PAD 637: Social Network Analysis

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PAD 637 Social Network Analysis

• Week 10
• Structural Equivalence and Blockmodeling

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Wasserman and FaustChapter 9 Structural
Equivalence

• Chapter describes the meaning of position and
  role in Social Network Analysis
• Position collection of actors similarly embedded
  in networks of relations
• Role patterns of relations between actors or
  positions, modeled on the level of actors,
  subsets of actors, or the whole network
• Position is not cohesion, or subgroup. Most
  informative role/position analyses are
  multirelational

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Position and Role Analysis

• Role/Position analysis can alternate in order of
  performance
• Position first, then role requires grouping
  actors and then describing associations among
  relations
• Role first, then position requires describing
  associations among relations and then grouping
  actors
• Chap. 9 covers position, Chap. 10 and 11 cover
  role

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Structural Equivalence

• Definition two actors are structurally
  equivalent if they have identical ties to and
  from all other actors in the network (356)
• Structurally equivalent actors have identical
  ties to and from identical actors, on all R
  relations, and will have the same position
• i j when rows and columns in sociomatrix are
  identical for all relations, they are
  substitutable
• Graph Equivalence reflexive and reciprocal ties
  between i and j

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Positional Analysis

• Permutation and partition of sociomatrix will
  reveal subsets of actors who are structurally
  equivalent, same position
• These can be collapsed into an image matrix,
  showing ties to and from (and within) each
  position (nodespositions, tiesrelations between
  positions)
• Ties between nodes of different positions implies
  ties between positions, or graph homomorphism
• Blockmodel an image matrix(ces) and description
  of how nodes are assigned to positions

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Tasks in a Position Analysis

• Formally define equivalence specifying
  mathematical conditions. Structural equivalence
  is one definition among many
• Measure of equivalence quantity to decide if
  subsets are equivalent according to definition
• Representation of assignments to equivalence
  classes/positions (partition)
• Assessment test for goodness-of-fit

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Measuring Equivalence Euclidean Distance

• Euclidean Distance (i and j) distance between
  rows i and j and columns i and j. Distance 0
  if entries in rows and columns is identical,
  grows with greater diversity
• In D, entries measure equivalence of row actor
  and column actor
• If multiple relations exist, distance measures
  equivalence across all relations

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Measuring EquivalenceCorrelation

• Similar to distance, measures correlation
  coefficient
• Arranged in matrix C1, shows Pearson
  product-moment correlation for a relation
• Perfect equivalence 1
• Multiple relations requires sociomatrices and
  transposes
• Other measures include similarity/dissimilarity

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Representation

• Partition equivalent actors into subsets,
  displayed in tree diagram or dendogram (379)
• CONCOR convergence of iterated correlations
  (eventually all 1 and -1) of C1, C2, Ct ?
  correlation of correlations of
• Subsets can be split into finer partitions
• Problems 1) form of partition is defined by the
  procedure, not the network 2) partition does not
  resemble intuitive hypotheses about network
  position 3) formal properties are not clear

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Representation

• Hierarchical Clustering find collections of
  actors equivalent at a, threshold value
• Complete link all pairs are the same as
  criterion value, successively uses less values to
  define clusters
• Displayed in a dendrogram (383), partitions
  increase as criteria increases
• Partitions used should match the number of
  classes described by a theory
• Input is matrix of structural equivalences,
  either distance (D) or correlations (C1)
• Decision of of number of subsets can be
  arbitrary, once actors are grouped together, this
  cannot be undone

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Representation Ties Between and Within

• First permute sociomatrix, actors in same
  positions become adjacent
• Summarize with density matrix, positions instead
  of actors in rows and columns
• Convert to image matrix based on density rule, if
  density is larger than a, note as 1, if not, note
  as 0
• Reduced Graph nodes as positions, ties either
  reflexive or between positions

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Chapter 10 Blockmodels

• Defined as a partition into B positions, and
  description of ties within and between positions,
  a hypothesis?
• Also represented by image matrix B, a B x B x R
  array
• Two components mapping that assigns actors to
  positions, and the matrix that specifies ties
  within and between

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Building Blockmodels

• Perfect (fit) equivalence (all 1s and 0s) is
  rare, so criteria are used to determine coding
• Zeroblock (lean), defined as a block with all
  zeros, but oneblock can have 1s and 0s
• Oneblock, all ties in block must be present
• a density threshold, grand density across all
  relations or specific density, for each relation
• These should be viewed as a spectrum, ranging
  from zeroblock, density, to oneblock

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Blockmodeling Valued Relations

• Max value small values 0, large 1, defined
  on e
• Mean value average value of each relation,
  similar to a density
• Consider how relation is measured before applying
  criteria, if values range from positive to
  negative, the mean will combine these
• Max might be better

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Interpretation Three methods

• Model can be externally validated if
  characteristics of actors are different between
  positions, do attributes determine structure or
  does structure determine attributes?
• Description of individual positions and relations
  (isolates, transmitters, receivers, carriers)
  Burt employs this with typology (414)
• Image matrix, to test the model against a theory
  of patterns within and between positions ? Ten
  possible simple image matrices (421), Five
  ideal image matrices for multiple positions

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DiMaggio Structural Analysis of Organizational
Fields

• The shift from Environments to Fields has three
  advantages 1) more useful to know organizational
  sources of characteristics rather than
  environmental sources 2) environmental variables
  may be positionally determined 3) permits
  examination of interorganizational structure as
  field bridges organizational and societal levels
  of analysis

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Methods of Partitioning

• Naturalistic use of a priori or categorical
  descriptions
• Attributes qualities of actors (but this
  confused IV and DV)
• Cohesion maximal interaction and clique
  formation
• Equivalence
• Cohesion is similar to exchange theory
  interaction increases affiliation and similarity,
  Equivalence is more like role theory, actors are
  similar based on their roles, patterns of ties to
  others in the network

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The Resident Theatre Field

• Blockmodeling fulfills seven requirements of a
  strategy for blockmodeling (last slide)
• Data mail questionnaires sent to 165 theatre
  operating officers (67.3 response rate)
• Three Questions
• Who do you ask for advice?
• Who would you eat dinner with at conference?
• Who do you admire?
• Three relations Advice, Association, Admiration
• Five answers only for each question

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Analysis Four-Block Interval

• CONCOR develops three matrices, four and eight
  block partitions based on structural equivalence
• Density matrices used to determine
  dichotomization
• Block A is superior, everyone sends to them and
  itself
• B and C send to A and B, but not to C, while D
  sends to A, B, C, but not itself. E only sends to
  A.
• E is a residual block, the leftovers

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Analysis Eight-Block Interval

• Compared against image matrices of domination,
  coalition, and no contact
• Resembles dominance but not perfectly, C1, A1,
  and B2 are a coalition
• A1 and A2 are historically dominant, B are not
  TCG members, might be in a different class of
  opposing theatres. D1 is avant-garde, most
  innovative
• Stratification of rewards is affected by ones
  hierarchical position
• Blocks do not divide based on pure cohesion,
  would not identify D1, which isnt reflexive
• Who you know determines success

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Problem Solving with Blockmodeling

• Knowing an organization field requires
  qualitative, ethnographic research. As an
  outsider, its hard to get a comprehensive
  understanding of such information especially from
  a large field
• Blocking can supplement this, a macroscope,
  adjusted for levels of refinement
• identify structural anomalies
• Organizations that float between blocks, dont
  fit in squarely. What properties cause this?
• Identify niches, positions that occupy a
  particular ecological combination of resources.
  Superior to attribute based definitions that are
  untestable.
• Blockmodeling define niche along axiom of
  isomorphism, and exploits it