Network Positions and Roles

1
Network Positions and Roles

• Chapter 12 in Wasserman and Faust

2
Chapter 12 Overview

• Concerned mainly with
• Defining equivalences
• Examining how closely actors adhere to these
  definitions
• Focus on Individual Actors Roles (local) vs.
  Chapter 11 which concentrated on role structure
  and comparing role structures for groups
  (global)

3
Social Role vs. Social Positions

• Position collection of actors
• Role How occupant of a position relates to other
  occupants/positions
• Role focuses on ties among actors rather than the
  collection of actors

4
Levels of Role Structures

• Global examines an entire group of actors, often
  quite abstractly defined
• Local examines subsets of actors, often
  similarly positioned (approximately structurally
  equivalent) actors
• Individual or Ego examines the patterns and
  regularity in ties to and from a particular actor

5
Kinds of Equivalence

• Structural
• Automorphic and isomorphic
• Regular Equivalence
• Local Role
• Ego Algebras

6
Structural Equivalence

• Actors with ties to exact same other actors
• Seven Sets of Structurally equivalent actors
• B(SE)1 (1)
• B(SE)2 (2)
• B(SE)3 (3)
• B(SE)4 (4)
• B(SE)5 (5,6)
• B(SE)6 (7)
• B(SE)7 (8,9)
• In this case 5 and 6 are structurally equivalent
  (identical ties to 2) , as are 8 and 9 (identical
  ties to 4)

7
Limitations of a Strict Structural Equivalence
Approach

• Need for identical ties can be too limiting (i.e.
  in order for two managers to be structurally
  equivalent they would need to oversee the same
  workers)
• Therefore it is impossible to compare even
  slightly different actors or to compare across
  networks (i.e. managers in two companies)
• Leads to arguments supporting more generalized
  measures of equivalence requires defining roles
  and positions in terms of patterns and types of
  ties

8
Automorphic and Isomorphic Equivalence

• Isomorphism Two graphs are isomorphic if there
  is a one-to-one mapping of points on these graphs
  that preserves adjacency
• Automorphism Works with similar logic, however
  within one graph (or directed graph)
• More general than Structural Equivalence
  Structurally Equivalent actors are always
  Automorphically Equivalent, however the opposite
  is not true
• Automorphically Equivalent actors are identical
  in graph theoretic properties (indegree,
  outdegree, measures of centrality, clique
  memberships, etc)

9
Automorphism

• Assuming the relation here is supervises the
  work of we can see 2 and 4 are both supervised
  by 1 and supervise two others themselves
• 3 however is not Automorphically Equivalent
  because it only supervises one other actor (AE
  requires same indegree and outdegree)
• Five Sets of Automorphically Equivalent actors
• B(AE)1 (1)
• B(AE)2 (2,4)
• B(AE)3 (3)
• B(AE)4 (5,6,8,9)
• B(AE)5 (7)

10
Regular Equivalence

• Broader than structural or automorphic
  equivalence
• Doesnt require identical ties to identical
  actors (Struc. Equiv) or even structural
  indistinguishability (Auto/Iso Equiv.)
• Regular Equivalence involves actors who have
  identical ties to equivalent actors
• i and j are regularly equivalent when i has a tie
  to k, j has a tie to l, and k and l are regularly
  equivalent
• A network or graph can have several partitions
  that meet the Regular Equivalence standard

11
Regular Equivalence

• Requires actors to have ties to other Regularly
  Equivalent actors
• Three Sets of Regularly Equivalent actors
• B(RE)1 (1)
• B(RE)2 (2,3,4)
• B(RE)3 (5,6,7,8,9)

12
Types of Possible Partitions for Regular
Equivalence

• Possible Partitions that meet Regular Equivalence
  standards include Stuctural and Auto or
  Isomorphic Equivalence
• The Maximal Partition is the partition with the
  fewest equivalence classes or the coarsest
  partition

13

• Maximal Partition is
• B(RE)1 (1, 2, 3, 4)
• Alternate Partition is
• B(RE)1 (1,3,4)
• B(RE)2 (2)
• For alternate partition, members of Regular
  Equivalence Class 1 have the ties to a member of
  RE Class 2, and vice versa

14
Regular Equivalence Block Models

• Can be constructed for Regular Equivalence
  Classes, in the same way that it can be done for
  Structural Equivalence Classes
• Relatively Recent Development, and not very
  widely used

15
Image Matrix for Regular Equivalence Blockmodel
16
Using Multiple Network Analytic Tools for a
Single Social Network

• By Patrick Doreian

17
Problems Facing SNA Scholars

• Inherent Complexity of Structural Data
• Comparing and Evaluating Various Network
  Measurements and Tools
• 1) The State of the Art is in its infancy
• 2) Disagreement about what structure
  actually is

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Strategies to Address this Issue

• Different Scholars, using their particular
  approaches and perspectives, analyze similar data
  for comparison
• Individual Scholars can analyze one data set with
  many approaches
• Construct toy networks with certain structural
  characteristics, and examine whether these
  characteristics are measured or manifested

19
Comparative Analysis of Single Set of Data

• From Doreian and Albert (1986)
• Network of 14 Prominent Political Actors in a
  Midwestern County
• 7 County Council Members, the County Sheriff,
  Auditor, Executive, Prosecutor, a City Mayor and
  two former Council Members

20
MDS Scaling

• Partitioning into camps around Executive (A) and
  the Auditor (B) - seems to support Hypothesis 1
• Cluster around A supported the jail construction,
  cluster around B did not seems to support
  hypothesis 2

21
Doreian and Alberts Hypotheses

• Focus on the construction of a jail
• In the wake of a power power struggle between
  County Executive and County Auditor
• The network would be partitioned into two camps
  surrounding the Executive and Auditor
  respectively
• Votes for/against jail construction would be
  correlated to the partition in Hypothesis 1

22
Overlaying Political Ties onto MDS

• Density within Alliances i.e. 11 of 14 ties in
  Alliance A are internal to the Alliance, whereas
  only 1 goes to Alliance B
• The Former Council President (L) has numerous
  links to both Alliances
• L and K are unaligned, but L plays an important
  structural role

23

• The Unique position of the Former Council
  President is illustrated particularly by the
  Betweeness Centrality score

24
First Order Stars

• First order star is the group of actors and ties
  surrounding a defined Ego
• Examining the overlap of First Order Stars offers
  another perspective on Network Location
• Creates Same basic Alliance structure, further
  emphasizes Ks uniqueness and lack of involvement

25
Using Equivalence to Define a Spatial Mapping

• Doreian claims hes using structural equivalence
  but that strict equivalence seldom holds and is
  relaxed to some degree of equivalence which, when
  measured, is subject to a clustering or MDS
  analysis.
• Uses two methods to map equivalence Kruskal and
  Guttman-Lingoes algorithms

26
Similar Mappings from Kruskal and Guttmann-Lingoes
27

• Using Concor
• Alliance A is on the left, Alliance B and the
  Unaligned are on the right
• The Unaligned are structurally distinguished from
  Alliance B

28

• Using STRUCTURE algorithm
• Similarly you get unaligned (L,K) and two
  broader Alliance Blocks

29
Structural (STRUCTURE) vs. Regular Equivalence
(REGE)
30
Regular Equivalence Classes

• B(RE)1 (L)
• B(RE)2 (B,H,M,D,F)
• B(RE)3 (K,E,N,G)
• B(RE)4 (I,J,C,A)
• Class 1 is Unique
• Class 2 Bridge Across Alliances
• Class 3 Entirely within alliances (or within the
  unaligned camp)
• Class 4 Bridge within Alliances (A is
  problematic)

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Clique Structure Seems to Coincide with MDS
32
Conclusions

• The various techniques seem to offer the same
  general picture of network structure with
  occasional differences in detail that require
  further study
• Call for a consensus on the structural properties
  of critical interest and the extent to which
  our tools succeed in measuring that structure.

33
The Structure of Social Protest 1961-1983

• By Peter Bearman and Kevin Everett

34
Social Protest Structure

• Goal 1 Descriptive
• - In each time period various groups engage in
  protest
• - Examines repertoires of protest, as well as
  which groups influence these repertoires
• - Chart a structural model of social protest
  over time
• Goal 2 Empirical
• Tests claims of New Social Movement Theory
  namely
• - Identity has replaced interest in
  determining
• - The role of Organized Labor has declined

35
Data

• Data Social Protest information on 397 protest
  events coded from the Washington Post
• 300 groups protested at least once, only 100
  protested more than once also includes multiple
  group protests
• Time Periods were 1961-1963, 1966-1968,
  1971-1973, 1976-1978, and 1981-1983
• Covers Broad Spectrum Civil Rights movement,
  Anti-War movement and Conservative Movements
• Issues were aggregated into Domains i.e.
  Womens Reproductive Rights, Anti-War, Human
  Rights, etc.

36
Method

• Groups linked to Issues in two mode matrix (GI
  Matrix)
• Matrix Multiplication (GI Matrix and its
  Transpose) used to create a Group to Group Matrix
  (GG Matrix)
• Resulting GG Matrix often to sparse (particularly
  in later time periods) so they aggregated into
  movements

37

• GG Matrix examined with CONCOR
• For Image Matrix relatively high threshold,
  densities that were twice the expected were coded
  1 (more stringent than simply higher than chance)

38
1961-1963
39
1966-1968
40
1971-1973
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1976-1978
42
1981-1983
43
Centrality and Repertoire

• Examining each time period Bearman Everett
  attempt to show the impact of Centrality on the
  role groups play in influencing repertoires of
  protest
• Central Blocks tend to represent the Dominant
  Repertoire in each time period
• Groups on the Periphery tend to experiment and
  use non dominant repertoires
• General trend (1961-1983) toward marches and
  rallies, and away from sit-ins and pickets

44
Repertoires over Time

• Observed use of tactic in comparison to expected
  distribution
• i.e. block 1 in 1961-1963 was 5.5 times more
  likely to use a sit in than expected

45
New Social Movement Theory Assertions

• Increase in Identity focus (New Social Movements)
  vis a vis Interest focus (Old Social Movements)
• Decline in the Importance of Organized Labor
• Neither seem supported by the data at left

46
The Japanese Corporate Network A Block Model
Analysis

• By Michael Gerlach

47
The Japanese Corporate Environment

• Due to Japanese business success, theres been
  much study of those characteristics seen as
  unique to the Japanese Corporate environment
• Highly integrated corporate environment, both
  within but also across sectors

48
Unique Inter-corporate Bonds

• Keiretsu Highly visible clique like patterns
  based on inter-corporate alliances
• Six Major Keiretsu Mitsui, Mitsubishi, Fujo,
  Sanwa, Sumitomo, and Dai-Ichi Kangyo
• Zaibatsu Prewar Family owned holding companies
  that dominated much of Japanese Corporate
  environment

49
Questions that arise

• Do nominal structures (Keiretsu and/or Zaibatsu)
  relate to traditional network measures or
  structures?
• Do these groups constitute the main element of
  structure in the Japanese Corporate Network?
• Is the role of Banks and Financial Institutions
  the same as in the US network or unique to Japan?
• Do Nominal Structures (Keir. And Zaib.) or links
  to Financial Institutions explain more of the
  network structure?

50
Approach to Analysis

• Network is partitioned into blocks based on
  common patterns of relationships
• - inter-corporate relationships
• - financial centrality
• - Industrial interdependence

51
Keiretsu vs. Zaibatsu
52
Financial Centrality
53
Ties Examined

• Bank Borrowing Important, banks provide half of
  all capital for Japanese corporations (double US
  ratio)
• Equity Shareholding Tends to be viewed as an
  expression of business relationships in Japan
• Board of Directors Crossover Less common in
  Japan than in US (sparse network), but where it
  occurs tends to imply business relations

54
Steps of Analysis

• Three 60x60 matrices were analyzed to spatially
  map the network
• CONCOR partitioning was used to create blocks
• Ties between and within blocks were analyzed for
  density
• Blocks were compared to hypotheses to see
  whether they reflected competitive firms within
  sectors, firms in vertically interdependent
  sectors or keiretsus

55
Spatial Mapping
56

• Industrial blocks seem to correlate by Keiretsu,
  where as financial blocks do not seem to

57
Resulting Density Matrices
58
Equity Shareholding

• Amounts to shared ownership
• Most common for Financial sector to own shares in
  industrial and vice-versa
• Industrial Sector firms do not seem to share
  equity

59
Shared Directorships

• Relatively uncommon in all sectors, however most
  common between financial sector and industrial
  sector (top right)

60
Combined Directorship/Equity Matrix

• Done using Boolean Addition of the Matrices

61
Equivalence of Competitors
62
Equivalence of Vertically Integrated Firms
63
Equivalence amongst Keiretsus
64
Conclusions

• Financial Institutions hold similar network
  positions to each other, but very different from
  industrial members
• The key to this differentiation is the central
  role played by financial institutions
  (illustrated in the centrality of Financial
  Institutions in Equity and Directorship ties)
• These ties tend to go from financial institutions
  to industrial corporations rather than to other
  institutions
• Structurally Equivalent positions are held by
  firms in diverse industries
• Block composition appeared to mirror historical
  affiliations (zaibatsu and keiretsu)