Centrality

1
Centrality PowerThe Centrality Trilogy

• Ian Anderson
• PAD637 Rethemeyer
• Week 5

2
Outline The Centrality Trilogy

• Episode IV A New Hope (Freeman)
• Sets out a framework for different conceptions of
  centrality (9 different measures!)
• Episode V The Exchange Networks Strike Back
  (Cook, Emerson, Gillmore)
• Shows limits of equating power with centrality
• Episode VI The Return of the Centrality Measure
  (Bonacich)
• Proposes a generalization of centrality measure
  to account for traditional powercentrality and
  Cook et als results

3
Freeman - Definitions Where We Started A Long
Long Time Ago

• Degree - of other points adjacent (directly
  connected by an edge) to the given point
• Reachable a path must exist between points
• Connected graph every point is reachable from
  any other point in the graph
• Distance measured by of edges on path
• Geodesic shortest path connecting 2 points
• Between points falling on the only geodesic or
  on all geodesics linking a pair of points
• No unanimity on exactly what centrality is
  theoretically or how it is to be measured 3
  major competing theories
• A key assumption throughout that Power
  Centrality

4
Point Centrality

• Common ground in all research recognizes that the
  center of a star network is most central
• 3 distinct structural properties of this point
• Maximum possible degree
• Falls on geodesics between the most points
• Minimally distant from all other points

5
Point Centrality 1 Centrality Degree

• Here centrality is not so much its own concept,
  but rather is simply another way of saying degree
• Logically, a point with high degree has many
  connections, and therefore is central to the
  network (and powerful as well)
• Ex In a communications network this person acts
  as the focal point of communication.

6
Point Centrality 2Centrality Betweenness

• Centrality is based on the frequency which a
  point falls on geodesics
• The measure used is that of betweenness
• When a person is strategically located on these
  paths, they are central and have gate keeping
  power (withhold/give info, goods)
• Ex In a communications network this person has
  the power to control communication by withholding
  or distorting information

7
Point Centrality 3Centrality Closeness

• Distance is used as a measure of centrality
• Closeness allows quick access to resources at the
  other points
• Here power is a function of control, but instead
  is a function of the extent to which it can avoid
  control of others
• In other words power/centrality stems from
  independence (no need for intermediaries)
• Ex In communications network, central point is
  not dependent of others to relay messages. Also a
  message originating with central point will
  spread the message quickest with least cost

8
Graph Centrality

• Graph centrality looks at entire networks and
  concept of compactness of graphs
• Here there are 3 parallels to the 3 point
  centrality concepts

9
Graph Centrality Conception 1

• Mirrors degree based point centrality
• Argument that centrality of network should relate
  to the tendency of one point to be outstandingly
  central
• Measure derived from differences between
  centrality of most central point and centrality
  of all other points in the network (relative
  dominance of a single point)

10
Graph Centrality Conception 2

• Mirrors betweenness based point centrality
• Defined as average distance between the relative
  centrality of the most central point and that of
  all other points

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Graph Centrality Conception 3

• Mirrors closeness based degree centrality
• Little studied on how to properly measure this
• We do know that it is a complicated sum of
  inverse proportionate distances, and is some sort
  of index of homogeneity of distance

12
Conclusions What We Know

• Distance based measures cannot be used in
  disconnected graphs
• The star or wheel graph is always given maximum
  centrality score by all measures
• The circle and complete graphs are always given
  minimum centrality score by all
• Between the max and min, the three measures vary
  greatly in rankings of other graphs
• The greatest range in variation for both point
  and graph centrality goes to the betweenness
  measures, showing this to be the finest grained
  measure
• The smallest range in variation for both point
  and graph centrality goes to the degree measures,
  showing this to be the coarsest grained measure

13
Cook, et al Exchange Networks Centrality ?
Power?

• Examines two possible theoretical bases for power
  in exchange networks
• Point centrality
• Power-dependence
• Shows that centrality does not always accurately
  predict power, and therefore previous theories of
  centrality power need to be revisited

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Background What are Exchange Networks?

• Exchange Networks consist of (Emerson)
• Set of actors
• Distribution of valued resources among actors
• Set of exchange opportunities with other actors
• Set of historically developed and utilized
  exchange opportunities (exchange relations)
• Set of network connections linking exchange
  relations into a single network structure
• Connections
• Positive exchange in one relation is contingent
  on exchange with other
• Negative - exchange in one relation is
  contingent on nonexchange in the other

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The Power in Position?

• Most networks are thought to be mixed (positive
  and negative connections)
• Often there are empirical boundaries to an
  exchange network, but these are unknown to the
  actors
• Therefore membership does not matter as much as
  position
• Positions are the same if actors have
  structurally similar locations in a network
• Position is important because
• It simplifies large complex networks
• It is an important determinant of behavior in
  exchange networks

16
Another Look at Point Centrality

• Recall Freemans 3 types of point centrality
  measures degree, betweenness, and closeness
• Degree measures are weak because they are too
  localized (only looks at direct links)
• Therefore the degree measure is thrown out for
  the study leaving betweenness closeness
  measures of network-wide power
• Point centrality is important based on its
  relation to power and influence
• The first hypothesis lays out this concept

17
An Alternative Explanation?

• But what if traditional wisdom is incorrect?
• Using the concept of dependence as it relates to
  power, alternative hypotheses are developed
• H2 Intermediary positions will have more power
  and this power will grow over time
• H3 Intermediary will leverage power over
  periphery 1st, and then this power can be used to
  leverage the center
• H4 Finally, the intermediary will exercise equal
  power over the center and periphery
• The study uses the structure in 1c to demonstrate
  this (traditionally D would have most power, but
  they predict that E will)

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The Experiment

• Used computer terminals in different rooms,
• limiting who could exchange with who (structure
  of 1c)
• limited how often exchanges could take place (1
  transaction per period, creating a negatively
  connected network)
• Power measured by number of points able to
  negotiate (points then converted into )

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The Results

• Results of the study show support for H2-H5
• So the hypotheses based on power-dependence
  theory better explained the power than
  traditional centrality measures
• The study was replicated on larger complex
  networks using computer simulations, and again
  the alternate hypotheses were supported
• Closeness betweenness measures of centrality
  did not accurate predict the locus of power in
  these negative relational networks

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Implications

• If the notion that centrality power is to be
  continued, then centrality must be
  reconceptualized more generally (preferred) or
  recognize it limits to only certain types of
  networks
• The power-dependence theory demonstrated here
  stems from micro sized networks and difficultly
  will ensue in trying to use on large complex
  networks (must raise power-dependence theory to
  macroscopic level)
• One weakness is that power-dependence relations
  are fundamentally dyadic, must find a way to
  raise this to structural level

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Vulnerability Answer to Network-wide Power in
Power-Dependence theory?

• Vulnerability how is structure weakened by the
  removal of a point or line?
• Networks are shown to be vulnerable at point that
  is most powerful
• Vulnerability in a negatively connected network
  locates the point of minimum dependence (maximum
  network wide power)

22
Bonacich Family of Centrality Measures

• Looks at the discrepancy between the previous 2
  articles
• Seeks to answer how to conceptualize centrality
  given Cook et als showing that Centrality ?
  Power in all cases
• Proposes solution a family of centrality
  measures represented by the function c(a,ß).
• Later in the article he points out that c(a,ß) is
  essentially a form of Freemans closeness point
  centrality because it is large when the
  connecting paths are highly weighted short paths

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What is ß?

• ß reflects the degree to which an individuals
  status is a function of statuses of those to whom
  he or she is connected
• If ß is positive, then it is traditional
  centrality measure like set out by Freeman
  (example is a communications network). Status is
  increased by contact with others with high
  status.
• If ß is negative, then each units status is
  reduced by higher status of those it is connected
  to (example being an exchange network). Power
  comes from being connected to the powerless in
  bargaining situations (lack of competition)
• ß essentially is the distinction that Cook et al
  made between positive and negative connected
  networks
• ßs magnitude measures the degree to which
  distant ties are measured

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A Second Look at the Experiments

• Bonacich has this function and a measure of
  centrality is next proposed in which a units
  centrality is its summed connections to others,
  weighted by their centralities.
• Using his function, he recreates experiments done
  by Cook et al and shows that it better predicts
  results than even Cook et al did in their
  original paper

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Conclusion

• Some might criticize c(a,ß) as too ambiguous
  since it gives very different centralities
  depending on value of ß
• But highlights both difference between and
  connected networks and depending on whether
  global or local ties are to weighted more. The
  function c(a,ß) attempts to capture all of this
  information.

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Fernandez Gould Brokerage Power

• Uses a study of National Health Policy network to
  examine relationship between brokerage positions
  influence
• Concludes that power of government orgs depends
  on their ability to link disparate actors in a
  communications network while staying policy
  neutral

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What is Brokerage?

• Defined as a relation in which one actor
  mediates the flow of resources or information
  between two other actors who are not directly
  linked
• 5 Types
• Liaison all 3 actors in separate groups
• Representative member of subgroup communicates
  with outsiders
• Gatekeeper member screens or gathers resource
  from outside and distributes internally
• Itinerant or Cosmopolitan a third member acts
  as intermediary to two members inside org
• Coordinator all 3 actors in same group

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Why Indirect Links Matter

• The number, diversity, and dispersion of actors
  makes it nearly impossible to maintain regular
  communication ties with all other orgs
• Actors in brokerage positions may bring together
  other actors who normally have no need to
  interact except in an unexpected relation to a
  given policy of shared interest
• This allows information and resources to flow
  easily across many diverse actors

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Findings of the Study

• Organizations that occupy one brokerage area are
  likely to occupy other types as well
• Total brokerage is highly correlated to liaison
  and cosmopolitan types, which shows that the
  broker tends to be outside of groups
• They find that their predictions were correct
• Gov orgs cannot have power in liaison and
  cosmopolitan brokerage positions unless they are
  seen as impartial on policy (an honest broker)
• With gov actors, gatekeeper and representative
  brokerage power is actually increased by taking
  stands on issues

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Findings Brokerage Centrality

• The partial brokerage scores used in the study
  measure control of communication paths, and
  therefore is a variant of Freemans betweenness
  centrality
• They demonstrate that cosmopolitan and liaison
  brokerage are systematically different from
  centrality by examining the study with centrality
  measures, getting different results.
• So while brokerage and centrality may both relate
  to power, this does not mean that brokerage and
  centrality completely relate to one another