Affiliation Networks

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Affiliation Networks

• Jody Schmid and Anna Ryan
• 10/25/07

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• Traditional social science studies look at the
  attributes of individuals (monadic attributes).
• Network analysis studies the attributes of pairs
  of individuals (dyadic attributes).

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• Affiliation networks are a special kind of two
  mode social network that
• Look at the affiliation of a set of actors with a
  set of social occasions (or events).
• Look at collections of actors or subsets of
  actors (versus ties between pairs of actors)
• View connections among members of one of the
  modes as based on linkages established through
  the second mode

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Affiliation networks are relational in three ways

• They show how actors and events are related
• They show how events create ties among actors
• They show how actors create ties among events

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Definitions

• Events social clubs, treaty organizations,
  boards of directors, etc. Events do not need to
  consist of face-to-face interactions among actors
  at a physical location and a particular point in
  time.
• Co-membership or Co-attendance focus on the
  ties between actors (co-membership relation).
• Overlapping or Interlocking events focus on the
  ties between events (overlapping relation).

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Background, Applications and Rationale

• Affiliation networks recognize the importance of
  individuals memberships in collectivities.
• Actors are brought together through their joint
  participation in social events. These events
  provide them with opportunities to interact and
  increase the likelihood that they will form
  pairwise ties.
• When an actor or actors participate in more than
  one event, a link is established between events
  and may allow the flow of information between
  groups, and the coordination of the groups
  actions.

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• Affiliations of actors with events are a direct
  linkage between actors through memberships in
  events, or between events through common
  memberships. Example Sonquist and Koenig
  (1975)
• Affiliations provide conditions that facilitate
  the formation of pairwise ties between actors.
• Example Kadushin (1966)
• Feld (1981)
• Affiliations enable us to model the relationships
  between actors and events as a whole system.

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Methods

• There are three methods for studying actors and
  events simultaneously
• Affiliation network
• Bipartite graph/Sociomatrix
• Hypergraph
• Each contains exactly the same information, and,
  as a result, any one can be derived from the
  other.

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Affiliation network matrix

• An affiliation network matrix is the most
  straightforward.
• It records the affiliation of each actor with
  each event.
• Each row of A describes an actors affiliation
  with the events and each column of A describes
  the membership of the event.

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The actors are the children and the events are
the birthday parties they attended. If a row
equals zero, then a child attended no events.
If a column equals zero, then the event had not
actors affiliated with it.
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Bipartite Graph

• Partitions the nodes into two subsets. Since
  there are g actors and h events, there are g
  h nodes.
• The lines on the graph represent is affiliated
  with from the perspective of the actor and has
  as a member from the perspective of the event.
• No two actors are adjacent and no two events are
  adjacent. If pairs of actors are reachable, it
  is only via paths containing one or more events.
  Similarly, if pairs of events are reachable, it
  is only via paths containing one or more actors.

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• The lines on the graph represent
• is affiliated with from the perspective of the
  actor
• has a member from the perspective of the event.

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Advantages and Disadvantages

• Advantages they highlight the connectivity in
  the network, as well as the indirect chains of
  connection. In addition, data is not lost. We
  always know which individuals attended which
  events.
• Disadvantage they can be unwieldy when used to
  depict larger affiliation networks.

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The bipartite graph can also be represented as a
sociomatrix.
g 6 children h 3 parties 6 3 9
rows and 9 columns The sociomatrix is the most
efficient way to present information and is
useful for data analytic purposes.
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The bipartite graph can also be represented as a
sociomatrix.
g 6 children h 3 parties 6 3 9 rows and 9
columns The sociomatrix is the most efficient
way to present information and is useful for data
analytic purposes.
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Advantages and Disadvantages

• Advantage it allows the network to be examined
  from the perspective of an individual actor or an
  individual event because the actors affiliations
  and the events members are directly listed.
• Disadvantage it can be unwieldy when used to
  depict large affiliation networks.

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Hypergraph

• Looks at affiliation networks as collections of
  subsets of entities in which each event describes
  the subset of actors affiliated with it and each
  actor describes the subset of events to which it
  belongs.

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Advantage allows the network to be examined from
the perspective of an individual actor or an
individual event because the actors affiliations
and the events members are directly listed.
Disadvantage it can be unwieldy when used to
depict large affiliation networks. Hypergraphs
have been used to describe urban structures and
participation in voluntary organizations.
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Properties of One-mode Networks

• Centrality and centralization
• Density
• Reachability, connectedness, and diameter
• Cohesive subsets of pairs of actors

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Centrality and Centralization

• Centrality addresses the different aspects of
  importance or visibility of actors within a
  network.
• Centralization measures the extent to which a
  particular network has a highly central actor
  around which highly peripheral actors collect.

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Centrality

• In one-mode dyadic networks, actors are central
  if
• They are active in the network (motivating degree
  centrality).
• They can contact others through efficient (short)
  paths (motivating closeness centrality).
• They have the potential to mediate flows of
  resources or information between other actors
  (motivating betweenness centrality).
• They have ties to other actors that are
  themselves central (motivating eigenvector
  centrality).

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Issues With Centrality In Affiliation Networks

• Affiliation networks are non-dyadic. The
  centrality of an actor should be a function of
  the collection of events to which it belongs and
  the centrality of an event should be a function
  of the centrality of its collection of members.
• The linkages between events created by actors
  multiple affiliations, and between actors created
  by events collections of members are important.
  Actors are always between events and events are
  always between actors. Therefore, some form of
  betweenness centrality is appropriate for
  studying affiliation networks.
• This leads to distinctions between primary and
  secondary actors, where secondary actors are more
  likely to participate in events where primary
  actors are present. Primary actors, on the other
  hand, participate regardless of the participation
  of secondary actors. In terms of centrality,
  central actors participate regardless of the
  participation of less central actors. The
  reverse is not true.

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Measures of Centrality

• Degree
• Closeness
• Betweenness
• Eigenvector
• Flow-betweeness
• Degree and eigenvector centrality are the most
  commonly used to study centrality in affiliation
  networks.

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Degree

• Actors are important because of their level of
  activity or their number of contacts.
• Events are important because of the size of their
  membership.
• Criticism of degree centrality it does not
  consider the centrality of the actors or events
  to which an actor or event is adjacent. Two
  actors may be adjacent to the same number of
  others, but an actor is more central if it has
  ties to actors that are central.

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Closeness

• Is based on the geodesic distances between nodes
  in a graph. Geodisic distance is defined as the
  length of the shortest path linking two nodes.
  It is not applicable to valued relations.
• The closeness centrality of an actor is a
  function of the minimum distances to its events,
  and the closeness centrality of an event is a
  function of the minimum distances to actors.

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Betweeness

• Focuses on whether actors sit on geodesic paths
  between other pairs of actors.
• Unlike closeness, betweenness centrality has a
  built-in sense of exclusivity or
  competitiveness.
• Betweenness centrality of an event increases to
  the extent that its members belong to no other
  events or pairs of actors share only one event in
  common.
• This also holds true for actors. An actor gains
  betweenness points if it is the only member of an
  event and for all pairs of events to which it
  belongs.

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Eigenvector

• The centrality of an actor should be proportional
  to the strength of the actors ties to other
  network members and the centrality of these other
  actors.
• Eigenvector centrality is a weighted degree
  measure in which the centrality of a node is
  proportional to the sum of centralities of the
  nodes it is adjacent to.
• One criticism of eigenvector centrality is that
  it is affected by the differences in the sizes of
  events. Two approaches deal with this problem
  (a) standardize the event overlap measure (2)
  remove from the centrality index that component
  which is due to the degree (size) of the event.

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Flow Betweeness

• Flow betweenness is applicable to valued
  relations.
• For a pair of actors, the value of the relation
  might be their amount of interaction and the
  range of different settings in which they
  interact.
• Flow centrality considers all paths between
  nodes, not just geodesics.
• It is appropriate for both graphs and valued
  graphs.

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Density, Reachability, Connectedness, and
Diameter

• Affiliation networks are important because
  affiliations create connections between actors,
  through membership in shared events, and between
  events, through shared members. Because ties
  between actors or between events are potential
  conduits of information, the connectedness of
  the affiliation network is important.
• We can determine if an affiliation network is
  connected by looking at whether each pair of
  actors and/or events is joined by some path, as
  well as by the diameter. Considering the valued
  relations allows us to study cohesive subgroups
  of actors or of events.

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Density

• Density is a function of the pairwise ties
  between actors or between events.
• The number of overlap of events is, in part, a
  function of the number of events to which actors
  belong.
• The number of co-membership ties is, in part, a
  function of the size of events. An actor only
  creates ties between events if it belongs to both
  or multiple events.
• For a dichotomous relation, density is the
  proportion of ties that are present.
• For a valued relation, density isthe average
  value of the ties.
• Example McPherson and Smith-Lovin (1982) found
  that, because men typically belong to larger
  organizations then women, men have the potential
  to establish more useful network contacts.

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Reachability

• Reachability can be studied using a bipartite
  graph, with both actors and events represented as
  nodes.
• In a bipartite graph, no two actors are adjacent
  and no two events are adjacent. If pairs of
  actors are reachable, it is only via paths
  containing one or more events. Similarly, if
  pairs of events are reachable, it is only via
  paths containing one or more actors.

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Reachability, connectedness and diameter
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Diameter

• The diameter of an affiliation network is the
  length of the longest path between any pair of
  actors and/or events.

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• Connectedness and reachability can also be
  studied through the affiliation matrix and
  sociomatrices.
• An affiliation network that is connected in the
  graph of co-memberships among actors is
  necessarily connected in the graph of overlaps
  among events (if no event is empty). This holds
  true in the reverse.

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Example The affiliation network of six children
and six birthday parties

• Connected there exist paths between all pairs
  of children, all pairs of parties, and all pairs
  of children and parties. All children attended
  at least one party, all parties contained at
  least one child, and all children attended at
  least one party with Ross. As a result, all
  children are reachable to/from Ross and all
  parties are reachable to/from Ross. Although the
  paths between pairs of children and/or parties do
  not need to contain Ross, it is possible to reach
  any child or party through paths that do include
  Ross. Note a connected affiliation network
  does not need to contain an actor who is
  affiliated with all events.
• Diameter All pairs of parties in the network
  are reachable through paths of length 2 or less.
  This is not true of all pairs of actors.
  Example the shortest path (geodesic) from Drew
  to Keith is Drew, Party 2, Ross, Party 3, Keith
  (four linesthe longest geodesic length 4, the
  diameter of this affiliation network is equal to
  4).

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Cohesive subsets of actors or events

• A clique is a maximal complete subgraph of three
  or more nodes. In a valued graph, a clique at
  level c is a maximal complete subgraph of three
  or more nodes, all of which are adjacent at level
  c. In other words, all pairs of nodes have
  lines between them with values greater than or
  equal to c. We can locate more cohesive
  subgroups by increasing the value of c.
• Actors in the co-membership relation a clique
  at level c is a subgraph in which all pairs of
  actors share memberships in no fewer than c
  events.
• Events in the overlap relation a clique at
  level c is a subgraph in which all pairs of
  events share at least c members.

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Taking Account of Subgroup Size

• Both the co-membership relation for actors and
  the overlap relation for events in one-node
  networks that are derived from an affiliation
  network are based on frequency counts.
• As a result, the frequency of co-memberships for
  a pair of actors can be large if both actors are
  affiliated with many events, regardless of
  whether or not these actors are attracted to
  each other. This is also true for events in that
  the overlap between events may be large because
  they include many members even if they do not
  appeal to the same kinds of actors.
• Some authors argue that it is important to
  standardize or normalize the frequencies to
  study the pattern of interactions.

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• Odds ratio One measure of event overlap that is
  not dependent on the size of events is the odds
  ration. If the odds ration is greater than 1,
  then actors in one event tend to also be in the
  other, and vice versa. If it is less than 1,
  then they do not tend to be in the same events.
  It is also possible to take the natural logarithm
  of the equation, but this is not recommended when
  g is small.
• Bonacich (1972) proposed a measure, which is
  analogous to the number of actors who would
  belong to both events, if all events had the same
  number of members and non-members. He creates
  correlation coefficients, and calculates the
  centrality of the events based on their overlap.
• Faust and Romney (1985) normalize the matrix
  for actors and events so that all row and column
  totals are equal. This is equivalent to allowing
  all actors to have the same number of
  co-memberships or all events to have the same
  number of overlaps.

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Simultaneous Analysis of Actors and Events

• Network data typically take the form of a square
  binary adjacent matrix where each row and column
  represents a social actor. Graphical models
  permit the visualization of networks and call
  attention to structural properties that may not
  be apparent otherwise.
• One downside of graphical models is that, in
  recording linkages that unite pairs of actors, we
  lose the ability to distinguish between patterns
  of ties that link pairs and those that link
  larger collections of actors. As a result, we
  limit our ability to uncover potentially
  important structural features of social linkage
  patterns.
• Example friendship ties between friends. One
  set of friends has alternated friendships so that
  A and B were initially friends, then B and C, and
  then A and C. Another set of friends has always
  been a tight threesome. The multiple
  relation in the second group has a different
  structural form than the first group in ways that
  have important consequences for the behaviors of
  the individuals involved.

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• We need information not just about the social
  relationships that link pairs of actors, but
  about how actors are linked together into
  collectivities of any size, or what Wasserman and
  Faust (1993) call two mode network data.
• Two-mode network data embody a structural
  duality, or they can be studied from the
  perspectives of either the actors or the events
  (events can be described as collections of
  actors affiliated with them and actors can be
  described as collections of events with which
  they are affiliated). In other words, we can
  study the ties between actors, the ties between
  events, or both.

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Issues

• The representation of two-mode data should
  facilitate the visualization of three kinds of
  patterning
• the actor-event structure
• the actor-actor structure
• the event-event structure
• Simplicial complexes, hypergraphs, and bipartite
  graphs are useful, but do not display all three
  kinds of relations both clearly and in a single
  model that facilitates visualization.
• Simplicial complexes and hypergraphs provide two
  images one shows how actors are linked to each
  other in terms of events and the other how events
  are linked in terms of their actors. However,
  neither image provides an overall picture of the
  total actor-actor, event-event, and actor-event
  structure.
• Bipartite graphs provide a single-image for two
  mode data, but only display the actor-event
  structure. They do not provide a clear image of
  the linkages among actors or among events.

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ExtensionsGalois Lattices

• Galois lattices, on the other hand, meet all
  three requirements in a clear, visual model.

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Shortcomings of Galois Lattices

• The visual display may become complex as the
  number of actors and/or events becomes large.
• There is no unique best visual. The vertical
  dimension represents degrees of subset inclusion
  relationships among points, but the horizontal
  dimension is arbitrary. As a result,
  constructing good measures is somewhat of an
  art.

Unlike a graph which uses properties and
concepts from graph theory to analyze a network,
these properties of Galois lattices are not well
developed.
















































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ExtensionsCorrespondence Analysis

• Correspondence analysis is a data analysis
  technique for studying the correlations among two
  or more sets of variables. It represents two
  theoretical concepts
• Simmels observation that an individuals social
  identity is defined by the collectivities to
  which he or she belongs. Reciprocal averaging is
  a formal translation of this insight
  (geometrically, an actors location in space is
  determined by the location of the events with
  which he or she is affiliated).
• The duality of relationships between actors and
  events is captured by the fact that actors can be
  viewed as located within the space defined by the
  events and events can be viewed as located within
  the space defined by actors.

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• Correspondence analysis is a method for
  representing both the rows and columns of a
  two-mode matrix, and results in a map in which
• Points representing the people are placed
  together if they attended mostly the same events.
• Points representing the events are placed close
  together if they were attended by mostly the same
  people.
• People-points and event-points are placed close
  together if those people attended those events.
• Correspondence analysis includes an adjustment
  for marginal effects. As a result, people are
  placed close to events to the extent that (a)
  these events were attended by few other people
  and (b) those people attended few other events.

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Advantages and Disadvantages

• Advantages it allows the researcher to study the
  correlation between the scores for the rows and
  the columns. Using reciprocal averaging, a
  score for a given row is the weighted average of
  the scores for the columns, where the weights are
  the relative frequencies of the cells. These
  averages are graphed.
• Disadvantages The data values have a limited
  range (0 or 1). As a result, they are difficult
  to fit using a continuous distance model of low
  dimensionality. Two-dimensional maps are almost
  always severely inaccurate and misleading.
• It is designed to model frequency data. The
  numbers do not represent distances and there is
  no way on a two-dimensional map to determine who
  attended what events.
• Distances are not Euclidean, yet human users
  often interpret them that way.

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