Wasserman and Faust Centrality and Prestige

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Wasserman and Faust Centrality and Prestige

• Edward W. Cardinale
• October 4, 2007

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Overview

• Introduction
• Centrality and Prestige
• Non-Directional Relations
• Directional Relations
• Closing

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Introduction Centrality and Prestige

• Most of measure come from Freeman
• Most Important actors
• Many definitions of IMPORTANCE and PROMINENCE
• Actor Location in the Network
• Actor Indices degree, closeness, betweenness,
  information and rank
• Group Indices

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5.1 Prominence Centrality and Prestige

• Not concerned with signed or multirelational
• Prominence ties of the actor make it visible
• Measured by direct ties and indirect paths
• Direction Choices made vs Choices received
• Prominence is based on pattern in sociomatrix
• Nondirectional vs Directional
• Prominence has two classes
• Centrality and Prestige

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Actor Centrality

• Central actors have many ties - INVOLVED
• Not concerned with receiving or transmitting
• Non-directional
• Most access or most control or most active will
  be the central to the network
• NOTATION Centrality - C

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Actor Prestige

• Prestigious actor recipient of extensive ties
• Directional
• Not the best term. Example Hates
• Prestige Status Deference Populartiy
• Measures of Rank in the Network
• NOTATION Prestige - P

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Group Centralization Group Prestige

• Group Level Measures are used to compare
  different networks easily
• Group measure are a combination of actor measures
• The larger the group more likely a single actor
  will be central
• Index of Centralization
• Equation (5.1) Page 177
• 0 All actors have the same centrality index
• 1 One actor dominates
• Shows how unequal actor are and dispersion

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Group Centralization Group Prestige

• Group level prestige
• The denominator is difficult to calculate
• Variance can used to measure prestige
• Compactness
• Centralized graphs are compact
• Centrality calculated based on closeness (small
  distance between nodes

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5.2 Nondirectional Relations

• Married to is example of nondirectional
• Most important actor are at the center of the set
  of actors
• New terms explored
• Maximum degree
• Betweeness
• Closeness
• Information
• 15th Century Florence

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Degree of Centrality

• Degree Centrality is based on degree
• Star vs. Circle
• Actor Level
• Equation (5.3) Page 179
• Star vs. Circle vs. Line
• High Centrality Where the Action is
• Ego Density actual tiesmax ties of an actor
• Span - ties that involve a actor

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Degree of Centrality

• Group Level
• Dispersion C Equation 5.5 on page 180
• Variance S Equation 5.6 on page 180
• Average Standardized Degree
• Density is the most widely used group level index
• Density is the average standardized degree
  indices
• Value ranges from 0 -1 or empty complete
• Group size increase density tends to decrease
• Best to report S and C with Average Degree and
  Graph Density

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Closeness Centrality

• Measures of how close an actor is to all others
• Short communication paths, minimum distance Ex
  Star
• Actor Level
• How close? Distance from one node to another
• Depends on direct and indirect ties
• Closeness to the whole network
• Equation (5.7) Equation (5.8)
• C value of 0 -1, 0 Isolate and 1 adjacent to
  all others
• Jordan Center Group of nodes with smallest max
  distance

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Closeness Centrality

• Centroid? used in tree graphs
• Considers all branches from a node
• The weight of node is the max weight of a branch
  from a node
• Centroid is the subset of all nodes that have the
  smallest weight
• Group Level
• Equation (5.9) p.186 and Equation (5.10) p.187
• Report the average closeness and the variance

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Betweeness Centrality

• Controls the path / actor in the middle
• Star vs. Line
• Actors are central if they are in between other
  actors, strategic points
• Actor Level
• With betweenness comes stress
• All geodesics have an equal probability
• Equation (5.12) p. 190 Thank God for UCINET!
• Star vs Circle vs Line

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Betweeness Centrality

• Group Level
• Equation (5.14) p. 192
• 0 All actor have equal betweeness
• 1 star graph

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Information Centrality

• Build on betweenness
• Not all paths have an equal probability
• The more degrees an actor has the more likely it
  will be used
• Actor may chose longer paths hide or shield
  information
• Stephenson and Zelen Looks at combine path
• All geodesics (Value 1), All paths(value less
  than 1)
• Information of the Path is the inverse of length

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Information Centrality

• Actor Level
• Information of all paths with a Actor
• Equation (5.18) page 196
• Star vs Circle vs Line
• All actor must sum to unity
• Group Level
• Average of information across actors
• Very difficult to compare network
• Variance of Information Equation (5.20)

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5.3 Directional Relations

• We can weigh choices made and choices received
• Imports and Export are different
• Centrality indices choices made
• Prestige indices choices received
• Four centrality indies degree, closeness,
  betweeness and information
• Followed by measure of pretige

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Centrality

• Degree
• Same as nondirectional except it only focus on
  the outdegree from an actor
• Closeness
• Influence range and distance to actor
• Sum of all reachable actors /Average distance of
  all actors
• Equation (5.22) page 201
• Betweeness and Information
• Only used in for nondirectional relationship

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Prestige

• Degree Prestige
• Based on a measure of indgrees
• Equation (5.24) Page 203
• Number of indegrees/g-1
• Proximity Prestige
• Proximity is closeness based on distance to each
  actor
• Equation (5.25) Page 204
• Group level measures average and the variance

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Prestige

• Status or Rank Prestige
• Looks at the prominence of the actors that are
  doing the choosing
• Prestigious friends then you are prestigious
• Status score is determined for each actor
• Hubs and bridges
• GRADAP can compute prestige

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Closing

• Looking at all of these indices can produce very
  different results
• Think basics mean, median and mode
• Multirelational data calculate for each realtion