Social Network Analysis

Introduction

What is Network Analysis?

• Social network analysis is a method by which one can analyze the connections across individuals or groups or institutions. That is, it allows us to examine how political actors or institutions are interrelated.

Network Analysis

• The advantage of social network analysis is that, unlike many other methods, it focuses on interaction (rather than on individual behavior).

• Network analysis allows us to examine how the configuration of networks influences how individuals and groups, organizations, or systems function.

Network Analysis

• It can be applied across disciplines—there are social networks, political networks, electrical networks, transportation networks, and so on.

History of (Social) Network Analysis

• First, let’s discuss the history of network analysis, to give an idea of what sorts of questions can be posed. Then, we’ll discuss some basic concepts.

• Much early research in network analysis is found in educational psychology, and studies of child development. Network analysis also developed in fields such as sociology and anthropology.

History of Social Network Analysis

• In the 19th century, Durkheim wrote of “social facts”—or phenomena that are created by the interactions of individuals, yet constitute a reality that is independent of any individual actor.

History of Social Network Analysis

• At the turn of the 20th century, Simmel was one of the first scholars to think in relatively explicit social network terms. He examined how third parties could affect the relationship between two individuals—and he examined how organizational structures or bureaucracies were needed to coordinate interactions in large groups.

• (See “The Number of Members in Determining the Sociological Form of the Group”)

Early History

• One of the first examples of empirical network research can be found in 1922, in Almack’s “The Influence of Intelligence on the Selection of Associates.” Almack asked children in a California elementary school to identify the classmates with whom they wanted as playmates. He then correlated the IQ’s of the choosers and the chosen, and examined the hypothesis that choices were homophilous.

Early History

• In 1926, Wellman recorded pairs of individuals who were observed as being together frequently. She also recorded trait (or attribute) data, including the student’s height, grades, IQ, score on a physical coordination test, and degree of introversion versus extraversion (based on teacher’s ratings). She then examined whether interaction was homophilous.

• (see “The School Child’s Choice of Companions”, Journal of Educational Research 14: 126-132.)

Early History

• In 1928, Bott took an ethnographic approach examine the behavior of preschool children in Toronto. She identified five types of interaction: talking to one another, interfering with one another, watching one another, imitating one another, or cooperating with one another. She then used “focal sampling”, observing one child each day.

Early History

• Note that Bott’s work also was a harbinger of the network research which was to follow, in that she organized her data into matrices, and discussed her results in terms of the linkages between individuals.

Early History

• In “The Companionships of Preschool Children”, Hagman (1933) both observed interaction throughout the term, and interviewed children to measure their recollections of their interactions earlier in the term.

• (University of Iowa Studies in Child Welfare)

Early History

• Note that these studies raise several issues– How to link attributes (such as IQ) to interaction

– The difference between observational approaches and relying on individual’s own accounts of their patterns of interactions.

– The many different ways in which individuals can interact.

– How to think about longitudinal aspects of interaction.

Early History

• In 1933, the New York Times reported on the new science of “psychological geography” which “aims to chart the emotional currents, cross-currents and under-currents of human relationships in a community”.

• Jacob Moreno analyzed the interconnections across 500 girls in the State Training School for Girls, and the interconnections of students within two NYC schools.

• Moreno concluded that many relationships were non-reciprocal—and that many individuals were isolated.

• Moreno’s quantitative method to map relationships is called “sociometry”.

Other Advances

• Festinger’s (1950) study of the influence of dorm room location indicated that individuals were more likely to associate with those who were similar to them—in this case, similar in terms of location. Festinger’s theory of propinquity posited that those who were physically close to each other were more likely to form positive associations. Specifically, the arrangement of dorms rooms could influence the formation of both weak and strong relationships.

Bennington College Study(1935-1939)

• Theodore Newcomb found that as Bennington college women were exposed to the relatively liberal referent group of fellow students and faculty, they became more liberal.

– “Becoming radical meant thinking for myself and, figuratively, thumbing my nose at my family. It also meant intellectual identification with the faculty and students that I most wanted to be like” (Newcomb, 1943, pp. 134, 131)

Bennington College Study

• Two follow-up studies indicated that the change was largely permanent—the women remained relatively liberal, likely in part because they picked new referent group (spouses, friends, co-workers) that reinforced those attitudes.

• In other words, attitudes have a “social-adjustment” function.

• We often choose reference groups that reinforce attitudes—but our attitudes are also changed by our reference groups.

1960s->

• After the 1950s, networks were less evident in social psychology...and more evident in sociology (particularly economic sociology), and (to a lesser extent) in anthropology.

• Developments in the last few decades include much attention paid to several concepts, including “the strength of weak ties”, and “small worlds”.

• Networks are also central to much of the research on ‘social capital’.

Some concepts

• Before we discuss “the strength of weak ties” and “small worlds”, let’s just go over some basic concepts.

• A node or vertex is an individual unit in the graph or system. (If it is a network of legislators, then each node represents a legislator).

• A graph or system or network is a set of units that may be (but are not necessarily) connected to each other.

Some concepts

• An “edge” is a connection or tie between two nodes.

• A neighborhood N for a vertex or node is the set of its immediately connected nodes.

• Degree: The degree ki of a vertex or node is the number of other nodes in its neighborhood.

Some concepts

• In an undirected graph or network, the edges are reciprocal—so if A is connected to B, B is by definition connected to A.

• In a directed graph or network, the edges are not necessarily reciprocal—A may be connected to B, but B may not be connected to A (think of a graph with arrows indicating direction of the edges.)

• Okay, now let’s discuss the meaning of the “strength of weak ties”....

The Strength of Weak Ties

• Granovetter’s “The Strength of Weak Ties” (considered one of the most important sociology papers written in recent decades) argued that “weak ties” could actually be more advantageous in politics or in seeking employment than “strong ties”, because weak ties allowed an individual to reach a higher number of other individuals.

The Strength of Weak Ties

• Granovetter observed that the presence of weak ties often reduced path lengths (distance) between any two individuals—which led to quicker diffusion of information.

Small Worlds---Intro

• Next, let’s consider the related concept of “small worlds”, another concept that has emerged in network analysis.

• But for some background, let’s discuss some different possible types of graphs, plus the concepts of “clustering” and “diameter”.

• Two possible graphs (almost at opposite ends of a spectrum) are “random graphs” and “regular graphs”. A “small world” can be thought of in-between a random and a regular graph.

BackgroundRandom Graphs

• In a random graph, each pair of vertices i, j has a connecting edge with an independent probability of p

• This graph has 16 nodes, 120 possible connections, and 19 actual connections—about a 1/7 probability than any two nodes will be connected to each other.

• In a random graph, the presence of a connection between A and B as well as a connection between B and C will not influence the probability of a connection between A and C.

BackgroundRegular Graphs

• A regular graph is a network where each node has the same number (k) of neighbors (that is, each node or vertex has degree k).

• A k-degree graph is seen at the left. k = 3 (each node is connected to three other nodes—that is, there are three nodes in each node’s neighborhood.)

Clustering Coefficients

• Clustering Coefficients were introduced by Watts & Strogatz in 1998, as a way to measure how close a node (or vertex) and its neighbors are from being a clique, or a complete graph within a larger graph or network.

• The clustering coefficient of a node is the number of actual connections across the neighbors of a particular node, as a percentage of possible connections. The clustering coefficient for the entire system is the average of the clustering coefficient for each node.

Clustering Coefficients

• This formula (on the right) is for the total number of possible connections for an undirected matrix. (Think in terms of a matrix—the total number of possible connections is half of the total # of cells, after subtracting the diagonal.)

A Very Simple Example

• Four legislators—whether they serve on at least one committee together.

• This is an undirected matrix—if legislator A serves with legislator B on a committee, then legislator B serves with legislator A on a committee.

A B C D

A 1 0 1

B 1 1 0

C 0 1 0

D 1 0 0

A Very Simple Example

• The possible number of connections in this matrix is 6.

• K=4 legislators.

• ½ * k * (k-1) = ½ * 4 * 3• = 6

A B C D

A 1 0 1

B 1 1 0

C 0 1 0

D 1 0 0

A Very Simple Example

• The clustering coefficient for legislator A is 2/3 – s/he is “connected to” two out of a possible 3 other legislators. The same is true of legislator B.

• Legislators C and D each have a clustering coefficient of 1/3.

A B C D

A 1 0 1

B 1 1 0

C 0 1 0

D 1 0 0

A Very Simple Example

• The average of those four clustering coefficients is .5.

• And note that across the entire network, .5 (3 of 6) of all possible connections are actually made.

A B C D

A 1 0 1

B 1 1 0

C 0 1 0

D 1 0 0

Clustering Coefficients

• This is the formula the clustering coefficient for the system. N=number of nodes. C=clustering coefficient for each node i.

Clustering Coefficient

• Note that the clustering coefficient for undirected graphs is a bit different than the clustering coefficient for directed graphs—there are twice as many possible ties, a non-reciprocated edge counts for one tie, and a reciprocated edge counts for two ties.

Clustering Coefficient

• So, in an undirected graph, if a node is connected to four other nodes—and among those four, only the first and the third are connected—the clustering coefficient is 1/6. (1 actual connection out of 6 possible connections.)

• Clustering refers to how connected your neighbors are to each other (relative to how connected they could be)

• Now let’s talk about network diameter.

Graph Diameter

• The graph diameter is the “longest shortest path” between any two vertices or nodes.

• The graphs above have diameters of 3, 4, 5, and 7, respectively.• The graph on the right has a relatively large diameter, because it takes

(at most) 7 edges to travel between one node to another. (the two nodes at the very bottom of the network are not very closely connected)

It’s a Small World, After All

• This is essentially the “six degrees of separation” idea—that the number of “steps” or “links” needed to connect any one arbitrarily chosen individual to any other is low (that is, networks have lower diameters than one would expect.)

• In Milgram’s 1967 “small world experiment”, individuals were asked to reach a particular target individual by passing a message along a chain of acquaintances. For successful chains, the average # of intermediaries needed was 5 (that is, 6 steps)—although note that most chains were not completed.

Small Worlds

• Brian Uzzi has focused on the importance of “small worlds”–networks that are both highly locally clustered and have short path lengths. A graph is small-world if its average clustering coefficient is significantly higher than a random graph constructed on the same vertex set (with the same number of edges), and if the graph has a short mean-shortest path length.

• These two characteristics are often mutually exclusive in random graphs—but do describe a wide variety of real-life situations.

Small Worlds

• The left is an example of a small-world graph.

• Note that it is highly clustered—a higher proportion (than one would expect randomly) of each node’s neighbors are actually connected to each other.

• It also has a small diameter, relative to the number of nodes.

Small Worlds

• See, for example:

• “Collaboration and Creativity: The Small World Problem” (also see the Newsweek Internationalarticle)

• “Small World Networks and Management Science Research: A Review”

Social Capital Research

• The importance of networks can also be seen in much social capital research.

• Social capital research often examines the connections across individuals—and the consequences of the number and type of those connections for groups/organizations and for individuals.

Social Capital Research

• For a review of this research, see “The Network Structure of Social Capital”

Network Research in Political Science

• The history of network analysis in political science is less substantial...

• One of the first uses of what we think of as network analysis was seen in the 1927 APSR: Rice examined ways to identify “blocs” in small legislative bodies. He focused on cohesion (a version of “clustering”) and on likeness.

• Other similar studies on cohesion occasionally followed. But political science’s traditional emphasis on individual, independent units meant that networks were less of a focus.

Network Research in Political Science

• And, of course, Huckfeldt and Sprague’s work on congruence and dissonance across discussion partners takes a network approach.

• More recently, networks have been receiving increased attention in political science—most obviously with the work of Jim Fowler (across disciplines). Much useful information can be found at the Social Network Blog(Program on Networked Governance).

Introductory Social Network Analysis

with Pajek

Based on slides by Q Mei (U Michigan)

Outline

• In Pajek– visualization and layouts– degree– connected components– snowball sampling– one mode projections of bipartite graphs– thresholding weighted graphs

• Fetching your Facebook network (or similar)– What is an egonetwork?

Using Pajek for Exploratory Social Network Analysis

• Pajek – (pronounced in Slovenian as Pah-yek) means ‘spider’

• website: http://vlado.fmf.uni-lj.si/pub/networks/pajek/ • wiki: http://pajek.imfm.si/doku.php

– download application (free)– tutorials– lectures– data sets

• Windows only (works on Linux via Wine, Mac via Darwine)

• helpful book: ‘Exploratory Social Network Analysis with Pajek’ by Wouter de Nooy, Andrej Mrvar and Vladimir Batagelj

Pajek: Interface

Drop down list of networks opened or created with pajek. Active is displayed

Drop down list of network partitions by discrete variables, e.g. degree, mode, label

Drop down list of continuous node attributes, e.g. centrality, clustering coefficients

we’ll use today

can be used for clustering

Source: Pajek, Free for noncommercial use - http://pajek.imfm.si/doku.php?id=download

Pajek: Opening a Network Fileclick on folder iconto open a file

Save changes to your network, network partitions, etc., if you’d like to keep them

Source: Pajek, Free for noncommercial use - http://pajek.imfm.si/doku.php?id=download

Pajek: Working with Network Files

• The active network, partition, etc is shown on top of the drop down list

Draw the network

Source: Pajek, Free for noncommercial use - http://pajek.imfm.si/doku.php?id=download

Pajek data format

*Vertices 261 "Ada" 0.1646 0.2144 0.50002 "Cora" 0.0481 0.3869 0.50003 "Louise" 0.3472 0.1913 0.5000..

*Arcs 1 3 2 c Black1 2 1 c Black2 1 1 c Black..

*Edges 2 3 1 c Black ..

Ada

Cora

Louise number of vertices vertex x,y,z coordinates (optional)

directed edges

undirected edges

from Ada(1) to Louise(3) w/eight “2” and color Black

between Ada(1) to Cora(2) w/weight “1” and color Black

Pajek: Let’s Get Started• Opening a network

– File Network Read• Visualization

– Draw Draw• Essential measurements

– …

Pajek: Opening a File• A simple graph and layouts in Pajek• Create a small network using a text editor in the Pajek

format (10-20 nodes).

• Open it in Pajek by either clicking on the yellow folder icon under the word "Network" or by selecting FileNetworkRead from the main menu panel

• A report window should pop up confirming that the graph has been read and the filename and location will be displayed in the 'active' position of the network dropdown list

Pajek: Visualization & Manual Positioning• Visualize the network using Pajek's DrawDraw

command from the main menu panel.• This will bring up the 'draw' window with its own menu

bar at the top• Reposition the vertices by clicking on them and holding

down the mouse button while dragging them to a new location. Continue doing this until you have shown that the graph is planar (no edges cross have to cross )

Pajek: Visualization & Layout Algorithms• Now let Pajek do the work for you by selecting from the

draw toolbar several layout algorithms under 'LayoutEnergy'.

• Why did you select the layout algorithm you did?• Did the layout leave any lines crossed? If you were to

do this assignment over, what order would you do it in?

A Directed Network

Ada

Cora

Louise

Jean

Helen

Martha

Alice

Robin

Marion

Maxine

Lena

Hazel Hilda

Frances

Eva

RuthEdna

Adele

Jane

AnnaMary

Betty

Ella

Ellen

Laura

Irene

• girls’ school dormitory dining-table partners (Moreno, The sociometry reader, 1960)

• first and second choices shown

Source: http://vlado.fmf.uni-lj.si/ pub/networks/data/esna/dining.htm

Node Centrality: Degree

• Node network properties– from immediate connections

• indegreehow many directed edges (arcs) are incident on a node

• outdegreehow many directed edges (arcs) originate at a node

• degree (in or out)number of edges incident on a node

– labels

outdegree=2

indegree=3

degree=5

Centrality: Degree

• More on degree and other centrality measures in the lecture…

• Degree: calculate it– Net Partitions Degree

• Visualize degree centrality– DrawDraw-Vector– If nodes are not the right size, use resize option

• Options Sizeof Vertices• Adjust the default size

Connected Components

• Strongly connected components– Any two nodes in the component

can be reached from each other by following directed edges B C D E A G H F

• Weakly connected components: every node can either reach or be reached from every other node by following directed edges

A

B

C

DE

F

GH

A B C D E G H F

• In undirected networks one talks simply about “connected components”

The bowtie model of the WebBroder et al. (1999)

• SCC (strongly connected component):

– can reach all nodes from any other by following directed edges

• IN – can reach SCC from any node in

‘IN’ component by following directed edges

• OUT– can reach any node in ‘OUT’

component from SCC• Tendrils and tubes

– connect to IN and/or OUT components but not SCC

• Disconnected – isolated components

Let’s try this on the dining table partners

• Suppose each girl will share her dish (and any dish that is shared with her), with only her 1st and 2nd choice– Find groups of girls who can all sample each other's dishes

• Net Components Strong• Draw Draw-partition

– Which girls will get to sample no other dishes but their own?• Net Partitions Degree Input

– Display the network of strongly connected components• Operations Shrink Network Partition• Draw Draw-partition

– Identify the bowtie structure • Net Partitions Bow-Tie

– What part of the bowtie is missing? What does this imply for the circulation of the different dishes?

Snowball Sampling

• Dining partners: You are a prince who just met an enchanting young lady at a ball, but she left at the stroke of midnight and left a shoe behind. Now you'd like to find the shoe's owner. All you know about her is that she lives in this particular girls' dorm. The headmistress won't let you talk to the girls, so the only way you can find your princess is to covertly ask the one girl you know, Ella, to introduce you to her two favorite friends. Once you know her friends, you can ask them to introduce you to their two favorite friends, etc. This is the snowball sampling technique we covered in class.

Highlight the vertices that you will reach using snowball sampling– Net K-Neighbors ...

• Which girls will you not find using snowball sampling starting with Ella (23)?

Bipartite networks

Going from a Bipartite to a One-mode Graph

• One mode projection– two nodes from the first

group are connected if they link to the same node in the second group

– some loss of information– naturally high

occurrence of cliques

Two-mode networkgroup 1

group 2

Let’s Try it on the Actors Network

• Download and open seminar_actorsAndMovies.net • Create a 2-mode partition (newer versions of Pajek do

this automatically) – NetPartition2-Mode

• Draw the network – DrawDraw-Partition (the two classes of nodes should be

colored differently)– If labels are not shown, add them:

Options Mark Vertices UsingLabels

• Experiment with different layout algorithms– “Layout” menu in the draw window

Actors Network (continued)

• Transform the network into a one-mode network– NetTransform2-Mode to 1-ModeRows

• Draw the network – DrawDraw

• Qualitatively compare the structure of the 2-Mode to the 1-Mode network. Is there a loss of information?

• Show the weights on each edge– OptionsLinesMark Lineswith Values

• What do the values represent? • Compute the unweighted degree of each node

– NetPartitionsDegreeAll• Draw the network using

– DrawDraw-vector• How is the degree represented?

• Add the vector value to each vertex (it will be the degree/(max possible degree)) – OptionsMark Vertices UsingVector Values

• Who are the most important actors using this measure?

Actors Network (continued)

• How does the boundary of the network (i.e. who is included) affect who is found to be most central? – Load the file seminar_ActorsAndMoviesPlusOne.net. It contains

one extra actor, Richard Gere. Repeat the above procedure.– In the 1-mode network of actors, is there a change in who is

most central?– What does this tell you about biases and boundaries in sample

selection?

• Remove all edges between actors who have costarred in fewer than 3 movies– NetTransformRemoveLines with valueLower than

• Which actors comprise the central core of this network

Grabbing Data from Social Network Platforms

https://linkurio.us/how-to-visualize-your-facebook-network/

http://konect.uni-koblenz.de/networks/ego-facebook

http://www.pewinternet.org/files/2014/02/How-we-analyzed-Twitter-social-media-networks.pdf

Pajek: Wrap Up

• Used frequently by sociologists – UCInet is comparable and arguably more user friendly (but not

free)

• Extensive functionality– But not extendable

• What we covered– visualization– node properties: degree– connected components– k-neighbors– converting two-mode networks to one-mode– thresholding the network