Networks: Basic Concepts

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Networks Basic Concepts

• Centrality

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Networks Basic Concepts

• In this discussion, well outline some basic
  concepts of network analysis, focusing on
  centrality.
  
• Well also fold into this discussion an overview
  of UCINET. UCINET is a software program that is
  commonly used with network analysis. While it
  does not handle some of the more recent ways in
  which networks can be analyzed (such as
  longitudinal or cross-sectional ERGM methods), it
  is a very user friendly way to obtain
  network-related measures, and to create visual
  depictions of networks. UCINET offers a free 30
  day trial.
• Much of the discussion of UCINET was drawn from
  the tutorial created by Hanneman at Riverside.

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Basic concepts

• Recallvertices or nodes are the units or actors
  in a network (or a graph or a system).
  
• Edges are the ties or connections between nodes.
• And the ego is the node under considerationany
  particular node that you might be thinking of.

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Basic Concepts Centrality

• Centrality is a measure of how many connections
  one node has to other nodes.
  
• Degree centrality refers to the number of ties a
  node has to other nodes. Actors who have more
  ties may have multiple alternative ways and
  resources to reach goalsand thus be relatively
  advantaged.

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Basic Concepts Degree Centrality

• Degree centrality for an undirected graph is
  straightforwardif A is connected to B, then B is
  by definition connected to A.
  
• Degree centrality for a directed graph or network
  has one of two forms.
  

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Degree CentralityDirected Networks

• One is in-degree centrality An actor who
  receives many ties, they are characterized as
  prominent. The basic idea is that many actors
  seek to direct ties to themand so this may be
  regarded as a measure of importance.
• The other is out-degree centrality. Actors who
  have high out-degree centrality may be relatively
  able to exchange with others, or disperse
  information quickly to many others. (Recall the
  strength of weak ties argument.) So actors with
  high out-degree centrality are often
  characterized as influential.

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Degree Centrality Individual and Network

• Consider the network on the left. Which nodes
  (actors) are more central than others?
  
• 2, 5, and 7 appear relatively central.

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Degree Centrality (directed networks)

• So, node 7 has an in-degree centrality absolute
  value of 9 (there are 9 other nodes connected to
  node 7). The normalized value is 100 (all
  possible other nodes are connected to node 7).
  The out-degree centrality has an absolute value
  of 3 (node 7 is connected out to nodes 2, 4, and
  5), and a normalized value of 33.33 (3 nodes is
  33.33 of the possible 9 nodes to which node 7
  could extend out.)
• The average outdegree is 4.9 (which means that
  each node has, on average, connections out to 4.9
  other nodes) the average indegree is also 4.9.
  Normalized, both measures are 54.44 (that is, 4.9
  / 9).

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Centrality Network Degree Centralization

• One can also calculate network indegree and
  outdegree centralization. These network measures
  represent the degree of inequality or variance in
  our network as a percentage of that in a perfect
  star network the most unequal type of
  network.
• A depiction of a star network is on the next
  slidenote that only one node is connected to any
  of the others, and that node is connected to all
  of the others.

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Star Network
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Degree Centrality Bonacich

• Another measure of degree centrality takes into
  account the problem that the power and centrality
  of each node (actor) depends on the power and
  centrality of the others.
• Bonacich used an iterative estimation approach
  which weights each nodes centrality by the
  centrality of the other nodes to which it is
  connected.
• So, node 1s centrality depends not only on how
  many connections it hasbut also on how many
  connections its neighbors have (and on how many
  connections its neighbors neighbors have, and so
  on.)

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Degree Centrality Bonacich

• When calculating out the Bonacich Power measures,
  the attenuation factor represents the weightan
  attenuation factor that is positive (between 0
  and 1) means that ones power is enhanced by
  being connected to well-connected neighbors.
• Alternatively, one could argue that actors who
  are well-connected to individuals who are not
  well-connected themselves are powerful, because
  others are dependent on them. In this case,
  one would use an negative attenuation factor
  (between 0 and -1), to compute power accordingly.

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Degree Centrality Bonacich

• Recall the graph presented above, in which actors
  5 and 2 were the most central. Calculating out
  Bonacich measures suggests that actors 8 and 10
  are also centralthey dont have many
  connections, but they have the right
  connections.
• However, taking the second approach (using a
  negative attenuation factor) identifies actors 3,
  7, and 9 as being strong because they have weak
  neighbors (who are dependent on them).

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Degree Centrality Bonacich

• As with all quantitative methods, its important
  to think about what you as a researcher are
  trying to measure before using the methods. In
  your particular context, are actors connected
  with other well-connected actors the most
  powerful? Or is it actors that are connected
  with those who are very dependent on them who are
  more powerful?

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Centrality Closeness Centrality

• Closeness is a measure of the degree to which an
  individual is near all other individuals in a
  network. It is the inverse of the sum of the
  shortest distances between each node and every
  other node in the network.
• Closeness is the reciprocal of farness.
• Nearness can also be standardized by norming it
  against the minimum possible nearness for a graph
  of the same size and connection.

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Centrality Closeness Centrality

• Closeness can also be calculated as a measure of
  inequality in the distribution of distances
  across the actors.
  
• These measures rely on the sum of the geodesic
  distances from each actor to all the others.
  However, in complicated graphs, this can be
  misleading.
• An actor can be very close to a relatively closed
  subset of a networkor moderately close to every
  actor in a large networkand receive the same
  closeness score. In reality, the two are very
  different.

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Centrality Eigenvector Closeness

• The Eigenvector approach to measuring closeness
  uses a factor analytic procedure to discount
  closeness to small local subnetworks.
  

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Closeness Influence Measures

• Another way to think of closeness is to move away
  from thinking just about the geodesic or most
  efficient (shortest) path from one node to
  anotherbut to also think about all connections
  of ego (that is, the one node in question) to all
  the others.

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Closeness Influence Measures

• There are several such measures Hubbell, Katz,
  Taylor, Stephenson, and Zelen.
  
• Hubbell and Katz methods count the total number
  of connections between actors (and do not
  distinguish between directed and non-directed
  data), but use an attenuation factor to discount
  longer paths. The two measures are very similar
  the Katz measure uses an identity matrix (each
  node is connected to itself) while the Hubbell
  measure does not.

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Closeness Influence Measures

• The Taylor measure also uses an attenuation
  factor, but is more useful for measuring the
  balance of in- versus out-ties in directed
  graphs. Positive values of closeness indicate
  relatively more out-ties than in-ties.

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Centrality Actor Betweenness

• BetweennessBetweenness is a measure of the
  extent to which a node is connected to other
  nodes that are not connected to each other. Its
  a measure of the degree to which a node serves as
  a bridge.
• This measure can be calculated in absolute value,
  as well as in terms of a normed percentage of the
  maximum possible betweenness that an actor or
  node could have had.

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Centrality Edge Betweenness

• In addition to calculating betweenness measures
  for actors, we can also calculate betweenness
  measures for edges.
  
• Edge betweenness is the degree to which an edge
  makes other connections possible.
  
• Recall the Knoke example we used earlier, and
  look at the edge from 3 to 6.

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Centrality Edge Betweenness
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Centrality Edge Betweenness

• That edge from 3 to 6 makes many other edges
  possiblewithout that edge, 6 would be relatively
  isolated.
  

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Centrality Levels of Hierarchy

• One can also identify levels of hierarchy. If
  one eliminates all the actors with no betweenness
  (that is, the subordinates), some of the
  remaining actors will then have 0
  betweennessthey are at the second level of the
  hierarchy. We can continue to remove actors,
  and measure the of levels of hierarchy exist in
  the network or system.
• Note that the Knoke data presented above is not
  very hierarchical.

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Centrality Flow Betweenness

• What if two actors want to have a relationship,
  but the path between them is blocked by a
  reluctant intermediary? Another pathwayeven if
  it is longermeans another alternative /
  resource. The flow approach to centrality
  assumes that actors will use all the pathways
  that connect them. For each actor, the measure
  reflects the of times the actor is in a flow
  (any flow) between all other pairs of actors
  (generally, as a ratio of the total flow
  betweenness that does not involve the actor).

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Basic Concepts

• This has been an overview of various perspectives
  on centrality, largely drawn from the UCINET
  tutorial. The UCINET tutorial also has a number
  of very useful review questions.