Cohesive Subgroups

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Cohesive Subgroups Core-Periphery Structures
PAD637 Week 6

• Oct. 23. 06
• Dong Chul Shim Hyun Hee Park
• Modified by Karl Rethemeyer

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Definition of subgroups
Wasserman Faust. Ch7

• Definition of sub-groups Cohesive subgroups are
  subsets of actors among whom there are relatively
  strong, direct, intense, frequent or positive
  ties. It formalizes the strong social groups
  based on the social network properties (p. 249)
• Usefulness of subgroups
• Finding cohesive informal groups (Howthorne
  study)
• Finding power elites or communication groups
  across the organizational boundary (Supreme court
  (Caldeira) national power elite (Moore))
• Subgroups and segregation (Social welfare study)
• Four general ways of defining cohesive subgroups
• Mutuality of ties
• The closeness or reachability of subgroup members
  
• The frequency of ties among members
• The relative frequency of ties among subgroup
  members compared to nonmembers

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Clique, Clan, Clubs Basic Mutuality Ties
Wasserman Faust. Ch7

• Clique
• n-Clique
• N-clan
• N-club

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Nodal degrees from the problem of vulnerability
Wasserman Faust. Ch7

• nodes should be adjacent relatively numerous
  other subgroup nodes
• K-plexes allowed to lack tie no more than k (it
  includes reflexes)
• K-Cores specifies the minimum number of missing
  links

Vulnerable 2-clique 1 lost tie causes it to
disintegrate
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Comparing within to outside subgroup ties
Wasserman Faust. Ch7

• cohesive subgroups should be relatively cohesive
  within compared to outside
• LS sets ties concept more inside than outside
• Lamda Set Connectivity concept. How many lines
  must one remove to disconnect two actors. Want
  this to be high.

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Dealing With Value and Direction
Wasserman Faust. Ch7

• Directional relations symmetrize the relations
• weakly connected n-clique.
• unilateral connected n-clique,
• strongly connected n-clique
• recursively connected n-clique.
• Recursive relationship, semi-path
• Valued relations Define cut-off values and
  then dichotomize. Question What strength
  connection?
• Example Choosing a level for dichotomization

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Other Approaches Permutations Visualization
Wasserman Faust. Ch7

• Permutation
• Multi Dimensional Approach (Clustering Analysis,
  Factor Analysis Caldeira (1988)

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Models of Core/Periphery structures by
Borgatti, Everett

• OVERVIEW
• The concept of a core/periphery structure
• Two models of the core/periphery structure based
  on the intuitive conception
• a discrete model
• a continuous model.
• The assessment of the fit to data

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A core/periphery structure
Models of Core/Periphery structures

• a core/periphery structure if the network can be
  partitioned into two sets
• a core whose members are densely tied to each
  other
• a periphery whose members have more ties to core
  members than to each other.

A network with a core/periphery structure
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Intuitive Conceptions
Models of Core/Periphery structures

• A group or network cannot be subdivided into
  exclusive cohesive subgroups or factions,
  although some actors may be better connected than
  others.
• A two-class partition of nodes (one class is the
  core and the other is the periphery) In the
  terminology of block-modeling, the core is seen
  as a 1-block, and the periphery is seen as a
  0-block. The blocks representing ties between the
  core and periphery can be either 1-blocks or
  0-blocks.
• A third intuitive view of the core/ periphery
  structure is based on the physical center and
  periphery of a cloud of points in Euclidean
  space.

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Discrete Model
Models of Core/Periphery structures

• The core/periphery (CP) Structure
  (Partition-based Approach)
• Core a cohesive subgraph in which actors are
  connected to each other in some maximal sense
• Periphery a class of actors that are more
  loosely connected to the cohesive subgraph but
  lack any maximal cohesion with the core.
• The core periphery have interchange the amount
  depends on the phenomenon

The adjacency matrix

• Core-core 1-block
• Core-periphery (imperfect) 1-blocks
• Periphery-periphery 0-block

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Discrete Model-Measure of fit to data
Models of Core/Periphery structures
Idealized C-P structure
Measure of Fit to Data

• An unnormalized Pearson correlation coefficient
• The measure achieves its maximum value when and
  only when A (the matrix of aij) and ? (the matrix
  of ?ij) are identical, which occurs when A has a
  perfect core/periphery structure.
• Idea Keep permuting until the structure is as
  close to core-periphery ideal as possible then
  stop

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Discrete Model
Models of Core/Periphery structures

• Testing a priori partition a permutation test
  for the independence of two proximity matrices
• Empirical example of male dominance in a troop of
  monkeys by Linda Wolfe
• Detecting core/periphery structure in data
• An empirical example of co-citations among social
  work journals by Baker (1992)
• Additional pattern matrices Example from my
  adult basic education class data

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Continuous Model
Models of Core/Periphery structures

• A limitation of the partition-based approach
• the excessive simplicity of defining classes of
  nodes
• specifying the ideal blockmodel that best
  captures the notion of a core/periphery structure
  is relatively difficult
• the number of classes
• A Continuous Model as an alternative approach
  coreness.
• Strength or probability of ties between node i
  and node j is function of product of coreness of
  each.
• the strength of tie between two actors is a
  function of the closeness of each to the center
• Similar to factor analysis and loglinear model of
  independence

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Continuous Model
Models of Core/Periphery structures

• Estimating coreness empirically
• An empirical example of co-citations among social
  work journals by Baker (1992)
• Coreness and centrality
• The multiplicative core/periphery model, when
  phrased as an eigenvector, is precisely
  Bonacich's (1987) measure of centrality.
• It is true that all actors in a core are
  necessarily highly central as measured by
  virtually any measure (except when the model fits
  vacuously).
• All coreness measures are centrality measures,
  but not every high centrality is part of the
  core.
• The key difference between a centrality and a
  coreness measures the pattern of ties in the
  network as a whole or not? The coreness is only
  interpretable to the extent that the model fits,
  but a centrality is interpretable no matter what
  the structure of the network.

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Peripheries of cohesive subgroups by
Borgatti, Everett

• Overview
• Peripheries in previous studies
• Defining Peripheries
• K-periphery
• C-P Measure
• Alternative Approach
• C-P Matrix

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Peripheries in Previous studies
Peripheries of Cohesive Subgroups

• No consideration is given to actors that do not
  belong to a given group in previous research
• Previous definition of the periphery the set of
  all vertices not in the core that are adjacent to
  at least one member of the core.
• A partition of all nodes in the network into
  three classes (1) the members of the subgroup,
  (2) the periphery "belonging to" that subgroup,
  and (3) the rest of the nodes in the network

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Defining Peripheries
Peripheries of Cohesive Subgroups

• K-periphery Defining peripheries based on
  distance how deep in the periphery are you?
• Definition Let G be any graph, and let C be a
  cohesive subgroup, called a core of G. Then the
  periphery P is G-C. If v ? P then we say v is in
  the k-periphery if v is a distance less than or
  equal to k lines from a member of C.
• C-P Measure Defining peripheries based on
  density
• Definition For v? V (any member of the network),
  if v ? C then C-P(v) 1, otherwise let q be the
  minimum number of edges incident with v that are
  required to make v part of C (how many
  connections would v have to possess to be
  considered a core member)
• and let r be the number of those edges that are
  already incident with v. Then C-P(v) r/q. We
  call C-P(v) the coreness of vertex v. This is the
  ratio of actual ties to ties needed to be
  considered core.

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CP Measure Peripheral Degree Index
Peripheries of Cohesive Subgroups

• An empirical example Zachary Karate Club data
  (Zachary, 1977)
• Limitations of C-P measure
• additional complexity
• Periphery-to-periphery interaction could still be
  quite high.
• Peripheral Degree index
• Pd(v) Number of peripheral actors connected to
  v Total number of peripheral actors
• Can use this to find actors who are
  periphery-centric

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An alternative approach
Peripheries of Cohesive Subgroups

• The generalizations of cliques that contain
  parameters that can be adjusted so as to relax
  the conditions of membership.
• Such generalizations produce hierarchies of
  clique-like structures, and these can be viewed
  as cores and peripheries.

• single 1-cliquec,d,e.
• 2-cliques b,c,d,e, c,d,e,f.
• 2-clique periphery b,f

• Another Example Taro data (Hage Harary, 1983)

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Using graphical analysis

• It may be that cliques are actually alternate
  cores
• There are some actors that belong to
  multiplecliques, but they overlap.
• Soone could analyze how core each actors toeach
  clique and then look for overlap
• One approach is to use two-mode data and
  thenvisualize it using correspondence analysis

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C-P Matrix Two mode data on coreness
Peripheries of Cohesive Subgroups
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Correspondence Analysis of Taro CP Matrix
Peripheries of Cohesive Subgroups
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Gregory Caldeira (1988) Legal precedent and
network analysis
Legal Precedents and Network Analysis

• Research question How state supreme courts
  communicate one another including the precedents
  of courts.
• Theoretical Backgrounds Regionalism
• Basic Hypothesis Supreme courts will refer and
  communicate more with neighborhood states

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Main application of Network analysis
Multidimensional approahces
Legal Precedents and Network Analysis

• Cluster Analysis provides the cluster trees
• Input reference percentage of each state
• Output Proximity among the states
• Discriminant Analysis provides the
  characteristics of the subgroups and forecast the
  group of new members
• Clique vs Block modeling (structurural
  equivalency)

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Methods
Legal Precedents and Network Analysis

• States and the District of Columbia (n51) in the
  calendar year of 1975 (Appellate cases)
• Standardized percentage scores 1) Valued model,
  2) asymmetric Matrix

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Example of Dendrogram
Legal Precedents and Network Analysis
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Legal Precedents and Network Analysis
Example of MDS Plot
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Example of Discriminant Analysis
Legal Precedents and Network Analysis
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Summary
Legal Precedents and Network Analysis

• Figure 1 (dendrogram) shows the hierarchical
  clustering of clique Geographically close states
  tend to be clustered together both in social
  cohesion model and block modeling.
• Figure 2 provides more the information 1) some
  states such as NY, CA, FL, IA are more prominent
  in influencing other states. 2) Some states are
  isolated (WY, MT, NEV). Geographically closely
  located states are positioned in the same
  dimension. For example, New England and Mid
  Atlantic states are positioned in upper dimension
  of the graphs.
• Figure 4 shows the regionalism more clearly (it
  is based on block modeling). In block modeling,
  first positioned states tend to be concentrated
  around north east while fifth positioned clusters
  are southern area.
• Discrimant analysis shows that similar states in
  terms of case load, innovation population wealth,
  legal capital tend to be clustered together.
  Legal professionalism and prestige have
  inconsistent results between block modeling and
  social cohesion approach

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Moore Stucture of National Power Elites

• Research question 1) The degree of integration
  among political elites in the United States. 2)
  The attributes of political elites
• Research backgrounds
• Mills and Domhoff Integrated power structure
• Pluralists fragmented Structure

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Application of subgroup approaches
Structure of National Power Elites

• Based on traditional Clique approaches All
  members should be connected to each other to be a
  subgroups Group are naturally small because of
  it.
• Circle Concept merging the cliques if they have
  overlapping members. (2/3 of members are
  integrated)
• Thus, the study used the two approaches first
  step is based on connectivity for finding
  cliques, and the second step is based on
  reachability for finding circles
• Comparison with centrality and block modeling

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Research Methods
Structure of National Power Elites

• American Leadership study 545 top position
  leaders in key institutions in 10 major
  positional sectors.
• Snow ball sampling
• Recognize the persons that they talk and work
  about the issue together.
• Symmetry method for non sample members
• Find circles by starting with strictly defined
  cliques and then looking for overlaps.

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Results (1) Integrated Network Structure
Structure of National Power Elites
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Results (2) Activity, Visibility and Reputation
the elite have influence
Structure of National Power Elites
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Result (3) Power Elite Origins
Structure of National Power Elites
Central circle members are not systematically
differentfrom elitesoutside thecentral circle.
Instead, all elites are different from
non-elites.