Social Network Analysis

1
Social Network Analysis

• Introduction

2
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.

3
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.

4
Network Analysis

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

5
History of (Social) Network Analysis

• First, lets discuss the history of network
  analysis, to give an idea of what sorts of
  questions can be posed. Then, well 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.

6
History of Social Network Analysis

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

7
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 individualsand 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)

8
Early History

• One of the first examples of empirical network
  research can be found in 1922, in Almacks 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 IQs of the choosers and
  the chosen, and examined the hypothesis that
  choices were homophilous.

9
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 students height, grades, IQ, score
  on a physical coordination test, and degree of
  introversion versus extraversion (based on
  teachers ratings). She then examined whether
  interaction was homophilous.
• (see The School Childs Choice of Companions,
  Journal of Educational Research 14 126-132.)

10
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.

11
Early History

• Note that Botts 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.

12
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)

13
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 individuals own accounts of their
  patterns of interactions.
• The many different ways in which individuals can
  interact.
• How to think about longitudinal aspects of
  interaction.

14
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-reciprocaland that many individuals were
  isolated.
• Morenos quantitative method to map relationships
  is called sociometry.

15
Other Advances

• Festingers (1950) study of the influence of dorm
  room location indicated that individuals were
  more likely to associate with those who were
  similar to themin this case, similar in terms of
  location. Festingers 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.

16
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)

17
Bennington College Study

• Two follow-up studies indicated that the change
  was largely permanentthe 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
  attitudesbut our attitudes are also changed by
  our reference groups.

18
1960s-gt

• 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.

19
Some concepts

• Before we discuss the strength of weak ties and
  small worlds, lets 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.

20
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.

21
Some concepts

• In an undirected graph or network, the edges are
  reciprocalso if A is connected to B, B is by
  definition connected to A.
• In a directed graph or network, the edges are not
  necessarily reciprocalA 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 lets discuss the meaning of the
  strength of weak ties....

22
The Strength of Weak Ties

• Granovetters 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.

23
The Strength of Weak Ties

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

24
Small Worlds---Intro

• Next, lets consider the related concept of
  small worlds, another concept that has emerged
  in network analysis.
• But for some background, lets 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.

25
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 connectionsabout 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.

26
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 nodesthat
  is, there are three nodes in each nodes
  neighborhood.)

27
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.

28
Clustering Coefficients

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

29
A Very Simple Example

• Four legislatorswhether they serve on at least
  one committee together.
• This is an undirected matrixif 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  
30
A Very Simple Example

• The possible number of connections in this matrix
  is 6.
• K4 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  
31
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  
32
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  
33
Clustering Coefficients

• This is the formula the clustering coefficient
  for the system. Nnumber of nodes. Cclustering
  coefficient for each node i.

34
Clustering Coefficient

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

35
Clustering Coefficient

• So, in an undirected graph, if a node is
  connected to four other nodesand among those
  four, only the first and the third are
  connectedthe 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 lets talk about network diameter.

36
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)

37
Its a Small World, After All

• This is essentially the six degrees of
  separation ideathat 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 Milgrams 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.

38
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 graphsbut do describe a wide
  variety of real-life situations.

39
Small Worlds

• The left is an example of a small-world graph.
• Note that it is highly clustereda higher
  proportion (than one would expect randomly) of
  each nodes neighbors are actually connected to
  each other.
• It also has a small diameter, relative to the
  number of nodes.

40
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

41
Small Worlds

• Click here to build your own small world graph.

42
Social Capital Research

• The importance of networks can also be seen in
  much social capital research.
• Social capital research often examines the
  connections across individualsand the
  consequences of the number and type of those
  connections for groups/organizations and for
  individuals.

43
Social Capital Research

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

44
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 sciences traditional
  emphasis on individual, independent units meant
  that networks were less of a focus.

45
Network Research in Political Science

• And, of course, Huckfeldt and Spragues work on
  congruence and dissonance across discussion
  partners takes a network approach.
• More recently, networks have been receiving
  increased attention in political sciencemost
  obviously with the work of Jim Fowler (across
  disciplines). Much useful information can be
  found at the Social Network Blog (Program on
  Networked Governance).

46
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

47
Instructional Sites

• Steve Borgattis site
• Note the Networks for Newbies presentation
  (Wellman) on the website
• From Sociology 712 (Moody) at Duke
• From Friedkins Intro to Social Network Methods
  (UCSB)
• From Martin and Montgomerys New Methods of
  Social Network Analysis
• Andrej Mrvars site