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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 nonreciprocal—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.
BackgroundRandom 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.
BackgroundRegular 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
A
B
C
1
A
B
1
C
0
1
D
1
0
D
0
1
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.
1
0
0
0

A Very Simple Example
A
B
C
1
A
B
1
C
0
D

The possible number of
connections in this
matrix is 6.
0
1
1
0

K=4 legislators.
0

½ * k * (k-1) = ½ * 4 * 3
=6
1

D
1
0
0
A Very Simple Example
A
B
1
A
B
C
1
C
0
1
D
1
0
D
0
1
1
0
0
0

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 Very Simple Example
A
B
C
1
A
B
1
C
0
1
D
1
0
D
0
1
1
0
0
0

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.
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 nonreciprocated 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
International article)

“Small World Networks and Management
Science Research: A Review”
Small Worlds
here to build your own “small
world” graph.
Click
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).
Additional Sources / Supplemental
Readings

Some Antecedents of Social Network
Analysis (Freeman)

An update on Strength of Weak Ties
(Granovetter)

New York Times, “Is MySpace Good for
Society? A Freakonomics Quorum”
Instructional Sites






Steve Borgatti’s site
Note the “Networks for Newbies” presentation
(Wellman) on the website
From Sociology 712 (Moody) at Duke
From Friedkin’s “Intro to Social Network Methods”
(UCSB)
From Martin and Montgomery’s “New Methods of
Social Network Analysis”
Andrej Mrvar’s site