Centrality and Prestige

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Centrality and Prestige

• HCC Spring 2005
• Wednesday, April 13, 2005
• Aliseya Wright

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Chapter Overview

• One of the primary uses of graph theory in social
  network analysis is the identification of the
  "most important" actors in a social network.
• To address this, this chapter looks at
• How to identify the important and the
  non-important actors.
• The most noteworthy definitions of importance
  along with the mathematical concepts that the
  various definitions have spawned
• Directional vs. Non-directional relations.
  (Non-directional relations allow you to analyze
  centrality, while directional relations give you
  the ability to analyze centrality as well as
  prestige)
• Prestige is usually tied to the number of
  "choices" an actor has which is related to the
  in-degree (as opposed to just the degree) of the
  actor.

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PROMINENCE Centrality and Prestige

• An actor is considered prominent if the ties of
  the actor make the actor particularly visible to
  the other actors in the network. (visibility is
  not only measured by direct ties, but also by
  indirect ties through intermediaries)
• However, it is not clear from the number of ties
  and choices alone whether an actor is important,
  so Knock and Burt distinguished two types of
  visibility centrality and prestige.

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Centrality

• With centrality, we are not concerned with
  whether prominence is due to the receiving or the
  transmission of many ties - what is important is
  that the actor is simply involved.
• For a non-directional relation, a central actor
  is involved in many ties.
• Sociological and economic concepts such as access
  and control over resources and brokerage of
  information are well suited to measurement and
  naturally yield a definition of centrality since
  the difference between the source and the
  receiver is less important than is simply
  participating in many interactions, therefore the
  actors with the most access or control will be
  the most central in the network.

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Prestige

• Prestige is a more refined concept in which the
  direction of the tie is important.
• Prestige increases when the actor becomes the
  object of more ties, but not necessarily when the
  actor itself initiates the ties.
• However, having a high in-degree is not always a
  measure of prestige when the tie is negative.
  Also if the tie is one such as "advises" then a
  high out-degree is now a measure of prestige.

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NON-DIRECTIONAL RELATIONS

• In order to find the most important actors, we
• will look for measures reflecting which actors
• are at the center of the set of actors. This can
• be found using several definitions of center
• including
• Maximum Degree
• Betweeness
• Closeness
• Information

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Degree Centrality

• ACTOR DEGREE CENTRALITY
• In this measure, the level of activity is equal
  to the degree. The more ties, the higher the
  centrality of the actor.
• The problem with this is that the measure depends
  on the size of the group with the maximum value
  of (g-1) which does not allow for standardization
  across groups of varying sizes.
• A related index to this is the ego index which
  relates the actual index of an actor the to the
  maximum numbers of ties that could occur.
• The span of an actor is the percentage of ties in
  the network that involve the actor or the actors
  that the primary actor is adjacent to.

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Degree Centrality

• GROUP DEGREE CENTRALITY
• A centralization measure that quantifies the
  range or variability of the individual actor
  indices.
• There are many formulas used to compute this
  ranging from the complex formula proposed by
  Freeman, to simpler ones that are based on the
  variance, however the most commonly used group
  level index is the density of the graph (the
  normalized average degree).
• Indices such as average degree and density are
  not really centralization measures.
  Centralization should quantify the range or
  variability of the individual actor indices,
  therefore the average degree or the graph
  density, which are quantifications of average
  actor tendencies rather than variability are not
  valid centralization methods.

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Closeness Centrality

• An actor is central if it can quickly interact
  with other actors. Actors that are very close can
  be effective in communicating information to
  other actors.
• ACTOR CLOSENESS CENTRALITY
• Sabidussi proposed that actor closeness should be
  measured as a function of geodesic (shortest
  path) distances. This type of centrality depends
  not only on direct ties, but also on indirect
  ties.
• Jordan Center the Jordan center of a graph is
  the subset of nodes that have the smallest
  maximum distance to all other nodes. To find this
  center, you take a gxg matrix of geodesic
  distances between pairs of nodes and then find
  the largest entry in each row. These distances
  are the maximum distances from every actor to
  their fellow actors. One then finds the smallest
  of these maximum distances. All nodes that have
  this smallest maximum distance are part of the
  Jordan center of the graph.
• The Centroid of a graph is based on the degrees
  of the nodes and is most appropriate for graphs
  that are trees. The centroid is basically the
  subset of all nodes that have the smallest weight
  where weight is defined as the maximum weight of
  any branch in the node.

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Closeness Centrality

• GROUP CLOSENESS CENTRALITY
• Freeman's general group closeness index is based
  on the standardized actor closeness centralities
  and reaches its maximum value of unity when one
  actor "chooses all other actors and the other
  actors have geodesics of the length 2 to the
  remaining g-2 actors (star graph). It is at a
  minimum when all geodesic lengths are equal
  (circle graphs)

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Betweeness Centrality

• Interaction between two actors may depend on the
  other actors in the set of actors. Theactors in
  the middle have some control over the path in the
  graph.
• ACTOR BETWEENESS CENTRALITY
• In defining this centrality, the following
  assumptions were made lines have equal weight
  and communications will travel along the shortest
  route.
• When there is more than one geodesic, all
  geodesics are equally likely to be used.
• This actor betweeness is simply the sum of these
  estimated probabilities over all pairs of actors
  not including the actor in question.
• The minimum is 0 when the actor fall on no
  geodesic, and the maximum is (g-1)(g-2)/2 which
  is the number of pairs of actors not including
  the actor all geodesics.
• Unlike the closeness index, the betweeness
  indices can be computed even if the graph is not
  connected.

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Betweeness Centrality

• GROUP BETWEENESS CENTRALITY
• Measures the heterogeneity or variability of
  betweeness in the entire set of actors.

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Information Centrality

• While Freeman's centrality measure based on the
  betweeness of actors on geodesics has found the
  most use because of its generality, it has the
  issue that it assumes that all geodesics are used
  with an equal probability. This assumption is not
  always justifiable.
• For instance, if we look at the actors in the
  geodesic, an actor with a high degree is more
  likely to be used than an actor with a low
  degree, which means that the probability of the
  geodesic containing the actor with a high degree
  is more probable.
• Also, it may not be reasonable to assume that
  just because a path is shorter that it the one
  used. In a communications network there maybe
  many reasons why that geodesic is ignored, for
  example in the case where many intermediaries are
  used in order to "hide" or "shield" information.
• So it makes sense to generalize the notion of
  betweeness centrality so that all paths between
  actors have weights depending on their lengths
  and that these lengths are considered when
  calculating betweeness counts.

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Information Centrality

• ACTOR INFORMATION CENTRALITY
• This version of centrality focuses on the
  information contained in all paths originating
  with a specific actor. The information index of
  an actor averages the information in these paths
  which in turn is inversely related to the
  variance in the transmission of a signal from one
  actor to another.
• GROUP INFORMATION CENTRALITY
• The summary group-level information index is the
  average of information across actors. This index
  has limits that depend on g, which make it
  difficult to compare across networks.

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Directional Relations

• With a directional relation, we can now
  distinguish between choices made and choices
  received.
• Centrality indices for directional relations
  generally focus on choices made while prestige
  indices focus on choices received (both direct
  and indirect)
• Degree and closeness are easy to apply to
  directional relations while betweeness and
  information are not because of their reliance on
  non-directed paths.

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Centrality (Directional Relations)

• DEGREE
• The calculation for this is the same as for a
  non-directional relation, except we use the
  out-degree of each actor.
• CLOSENESS
• The actor level centrality index based on
  closeness can be defined as the sum of the total
  distances from an actor to all of the other
  actors then dividing by the total maximum
  distance.
• One problem with this index is that it is not
  defined unless the digraph is strongly connected
  (there is a directed path from i to j for all
  actors i and j) otherwise the equation for
  closeness will be undefined.

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Prestige

• With directional relations, choices received are
  of interest, so measures of centrality may not be
  of as much concern as measures of prestige. There
  is (as of the writing of this text) little
  research that has been done on group-level
  prestige indices.
• DEGREE PRESTIGE
• This is measured by the in-degree of each actor.

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Prestige

• PROXIMITY PRESTIGE
• Is a measure of how close other actors are to a
  given actor
• The actor and group-level prestige indices on
  proximity or graph distances to each actor can be
  useful. Actors are judged to be prestigious based
  on how close or proximate the other actors in the
  set of actors are to them. However, one should
  also consider the prestige of actors that are
  proximate to the actor in question. If many
  prestigious actors choose an actor, then that
  should be given more weight than if many
  non-prestigious actors choose an actor. This
  naturally leads to the definition of
• STATUS OR RANK PRESTIGE
• This reflects the status or prestige of the
  actors doing the "choosing" This combines the
  number of direct choices to a specific actor with
  the status or rank of the choosing actors
  involved.

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Questions?