Borgatti, S'P'

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Borgatti, S.P. Everett, M.G. (1992). Notion of
position in social network analysis.Sociological
Methodology, 22, 1-35.

• Position is a central concept in social network
  analysis and structural theory
• Position plays a role in studying world systems,
  power in exchange networks, and social
  homogeneity

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Borgatti, S.P. Everett, M.G. (1992). Notion of
position in social network analysis.Sociological
Methodology, 22, 1-35.

• So what is the problem?
• Problem position is actually more than one
  concept. Borgatti and Everett critical of
  published works that define position in one way
  but draw conclusions as if it was defined in a
  different way.
• This paper sets to straighten out the
  conceptual definitions of position and provide
  examples as to when each definition is
  appropriate a corrective paper

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Positional Equivalences

• Actors (nodes) who are connected in the same way
  to the rest of the network are said to be
  equivalent and to occupy the same position
• The objective of positional analyses is to
  partition actors into mutually exclusive classes
  of equivalent actors who have similar relational
  patterns
• This contrasts a relational or cohesive approach.
  In the 1st case the clustering principle is
  similarity, in the 2nd it is proximity

4
A lt 1 2 3 4 5 5 4 3 2 1 gtB lt 0 1 2 3 4 4 3 2
1 0 gt

• What is to B as 4 is to A? What plays the role
  in B that 4 plays in A?
• Problem there are at least 2 different ways of
  interpreting connected in the same way to the
  rest of the network
• Answer literal, concrete is 4 while natural,
  more analogical response is 3

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• Difference between structural equivalence and
  automorphic equivalence (structural isomorphism)
• Binary relations are equal or identical if they
  contain the same ordered pairs, but they are
  isomorphic if there is a one-to-one
  correspondence between their multiplication
  tables
• Conceptually
• Equal triangles, Similar triangles
• sides same length proportional to each other
• Structural equivalence refers to a specific
  definition of position, NOT to the general
  principle of structural similarity

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Definitions and Notation

• The position of a node is a categorical attribute
  of that node, which can be thought of as its
  color or flavor
• P(a) is used to denote position of node a in a
  network
• A structural or graph-theoretic attribute is any
  attribute of a node or graph that makes no
  reference to the names or labels of the nodes in
  the graph
• The property of being no more than 3 links
  distant from any node is a structural attribute
  of a node
• BUT the property of being 3 links distant from
  Mary is not a structural attribute

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More on structural equivalence and
substitutability (Lorrain and White)
8

• More modern and simpler concept (Burt) is defined
  as a set of structurally equivalent nodes as a
  set of nodes connected by the same relations to
  exactly the same people. An actors position is
  defined by who he/she is connected to
• In nonvalued graphs, this definition says that 2
  actors in a network occupy the same position if
  and only if they have perfectly overlapping
  neighborhoods
• Implies identical ego networks, not only same
  individual actors, but same relationships

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Structural Equivalence formula
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Bill and Joe are structurally equivalent actors.
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• Structurally equivalent actors connected to
  exactly the same nodes are all identical with
  respect to structural variables such as
  centrality, eccentricity, degree, prestige, etc.
  
• Any graph-theoretic statement that can be said
  about one actor applies to the other

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• However, indistinguishable actors on
  graph-theoretic attributes are not necessarily
  structurally equivalent
• a, h are identical (graph-theoretic variables)
  but not structurally equivalent (different
  neighborhoods N)
• Implication structural equivalence is local it
  only matters who they are directly connected to,
  so we can compute on incomplete data sets.
• This differs from global variables like
  betweenness centrality
• Structural equivalence is then the collection of
  ego-network data. It is related to
  cohesion/proximity measures. For undirected
  graphs, structurally equivalent actors form
  cohesive subsets.

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• Borgatti and Everett note that much of the
  literature has generally regarded structurally
  equivalent actors as different than cohesive
  subsets, although they are not
• Structural equivalence is about location -
  proximity and similarity

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• Relations among coworkers in a formal
  organization. Each might be separate divisions
  or units
• a, g, m are similar in patterns of connection,
  but they are not all proximate
• a, b, c, d, e, f, are relatively proximate, but
  they are not similar
• d, e are structurally
• equivalent are both
• proximate similar

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• KEY
• CONCEPTUAL
• DIFFERENCE
• FROM
• STRUCTURAL EQUIVALENCE

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AUTOMORPHIC EQUIVALENCE STRUCTURAL ISOMORPHISM
(interchangeable terms)Another main concept of
position in network analysis

• Isomorphism a one to one mapping of one set of
  objects to another such that the relationships
  among the objects are also preserved
• The only possible differences between isomorphic
  graphs are the labels of the nodes and edges (if
  any)

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• Isomorphism a one to one mapping of one set of
  objects to another such that the relationships
  among the objects are also preserved
• The only possible differences between isomorphic
  graphs are the labels of the nodes and edges (if
  any)

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Why?

• Because we can map

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Isomorphic graphs

• Identical with respect to all graph theoretic
  attributes
• If a graph has 12 cliques of size 3 and 10
  cliques of size 4,
• Then all graphs isomorphic have the same
• The only possible difference between isomorphic
  graphs are the labels of nodes and edges (if any)
• If not for this labeling, graphs and their
  isomorphs would be indistinguishable
• All graphs then by definition are isomorphs with
  themselves

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Pick these graphs off the paper, shuffle and put
them back, you couldnt tell which was which
without labels
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But G, H differ from K
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Two actors occupy the same position if
isomorphic, Sets of isomorphic actors are called
orbits
Orbits (same position) a, c, h, j b, d, g,
i e, f
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Structurally equivalent (same relationship) a,
c b, d g, i h, j
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Summary

• Structural isomorphism is independent of
  proximity
• Isomorphic nodes may be adjacent, distant, or
  completely unreachable
• Structural equivalence and structural isomorphism
  are fundamentally different approaches to notion
  of position

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• Review Structural Equivalence
• Position seen literally as location in a labeled
  graph
• Identifies who an actor is connected to
• Network or labeled graph representative of
  underlying structure
• Structurally equivalent actors are both proximate
  and similar
• Review Structural Isomorphism
• Position seen as a location on an unlabeled graph
• Identifies the way in which nodes are connected
  to others
• Structurally isomorphic actors are only similar

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So, with definitions learned, which approach is
best for which application?

• Borgatti and Everett state that the choice
  between structural equivalence or structural
  isomorphism as measures of position depends on
  which of the labeled or unlabeled representations
  best correspond to ones intuitive concept of
  what structure means
• Borgatti and Everett critical of literature that
  confounded using structural equivalence for data
  but interpreted results as if structural
  isomorphism had been used instead
• Status/Role Systems
• Power in Experimental Exchange Networks
• Social Homogeneity

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Status/Role systems Example, the gives
information about medicine relation

• Nadel (1957) Merton (1959) Linton (1936) have all
  discussed social structure in terms of a pattern
  or network or system of relationships this
  emphasizes that what defines a role, such as
  nurse or doctor is precisely the
  characteristic set of relationships that actors
  who are nurses have with actors who are doctors,
  patients, suppliers, etc
• Society is a network of relationships among
  individuals and social structure is an underlying
  network of relationships among roles or positions
• If we are interested
• in modeling social roles
• in the sense of Nadel (1957)
• Merton (1959) Linton (1936),
• we must choose structural
• isomorphism over
• structural equivalence

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• Problem Researchers (like Caldeira 1988)
  operationalizing social roles and status have
  employed structural equivalence.
• Evan 1966, Burt 1979 economic systems as role
  systems
• Galaskiewicz and Krohn, Burt 1979 economic
  sectors, made up of structurally equivalent
  firms, leading to industries in the economy
• Criticism structurally equivalent firms cannot
  constitute sectors. Why? They may purchase from
  similar suppliers and sell to similar clients,
  but not to the same clients.
• Another example (Snyder and Kick 1979) looked at
  role theory and structural equivalence to define
  nations positions in the world economy
• Criticism two nations may occupy similar
  positions (as core nations) and have similar
  relations with peripheral nations but not the
  same relations

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Power in Experimental Exchange Networks

• Cook et al, and Markovsky et al. described power
  as structural attributes. In their experiments
  it doesnt matter who actors are connected to,
  but how they are connected
• Structural isomorphism is the right concept for
  modeling power in experimental exchange networks
  equality of power is achieved structurally and is
  unrelated to proximity
• In contrast, structural isomorphism is the wrong
  concept for modeling an infectious disease
  process, or rumor gossip. In these examples
  processes are not structural but depend on
  proximity

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Power in Experimental Exchange Networks
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Social Homogeneity

• Coleman, Katz, Menzel (1957) used a network
  approach to explain adoption of a new drug by
  physicians. Is probability increased that
  adoption of a drug will be increased if the
  physician knows another physician who has already
  adopted the drug. This is a cohesion approach
  beyond general drug knowledge from published
  studies, drug manufacturers, etc.
• Critique structural equivalence is not
  different than cohesion, but it was explored this
  way.
• You have to know what you are looking for, a
  relational approach as a predictor of
  transmission, or a positional approach. You can
  not use structural equivalence if interested in
  non cohesive or noninfectious transmissions
• But, if both structural similarity and cohesion
  are important for transmission, structural
  equivalence could be useful

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DiMaggio, Paul. (1986). Structural analysis of
organizational fields A blockmodel approach.
Research in Organizational Behavior, 8, 335-70.

• This is a paper on organizational studies
• Notes analytic shift from study of environments
  surrounding organizations to the study of
  organizational fields
• This study of organizational fields requires a
  technique for partitioning interorganizational
  networks into subgroups
• Approaches to partitioning are reviewed
• Blockmodeling or Structural Equivalence Analysis
  is recommended

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What is the best means of understanding complex
systems of organizations?

• The use of organizational fields
• (rather than environments)
• Organizational fields are networks, populations,
  sectors or domains

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• This articles purpose is to describe an analytic
  instrument useful for characterizing the
  structure of organizational fields
  Blockmodeling
• DiMaggio phrases blockmodeling as a means of
  analyzing relational data through the
  identification of structurally equivalent
  positions (the positions of sets of actors within
  populations of human actors)
• This includes analyzing data of directed ties
  between pairs of actors in a population

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• 3 sections
• Discussion of circumstances requiring
  partitioning organizational fields into subgroups
• Introduction of blockmodel or structural
  equivalence analysis
• Utility of using the blockmodel approach

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Fields vs. Environments

• Environmental studies look at a focal
  organization and its operating environment
• Field studies refer to sets of interacting
  organizations that together constitute an
  institutionalized domain or a sector of
  substantive analytic interest
• Examples of an institutionalized domain are
  health care, higher education, the aircraft
  manufacturing industry
• Examples of a sector interest might be all
  volunteer organizations in Indianapolis or
  health-related agencies in Philadelphia

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• Advantages to studying fields is, its less useful
  to learn about a particular organizations
  turbulent environment than to study the
  organizational sources of such turbulence
• Also the position that an organization occupies
  in its field is key to understanding its relative
  turbulence or stability
• Analysis at the field level permits the
  examination of interorganizational structure on
  field-level variables
• Scott and Meyer (1983) noted a trend of societal
  sectorialization where key actors in different
  functional areas (health, communications,
  education ,etc) are national systems of
  organization coordinated through vertical and
  horizontal network ties. So this is big stuff

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Approaches to Partitioning Organizational
Populations

• Without partitioning this matrix, we can
  characterize density of ties, distribution,
  dyadic or triadic relations, centrality,
  reachability, path distance, etc
• BUT we need partitioning into subpopulations for
  some applications
• Example Assess centrality vs. organizational
  effectiveness

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• Centrality is an important predictor of
  effectiveness, but this is not readily apparent
• Compare centrality of clique 1 and clique 2,
  similar scores
• But, clique 1 (defense industry) effective at
  lobbying appropriations
• Clique 2 (social equity causes) not effective at
  lobbying appropriations
• So we mistakenly concluded centrality of clique
  and lobbying effectiveness not related
• But if we split the cliques into subgroups, we
  see that in fact, centrality does make a
  difference, centrality is an important predictor
  of effectiveness, but has reverse effects for
  different groups

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Note here the importance of structural
equivalence - who actors are connected too,
makes a difference. (Defense industry vs. social
equity causes) The graphs are isomorphic, but
there is a big difference in lobbying
effectiveness for getting appropriations
DiMaggio Powell (1983), Baty, Evan, Rothermel
(1971) Boorman Levitt (1983) show partitions of
fields into structural positions are important
for studying flows of innovation, personnel, or
information. These can be actor-specific, so
structural equivalence can be useful for studying
transmission
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Methods for partitioning organizations into
subsets

• Naturalistic approaches uses descriptors,
  categorical definitions, which can miss aspects
  of structure central to analysis (ex. Economic
  studies between firms in different areas,
  manufacturing, finance, etc
• Classification on the basis of attributes
  Relational approaches may be advantageous over
  categorical approaches. Relational approaches
  assumes social relations delineate roles and
  define positions in a group. Categorical
  approaches assumes differences in attributes
  affect social relations

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Methods for partitioning organizations into
subsets

• Partition on the basis of structural cohesion
  uses the presence or absence of ties between
  pairs to partition the population into sets of
  actors that interact maximally or minimally with
  other members. This includes clique
  measurements, etc. Superior to naturalistic or
  attribute based classifications
• Partition on the basis of structural equivalence
  confusion here, does DiMaggio blend
  isomorphism with structural equivalence? He
  states partitioning a population into
  structurally equivalent positions is to divide
  the organization into subsets (blocks) so that
  they share similar relations with organizations
  in other blocks whether or not they are connected
  to one another. (Note I thought by definition
  structural equivalence would mean actors would
  have to have the same relationships, being both
  proximate and similar?? Also with structural
  isomorphism nodes can be completely unreachable,
  but doesnt structural equivalence require a
  connection?)

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• Structural-equivalence approaches are more likely
  than cohesion approaches to identify non-cliquing
  patterns
• A clique is ordinarily a special case of a
  structurally equivalent position. (Yet Borgatti
  stresses cohesion and structural equivalence are
  strongly related)
• Cohesion analysis is consistent with exchange
  theory, while structural equivalence is akin to
  role theory. Structural equivalence premise is
  that organizations with similar patterns of
  relations to other organizations will become more
  similar even if they do not act with one another
  directly
• Again, are we talking about structural
  equivalence or structural isomorphism? Is role
  defined by positional analysis, or to whom an
  actor is related?

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Summary criteria for mapping organizational
fields (sectors)

• Structural based on patterns of ties (not
  attributes)
• Capable of partitions yield identifiable
  subgroups
• Sensitive to cohesion
• Sensitive to structural equivalence- similar
  positions by virtue of relations with other
  organizations
• Capable of identifying domination (i.e.
  unreciprocated positive ties)
• Open-ended in definition of field (i.e. does not
  use categorical approaches) Note this is a
  boundary specification issue, how you define the
  field or sector boundaries of organizations
• Capable of basing partitions on different kinds
  of relations.
• The only theory that meets all criteria above is
  Blockmodel Analyses

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Blockmodel Analyses of Organizational Fields

• Author states this is the first study/application
  of blockmodeling to the analysis of a nationwide
  organizational field in a single institutional
  sector
• Definition, so far we still havent defined
  blockmodeling
• Blockmodeling is a model based on an asymmetric
  matrices in which cell entries represent ties
  sent from row-actors to column-actors, permitting
  the analyst to group members of a population into
  subsets or blocks on the basis of the relations
  among them.
• Blockmodels operationalize the notion of
  structural equivalence by grouping together
  actors who sit in the same position in a larger
  social structure. Actors allocated to the same
  blocks are those with similar relationships to
  the same other actors

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Illustration the Resident-Theatre field

• Survey of chief operating officers at 165 member
  Theater Communications Group (TCG) theaters,
  asking survey respondents about advice,
  associations, and admiration.
• The choice of measures of intraorganizational
  ties is crucial, as only the top managements of
  organizations are often involved in
  organizational fields (Knokle and Laumann 1982)
• Data on which managers sought for advice, chosen
  for interaction, and admired were modeled based
  on structural equivalence, on the basis of their
  received ties from other actors in the system

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Why was Blockmodeling useful in this study?

• The field was partitioned into several blocks of
  organizations and their managers. This
  identified elites the managers of theaters
  which are connected by ties of advice,
  association, and affect
• Relations among structurally equivalent sets of
  organizations identified, and the structure of
  the field shown to be largely hierarchical
• Partitioning on the basis of structural
  equivalence provides information about phenomena
  that might otherwise be inexplicable, (for
  example why some small theaters did well in
  getting NEA government grants and some large
  established theaters did poorly). No obvious
  division based on attributes only would have
  predicted the allocation of the different
  theaters to blocks in the way they were
• Core and periphery notions were operationalized
  through blockmodeling

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Other substantive applications of Blockmodels to
the study of organizational fields

• Blockmodeling can be a valuable supplement to
  qualitative approaches to the study of
  organizational fields where the researcher cannot
  possibly devote as much study to the field as can
  be applied to an individual organization, thus
  Block model network analysis can be a
  macroscope
• Intuitive judgment and inherent observer bias can
  be offset by feedback provided by data from all
  organizations in the field (again I make note of
  boundary specification issues)
• Blockmodeling can be helpful in the
  identification of niches and forms

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• Niche (Aldrich 1979) distinct combinations of
  resources and other constraints that suffice to
  support an organization form
• (Hannan Freeman 1977) A niche is any set of
  resources that supports a population
• A form is any set of organizations that occupy
  the same niche
• No two populations can continuously occupy the
  same niche

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• The power of the population-ecology perspective
  may be enhanced by defining relations among sets
  of actors
• Consumer market of 6 organizations (A-F) in the
  same industry (sellers) (Columns)
• 10 consumers (M-V) (Rows). Consumers
  differentiate forms as all organizations are
  identically constrained (via regulatory means,
  etc)

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• Blockmodeling the columns on row choices and the
  rows on column choices yields 2 consumer blocks
  and 2 blocks of sellers
• 2 niches and 2 forms have been identified via
  blockmodeling, which is an advantage of a
  structural equivalence approach to niche and form
  identification over attribute-based approaches

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Operationalizing intraorganizational diversity in
studies of interorganizational fields

• Shift from viewing organizations as unified
  goal-directed actors to the concept of
  organizations as tenuous coalitions that many
  participants (both internal and external) seek to
  use for their own purpose (Cyert March 1963)
• Organizations consist of individuals and subunits
  with quite different agendas and objectives is
  difficult to capture in an interorganizational
  framework
• There is inherent duality in an individuals
  participation in an interorganizational system as
  a representative of the organization and in
  pursuit of self-interests
• For example, a theater managing director (from
  previously mentioned study) may participate in a
  national trade organization to advance specific
  career opportunities, but is likely to promote
  his or her own organization in the process
• This is related to the Blockmodeling approach,
  because blockmodeling has the ability to analyze
  simultaneously multiple patterns of relations,
  (i.e. not only organization to organization, but
  ties of subunits or staff to staff in other
  organizations)

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Wasserman and Faust, chapters 9, 10.

• Introduces the theoretical background for
  studying social network roles and positions
• Many methods for the description of network
  structural properties are concerned with dual
  notions of social position and social role
• Position - refers to a collection of individuals
  who are similarly embedded in networks of
  relations
• Role refers to the patterns of relations which
  obtain between actors or between positions

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• Position differs from cohesive subgroups in that
  while position is based on similarity of ties
  among subsets of actors, actors occupying the
  same position need not be in direct or even
  indirect contact with one another
• Role focuses on associations among relations, as
  opposed to properties of actors (ex. Kinship)
• Role can be modeled at the level of
• Actor
• Subsets of actors
• Network level

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• Conceptual Definitions
• Lorrain and White role becomes identified with
  sets of relations
• White, Boorman, Breiger Each of the sets into
  which a population is partitioned is a position
• Burt a position in a network is the specified
  set of relations to and from each actor in a
  system

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Left /right side maps out actors into equivalence
classes Top/bottom horizontals will be role
analyses This chapter focuses on methods for
locating subsets of equivalent actors
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9.2 Definition of Structural Equivalence

• Two actors are structurally equivalent if they
  have identical ties to and from all other actors
  in the network
• Structurally equivalent actors have identical
  ties to and from identical actors, on all R
  relations
• i j denotes structural equivalence of actors i
  and j
• Bk denotes position. We let B be the number of
  positions in the network, and use the notation
  F(i) Bk to denote the assignment of actor i to
  position Bk
• If actors i and j are structurally equivalent, i
  j, then they are assigned to the same position
  thus
• if i j then F(i) F(j) Bk

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SociomatricesIf two actors are structurally
equivalent then their respective rows and columns
in the sociomatrix will be identical
Actors 1,2 and 3,4 are structurally equivalent
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• In defining structural equivalence consider
  different kinds of relations
• Dichotomous or valued
• Directional or nondirectional
• A relation in which self-ties (diagonals on the
  sociomatrix) are substantively meaningful

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• Multiple Relations
• For 2 actors to be structurally equivalent in a
  multirelational network, they must have identical
  ties to and from all other actors on all
  relations
• Valued Relations
• For 2 actors to be structurally equivalent on a
  valued relation they must have ties with
  identical values to and from identical other
  actors
• In dichotomous relations, with a tie either
  present or absent, structural equivalence is easy
  to determine
• Values relations are less clear cut

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• Self-ties and Graph Equivalence
• If self ties are substantially meaningful,
  diagonal entries in the sociomatrix should be
  included in calculating structural equivalence
  (ex. Memos sent within and between departments
  would have a lot of important reflexive ties)
• If self ties are meaningless, then ignore (ex.
  Communications among friends, you probable dont
  communicate with yourself)
• Graph equivalence, if i?i and j?j, then i?j and
  j?i
• Graph equivalence is more restrictive than
  structural equivalence and contains reflexive
  ties

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9.3.1 Simplification of Multirelational Networks

• Positional analysis allows the simplification of
  a network data set
• Simplified sociomatrices are called an image
  matrix.
• In an image matrix rows and columns refer to
  positions, rather than individual actors
• A blockmodel is an image matrix or a collection
  of image matrices along with a description of
  which actors are assigned to which positions

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Sociomatrix
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Permuted and Partitioned Sociomatrix
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Image Matrix
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9.3.2 Tasks in a Positional Analysis and the4
steps of complete positional analysis

• A formal definition of equivalence
• A measure of the degree to which subsets of
  actors approach that definition in a given set of
  network data
• A representation of equivalences
• An assessment of the adequacy of the
  representation

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9.4 Measuring Structural Equivalence

• Structural Equivalence is a mathematical
  property, which under its strict definition, is
  seldom realized in network analysis
• So we use measures to identify subsets of actors
  that are approximately structurally equivalent

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• 9.4.1 Euclidean Distance as a Measure of
  Structural Equivalence
• The distance measure of structural equivalence
  for actors i and j is the Euclidean distance
  between the ties to and from these actors
• This is the distance between rows i and j and
  columns i and j of the sociomatrix

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• What this means is that for two actors that are
  not structurally equivalent, the Euclidean
  distance will be large.
• For two actors i and j that are structurally
  equivalent, the Euclidean distance will be
  exactly zero
• Across multiple relations, when there are R
  relations, and Xikr is the value of the tie from
  actor i to k on relation R, then

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• Example Krackhardts high-tech managers
• Notice that no pairs of actors are structurally
  equivalent none of the off-diagonal distances
  0
• See page 24 in notes hand out (UCINET)

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9.4.2 Correlation as a Measure of Structural
Equivalence

• If two actors are structurally equivalent, then
  the correlation between their respective rows and
  columns of the sociomatrix will be equal to 1
• Using correlation is similar to using Euclidean
  distance
• Calculate the average values of row i and column
  i, where the calculation excludes diagonal
  elements
• This is the Pearson product-moment correlation
  coefficient
• Other measures can be a simple match coefficient
  that counts the number or proportion of ties that
  are identical between actors

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• How do you decide which measure of structural
  equivalence to use?
• If you want to measure similarity of patterns
  of ties, then correlation coefficient is
  preferred
• If you want a measure of the identity of ties,
  Euclidean distance is preferable

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• The problem of measuring degree of structural
  equivalence is the problem of measuring the
  similarity or dissimilarity of the ties to and
  from pairs of actors
• Euclidean distance reflects a smaller amount of
  structural equivalence than does a correlation
  coefficient, if the actors differ in the mean and
  variance of their ties
• Differences between Euclidean distance and
  correlation are especially acute when relations
  are valued

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• We can express the relationship between Euclidean
  distance, dij, and correlation, rij
• The means of the values in rows i and j are x?i
  and x?j
• The variances of the rows i and j are s2i and
  s2j
• So
• d2ij (g-2) (x?i -x?j )2 s2i s2j -
  2rij si sj

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9.5 Representation of Network Position

• The 3rd step in a positional analysis includes
  representation of positions and a statement of
  how the positions are related to each other
• Use of partitioning actors into subsets
• Use of hierarchical clustering
• Blockmodeling

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9.5.1 Partitioning Actors

• Partitioning actors based on structural
  equivalence can be done via CONCOR
• CONvergence of iterated CORrelations)
• Repeated calculations of correlations between
  rows or columns will eventually result in a
  boiled down matrix of
• CONCOR is a divisive hierarchical clustering
  method

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• The first application divides the actors into 2
  groups, with further applications getting into
  finer splits, and so on
• A dendrogram displays the results of a CONCOR
  partition indicating the degree of structural
  equivalence and identifying members
• Actors connected by branches low in the diagram
  are closer to being perfectly structurally
  equivalent,
• Actors who are joined only through paths high up
  the diagram are less structurally equivalent
• See WF p. 379

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• Problems with CONCOR
• Bi-partition, or a binary tree is a product of
  the procedure, not the network (i.e. splitting
  into two subset partitions each time)
• The resulting CONCOR partitions often have little
  resemblance to understood social position (?)

82
Partitioning Actors Using Hierarchical
ClusteringThe grouping of actors into subsets
that are relatively similar

• A positional analysis using hierarchical
  clustering would use a matrix with measures of
  structural equivalence
• Correlation matrix C1
• Matrix of Euclidean distances D
• Beware of errors in both CONCOR and hierarchical
  clustering. With the first partition, as actors
  are arranged in subsets, the initially separated
  actors remain there for each subsequent
  partition. So if two actors are split into
  different partitions early on, they cannot be
  undone and so the error can be compounded with
  further clustering throughout the analysis if
  initially partitioned incorrectly

83
Spatial Representations of Actor Equivalences

• Hierarchical clustering and CONCOR are discrete,
  in that they partition actors into mutually
  exclusive and exhaustive subsets
• An alternative model is a continuous or spatial
  model, like Multidimensional scaling (MDS)
• MDS seeks to represent similarities (or
  dissimilarities) among a set of entities in
  low-dimensional space, the more similar the
  entity, the closer in space

. .. . . .. .. . ... .. . . . .
. . ..
84
9.5.3 Ties Between and Within Positions

• There are 2 parts to represent positions in a
  network
• Assigning actors to positions
• Describing how positions relate to each other
• There are 3 common ways to represent the ties
  between positions
• Density table
• Image matrix
• Reduced graph
• 1st step for any analysis is to permute the rows
  and columns so actors assigned to the same
  position are adjacent in the sociomatrix
• (See WF p. 389)

85

• Density Tables
• A matrix that has positions rather than actors as
  its rows and columns
• The values in the matrix become the proportion of
  ties that are present from the actors in the row
  to the column
• p.389
• B1 to B2 There are 15 out of 24 possible ties
  present (6 x 4) so the density of that cell
  .15/24 .625

86

• Density table example
• Note B4 has all possible ties (p.389)

87
Image Matrices

• Uses a density rule to code the presence or
  absence of ties
• This rule specifies a tie as present if the
  density of ties from actors in one position to
  actors in another position is gt to density of the
  matrix as a whole
• Notation, Let ? be the density of ties

88
Reduced GraphsFurther reduces the image matrix
via density values in graphic formFrom
this. To this

• B3 B4
• B1 B2

89
WF Ch. 9 conclusions

• Structural Equivalence requires that equivalent
  actors have identical ties to and from identical
  others
• Structural Equivalence can only be met by actors
  who belong to the same population
• So comparisons across populations are precluded
• Actors who are structurally equivalent must be
  close to each other in a graph theoretic sense

90
WF Chapter 10 Blockmodels

• 10.1 Definition
• On a one-mode network with a set of dichotomous
  relations R, a Blockmodel consists of
• A partition of actors in the network into
  discrete subsets called positions
• For each pair of positions a statement of the
  presence or absence of a tie within or between
  the positions on each of the relations
• F(i) Bk
• If actor i is in position Bk
• bklr indicates the presence or absence of a tie
  from position Bk to position Bl on relation Xr,
  where bklr 1 if there is a tie from Bk to
  position B1, and bklr 0 if not

91

• A blockmodel has 2 components
• The mapping F that describes the assignment of
  actors to positions
• The Matrix B that specifies the presence or
  absence of ties
• The entries in the B x B x R matrix B is called a
  block
• A block containing a 1 is a oneblock, or a bond
  (presence)
• A block containing a 0 is a zeroblock (absence)

92
10.2 Building Blocks

• To code whether a block should be coded as a
  zeroblock or a oneblock use the following
  criteria
• Perfect fit (fat fit)
• Zeroblock (lean fit)
• Oneblock
• Density Criteria
• Maximum value - for valued data
• Mean value - for valued data

93

• Perfect Fit
• Ideal all actors in all positions will be
  structurally equivalent
• bklr 0 if Xklr 0, for all i ? Bk, j ? Bl,
  and
• bklr 1 if Xklr 1, for all i ? Bk, j ? Bl

94

• Zeroblock
• Zeroblock criterion states that the tie between
  two positions on a given relation is zero only if
  there are no ties from actors in the row position
  to the column position, otherwise it is a
  oneblock
• Quite rare
• Oneblocks would be the opposite

95

• Density Criteria
• Real world model since it is unlikely all blocks
  will be perfect oneblocks or zeroblocks, we can
  use a density of ties value to code as a oneblock
  or zeroblock
• bklr 0 if ?klr lt a
• bklr 1 if ?klr gt a

96
To review, steps for blockmodeling are

• Partition actors into structurally equivalent
  subsets (approximately)
• Describe the ties between and within positions
  via density (ratio of ties present to total
  that could be present)
• Ex. p. 402 Krackhardts high-tech managers,
  density is 0.452, so any submatrix with a
  density gt 0.452 will be coded as a oneblock,
  otherwise code as a zero block
• Create image matrix, reducing the density table
  to 1s and 0s

97
Another Example The Countries Trade Network

• 5 dichotomous relations
• 4 trade relations, imports by the column from the
  row country
• 1 diplomatic relation, indicating the row country
  has an embassy in the column country
• 3 relations are blockmodeled
• Manufactured goods
• Raw materials
• Diplomatic ties
• Steps
• Structural equivalence calculated using
  correlation coefficients
• Positions identified via hierarchical clustering
• Density tables and image matricies constructed
• See p.405
• Some values are 1, and some 0
• Oneblock or zeroblock criterion could be applied
• BUT this would result in a sparse model, so a
  better decision is to use density criteria ?

98
10.2.7 Valued Relations

• For a valued relation each submatrix of a
  sociomatrix corresponding to ties within or
  between positions may contain a range of values
• The maximum value criterion for valued relations
  is analogous to the zeroblock criteria for
  dichotomous relations (i.e. blocks that contain
  only small values are defined as zeroblocks the
  maximum sets the ceiling)
• Blocks that contain any large values are then
  defined as oneblocks
• If we define as the highest acceptable value for
  a zeroblock, then
• bklr 0 if Xklr lt ?, for all i ? Bk, j ?
  Bl, and
• bklr 1 otherwise

99
Mean value criterion

• Another possible criterion is the mean value
• If the mean value of a submatrix gt the mean
  value of the relation, then the block is defined
  as a oneblock, otherwise it is a zeroblock
• bklr 0 if Xklr lt Xr
• bklr 1 if klr gtXr

100

• Caution!
• If the relation is measured on a continuum where
  the small values indicate a substantively
  negative tie, while the large values indicate a
  substantively positive tie, the mean value Xklr
  could be a combination of both and ties
• In that case, use the max value criteria

101

• Interpretations of blockmodels
• Validation of a blockmodel using actor attributes
• Descriptions of individual positions
• Descriptions of the overall blockmodel
• Each has its limitations

102

• Attribute examples in countries trade network
  include
• Population annual growth rate
• GNP per capita annual growth rate
• Secondary school enrollment ratios

103

• Describing individual positions
• Digraph concepts applied to position
• Isolates (neither indegree nor outdegree)
• Transmitters (nodes with only outdegree)
• Receivers (nodes with only indegree)
• Carriers (nodes with both indegree and outdegree)
• Burt (1976) looked at if ties occur primarily
  within a position and whether they are directed
  to members of the position from others
• Isolate positions - neither give many ties nor
  direct many ties to other positions
• Sycophants - have more ties to members than to
  themselves, and do not receive many ties
• Brokers both receive ties both from members of
  other positions and from its own members
• Marsden (1989) extended Burts typology by
  distinguishing between the ties made by a
  position, the level of ties received by a
  position, and the positions ingroup preference

104
Caution!

• Indegree, outdegree, and within position ties
  give a useful description of positions that rely
  on ties to other positions. BUT a blockmodel is
  likely to have several relations, and so the
  label of a position may not be the same across
  substantively differing relations
• Example Krackhardts managers
• Relations of advice and friendship, no isolates
  or sycophants
• All three positions, (B1, B2, B3) are primary on
  the friendship position
• Intermediate levels in a hierarchy could be
  indistinguishable (since they would have both
  indegree and outdegree)
• Labels for kinds of positions therefore capture
  only limited information

105
Descriptions of the overall blockmodel

• Image matricies for 2-position Blockmodels
  represent theoretical properties (see WF p. 421)
• Image matricies with more than 2 positions get
  more complicated. Ideal images include
• Properties of cohesive subgroups consists
  primarily of intraposition ties
• Center-periphery structures core position that
  is internally cohesive, with one or more other
  positions having ties to the core position, but
  not to each other
• Centralized systems similar to core periphery.
  In centralized systems all ties are pointed
  toward or away from a single position
• Hierarchical systems unreciprocated ties are
  directed from each position to the position
  immediately above it (ex. chains of command)
• Transitive systems a transitive image could
  indicate dominance or deference between
  positions. In a fully transitive model, rows and
  columns of the image matrix can be permuted so
  that all the oneblocks are in either the lower
  left triangle or upper right

106
Cohesive subgroups
107
Center-periphery
108
Centralized
109
Hierarchy
110
Transitivity
111

• The End