-Regressive weights• and ψ is the matrix for the Moving Average weights

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

99-133

Network Autocorrelation Models

• Family of models for estimating how covariates relate toeach other through ties

• Special case of standard OLS regression• Dependent variable is a vertex attribute

• y = (I −ΘW )−1(Xβ + (I − ψZ)−1v)• where Θ is the matrix for the Auto-Regressive weights• and ψ is the matrix for the Moving Average weights

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

99-133

Network Autocorrelation Models

• Family of models for estimating how covariates relate toeach other through ties

• Special case of standard OLS regression• Dependent variable is a vertex attribute

• y = (I −ΘW )−1(Xβ + (I − ψZ)−1v)

• where Θ is the matrix for the Auto-Regressive weights• and ψ is the matrix for the Moving Average weights

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

99-133

Network Autocorrelation Models

• Family of models for estimating how covariates relate toeach other through ties

• Special case of standard OLS regression• Dependent variable is a vertex attribute

• y = (I −ΘW )−1(Xβ + (I − ψZ)−1v)• where Θ is the matrix for the Auto-Regressive weights

• and ψ is the matrix for the Moving Average weights

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

99-133

Network Autocorrelation Models

• Family of models for estimating how covariates relate toeach other through ties

• Special case of standard OLS regression• Dependent variable is a vertex attribute

• y = (I −ΘW )−1(Xβ + (I − ψZ)−1v)• where Θ is the matrix for the Auto-Regressive weights• and ψ is the matrix for the Moving Average weights

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

100-133

The Classical Regression Model

Xiβ

εi

yi

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

100-133

The Classical Regression Model

Xiβ

εi

yi

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

100-133

The Classical Regression Model

Xiβ

εi

yi

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

100-133

The Classical Regression Model

Xiβ

εi

yi

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

101-133

Adding Network AR Effects

Xiβ

εi

yiXjβ

εj

yj

Xkβ

εk

yk

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

101-133

Adding Network AR Effects

Xiβ

εi

yiXjβ

εj

yj

Xkβ

εk

yk

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

101-133

Adding Network AR Effects

Xiβ

εi

yiXjβ

εj

yj

Xkβ

εk

yk

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

102-133

Adding Network MA Effects

Xiβ

εi

yiXjβ

εj

yj

Xkβ

εk

yk

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

102-133

Adding Network MA Effects

Xiβ

εi

yiXjβ

εj

yj

Xkβ

εk

yk

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

103-133

Network ARMA Model

Xiβ

εi

yiXjβ

εj

yj

Xkβ

εk

yk

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

104-133

Network ‘Resonance’

εi

εj

εk εl

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

104-133

Network ‘Resonance’

εi

εj

εk εl

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

104-133

Network ‘Resonance’

εi

εj

εk εl

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

104-133

Network ‘Resonance’

εi

εj

εk εl

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

104-133

Network ‘Resonance’

εi

εj

εk εl

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

104-133

Network ‘Resonance’

εi

εj

εk εl

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

105-133

Inference with the NetworkAutocorrelation Model

• Usually observe y, X, and Z and/or Z, want to infer β,θ, and φ

• Need each I−W, I− Z invertible for solution to exist

• error in disturbance autocorrelation, v, assumed as iid,vi N(0, σ2)

• Standard errors based on the inverse informationmatrix at the MLE

• Compare models in the usual way (eg AIC, BIC)

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

105-133

Inference with the NetworkAutocorrelation Model

• Usually observe y, X, and Z and/or Z, want to infer β,θ, and φ

• Need each I−W, I− Z invertible for solution to exist

• error in disturbance autocorrelation, v, assumed as iid,vi N(0, σ2)

• Standard errors based on the inverse informationmatrix at the MLE

• Compare models in the usual way (eg AIC, BIC)

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

105-133

Inference with the NetworkAutocorrelation Model

• Usually observe y, X, and Z and/or Z, want to infer β,θ, and φ

• Need each I−W, I− Z invertible for solution to exist

• error in disturbance autocorrelation, v, assumed as iid,vi N(0, σ2)

• Standard errors based on the inverse informationmatrix at the MLE

• Compare models in the usual way (eg AIC, BIC)

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

105-133

Inference with the NetworkAutocorrelation Model

• Usually observe y, X, and Z and/or Z, want to infer β,θ, and φ

• Need each I−W, I− Z invertible for solution to exist

• error in disturbance autocorrelation, v, assumed as iid,vi N(0, σ2)

• Standard errors based on the inverse informationmatrix at the MLE

• Compare models in the usual way (eg AIC, BIC)

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

105-133

Inference with the NetworkAutocorrelation Model

• Usually observe y, X, and Z and/or Z, want to infer β,θ, and φ

• Need each I−W, I− Z invertible for solution to exist

• error in disturbance autocorrelation, v, assumed as iid,vi N(0, σ2)

• Standard errors based on the inverse informationmatrix at the MLE

• Compare models in the usual way (eg AIC, BIC)

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

105-133

Inference with the NetworkAutocorrelation Model

• Usually observe y, X, and Z and/or Z, want to infer β,θ, and φ

• Need each I−W, I− Z invertible for solution to exist

• error in disturbance autocorrelation, v, assumed as iid,vi N(0, σ2)

• Standard errors based on the inverse informationmatrix at the MLE

• Compare models in the usual way (eg AIC, BIC)

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

106-133

Choosing the Weight Matrix

• crucial modeling issue to choose the right form• standard adjacency matrix• row-normalized adjancecy matrix• structural equivalence distance

• Many suggestions given by Leenders 2002

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

106-133

Choosing the Weight Matrix

• crucial modeling issue to choose the right form

• standard adjacency matrix• row-normalized adjancecy matrix• structural equivalence distance

• Many suggestions given by Leenders 2002

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

106-133

Choosing the Weight Matrix

• crucial modeling issue to choose the right form• standard adjacency matrix

• row-normalized adjancecy matrix• structural equivalence distance

• Many suggestions given by Leenders 2002

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

106-133

Choosing the Weight Matrix

• crucial modeling issue to choose the right form• standard adjacency matrix• row-normalized adjancecy matrix

• structural equivalence distance

• Many suggestions given by Leenders 2002

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

106-133

Choosing the Weight Matrix

• crucial modeling issue to choose the right form• standard adjacency matrix• row-normalized adjancecy matrix• structural equivalence distance

• Many suggestions given by Leenders 2002

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

106-133

Choosing the Weight Matrix

• crucial modeling issue to choose the right form• standard adjacency matrix• row-normalized adjancecy matrix• structural equivalence distance

• Many suggestions given by Leenders 2002

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

107-133

Data Prep

• Dependent variable is a vertex attribute

• Covariates are in matrix form with one column perattribute

• Can include an intercept term by adding a column of 1s

• Weight matrices for both AR and MA terms in matrixform

• Can include multiple weight matrices (as a list) forboth AR and MA

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

107-133

Data Prep

• Dependent variable is a vertex attribute

• Covariates are in matrix form with one column perattribute

• Can include an intercept term by adding a column of 1s

• Weight matrices for both AR and MA terms in matrixform

• Can include multiple weight matrices (as a list) forboth AR and MA

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

107-133

Data Prep

• Dependent variable is a vertex attribute

• Covariates are in matrix form with one column perattribute

• Can include an intercept term by adding a column of 1s

• Weight matrices for both AR and MA terms in matrixform

• Can include multiple weight matrices (as a list) forboth AR and MA

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

107-133

Data Prep

• Dependent variable is a vertex attribute

• Covariates are in matrix form with one column perattribute

• Can include an intercept term by adding a column of 1s

• Weight matrices for both AR and MA terms in matrixform

• Can include multiple weight matrices (as a list) forboth AR and MA

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

107-133

Data Prep

• Dependent variable is a vertex attribute

• Covariates are in matrix form with one column perattribute

• Can include an intercept term by adding a column of 1s

• Weight matrices for both AR and MA terms in matrixform

• Can include multiple weight matrices (as a list) forboth AR and MA

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

107-133

Data Prep

• Dependent variable is a vertex attribute

• Covariates are in matrix form with one column perattribute

• Can include an intercept term by adding a column of 1s

• Weight matrices for both AR and MA terms in matrixform

• Can include multiple weight matrices (as a list) forboth AR and MA

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

108-133

Leenders 2002

SNA

LJasny

Intro

DataStructures

Descriptives

HypothesisTesting

109-133