Katie Anderson

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Week 8 Analyzing and Representing Two-Mode
Network Data

• Katie Anderson
• Geunpil Ryu
• Djoko Sigit Sayogo

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Ch. 8 Affiliations and Overlapping Subgroups

• Stanley Wasserman
• Katherine Faust

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Affiliation Networks

• Represents the affiliation of a set of actors
  with a set of social occasions or events.
• Affiliation matrix, A aij

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Affiliation Networks

• Two-mode networks Set of actors and set of
  events
• May analyze one-mode or two-mode networks
• Consists of subsets, rather than pairs
• Studies the dual perspectives of actors and
  events
• Due to these unique qualities, affiliation
  networks require special methods and
  interpretations.

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Theory

• Simmel (1950,1955) Social theorist
• Multiple group affiliations are fundamental in
  defining the social identity of individuals.
• Actors are brought together their joint
  participation in events.
• This interaction increases odds that direct
  pairwise ties will develop between actors.
• A linkage is also established between the two
  events when a person(s) participates in both.

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Notations

• Two-modes
• Set of actors N n1, n2, ng
• Set of events M m1, m2, mg

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Duality of Affiliation Networks

• The two perspectives
• Actors linked to one another by their affiliation
  with events
• Events linked by the actors who are their members
• Thus, two ways to view an affiliation network
• Actors linked by events or,
• Events linked by actors

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Alternate Representations

• Two-mode sociomatrix
• Bipartite graph
• Hypergraph
• All contain same information and may be derived
  from one another.

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Affiliation Network Matrix
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• Allison
• Drew Party 1
• Eliot
• Party 2
• Keith
• Ross Party 3
• Sarah
• Set of Actors Set of Events
• A bipartite graph

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Sociomatrix for the bipartite graph
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Subsets
M1 n1, n5, n6 M2 n2, n3, n5, n6 M3 n1,
n3, n4, n5 N1 m1, m3 N2 m2 N3 m2,
m3 N4 m1, m2, m3 N5 m1, m2
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Hypergraph
H (N, M)

• Sarah
• Drew
• Party 1 Ross Party 2
• Allison Eliot
• Keith
• Party 3

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Dual Hypergraph
H (N, M)

• Drew
• Party 2
• Ross Eliot
• Sarah
• Party 1 Party 3
• Allison Keith

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One-mode networks

• Ties between pairs of entities derived from the
  affiliation matrix, A, by processing the
  affiliation network data.
• Look at linkages implied by the second mode to
  get ties between pairs of entities in one mode
• Nondirectional
• Valued

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One-mode networks

• Actor co-membership
• Number of times two actors have 1s in the same
  columns gives the number of events they have in
  common.
• XN AA
• Event overlap
• Number of actors who are affiliated with each
  pair of events.
• XM AA

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Affiliation Network Matrix
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Actor co-membership matrix
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Event overlap matrix
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Properties of Affiliation networks

• Rates of Participation Number of events with
  which each actor is affiliated row totals of A
  of the entries on the main diagonal of XN
• Size of Events Column totals of A or the entries
  on the main diagonal of XM
• Equal to the degree of the node representing the
  event in the bipartite graph.

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Properties of one-mode networks

• Density
• Reachability
• Connectedness
• Diameter

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One-mode Interpretation

• Use caution with inferences
• For example, with a co-membership relation
• No information about which events were attended,
  the characteristics of the events, or about the
  identities of the other actors who attended the
  events

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Two-mode Analysis

• Least developed methods.
• Analyzes how the actors are linked to the events
  they attend and how the events are related to the
  actors who attend them.
• Galois Lattices
• Correspondence Analysis

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Introductory Overview in the Organizational
State

• Laumann Knoke (1987)

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State Policy Making Approaches

• Instrumentalists argue that state policies are
  determined by the class interests of capitalists
  and their agents.
• Power elites domain state policies
• State policies are shaped by the interest groups
  from a view of pluralist political science
• Corporatism which stress that groups play in a
  societal corporatist system gives special
  attention to organized interests and their
  relationship with the state
• From the managerial-elite perspective, the state
  is an autonomous social formation whose
  strategies emerge from the basic organizational
  imperatives coping with environmental
  uncertainties, resource scarcities, and
  socio-legal constraints

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Characteristics of State Policy

• Laumann Knoke assume that there are corporate
  entities which participate in policy decision and
  there are structural arrangement among those
  corporate entities
• State policies are the product of complex
  interactions among government and nongovernment
  organizations, each seeking to influence the
  collectively binding decisions for their own
  interests.
• For better understanding of shaping state policy,
  we need to incorporate actors behavior on a
  particular policy as well as relationship among
  actors into policy study

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The Necessity of Studying Event Structures

• Sociologists have focused almost on actors,
  relations among actors since they believe that
  events are consequences for the ways actors
  behave
• However, they neglect the characteristics of the
  events in which actors are active. For example,
  a crucial event might change a whole relation
  structure among actors.
• Moreover, some institutionalists represents steps
  toward a theory of the structure of events,
  but many of them are difficult to be
  operationalized empirically.
• Then, how can we incorporate events and their
  structure into actors relationship?

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Policy Event Actors Participation

• Institutionalist theories of government
  functioning provide at least one significant
  basis for identifying key events to study and for
  anticipating the linkages among them.
• To understand how national policy shape, one must
  take into account how organizations perceive and
  response to an opportunity structure for
  affecting policy outcomes that is created by the
  temporal sequence of policy-relevant events.
• If a specific policy event is embedded in the
  context of other antecedent, concurrent, and
  impending events, one must incorporate the entire
  structure of networks and events into policy
  study.

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Level of Analysis

• Quadrant A assumes that individual actor and
  event are considered to be independent from their
  social environment
• Quadrant B assumes that institutional structure
  within which events occur have an impact on
  individual choice
• Quadrant C assumes that networking structure of
  actors affect a particular event
• Quadrant D assumes that framework of the
  relationship between network structure and event
  structure

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Structuration of Policy Action Systems

• Submatrix A reveals a underlying structure among
  events
• Submatrix B illustrates relationship between
  events and actors
• Submatrix C defines the interrelationships among
  actors

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Examples of Different Network and Event
Structures
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Network Analysis of 2 Mode Data

• Borgatti Everett (1997)

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Visual Representation of 2 Mode Data
1. Correspondence Analysis

• Have a severely limited range (all zeros and
  ones)
• The distances are not euclidean

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Visual Representation of 2 Mode Data
2. Bipartite Graph

• No spatial positioning
• Difficult to get a sense for the structure of
  relationship

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Visual Representation of 2 Mode Data
3. Compromise Approach and MDS

• Using a minimizing function in terms of the
  number of crossed lines for better visualization
• Using concept of Euclidean distance

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Density Problem of 2 Mode Data

• Regular Density equation is not appropriate for
  2-mode data since no ties are possible within
  vertex sets (i.e., set of actors and set of
  events).

g Size of Actors ni Size of Actor Set n0 Size
of Event Set
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Centrality Problem of 2 Mode Data

• Nonlinear normalization can influence of rank
  order of the centralities

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Centralization Problem of 2 Mode Data

• With 2-mode data represented as a bipartite graph
  we could simply apply the centralization methods
  directly. However, we come across a problem of
  interpretation since we assume the two modes are
  fixed in size.
• For degree centralization, the denominator
  requires a new function if nonlinear
  normalization makes sense (n0ni-1)-(n0-1)/ni-
  (nin0-1)/n0

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Subgroup Problem of 2 Mode Data

• Bipartite graphs contain no clique, however they
  have too many 2-cliques and 2-clans
• When strong bridging node like e8 and e9 exist,
  e8 and e9 would be a core structure in a
  core/periphery structure.

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The Structure of Class Cohesion the Corporate
Network and its Dual

• Bearden Mintz (1987)

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Research Questions

• What types of individuals are the most crucial in
  organizing the business world?
• What is the relationship between institutional
  positions and class structure in modern
  capitalism?
• Using two mode data set corporate network vs.
  individual network

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Sampling

• Nonfinancial firms were sampled from ranked 200
  with sales of 1.5 billion in Dry Goods and
  Fortune Sales 500.
• Financial firms were sampled from Forbes
• 252 firms 200 largest nonfinancial firms 50
  largest financial firms 2 additional Firms
• 3,976 executive positions and the directors on
  the boards occupied by 3,108 people

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Corporate Network

• The corporate network is sparse 3 of all
  possible ties
• 90 of the firms are directly tied to at least
  one other firm
• Board members without executive position are
  responsible for the cohesion of the system
• Retired executives are the most important in
  solidifying the networks of corporate interlock
• Directors from smaller corporations are more
  likely to have elite connections and those
  directors have high tendency to sit on the board
  of major commercial banks as outsiders
• Elite status means membership in an elite
  social club or important policy planning group,
  attendance at an elite prep school.
• The boardrooms of the largest firms are major
  location for intersection of class and
  institutional interests while the system is held
  by outsiders representing two components of elite
  status and institutional positions

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Individual Network

• National, seminational, and regional components
  were found in the individual networks.
• National components are members of a national
  ruling class while the members of the regional
  components are local elites
• As regional component, the Chicago area in
  particular has been identified as a regional
  center with national ties.
• The national and seminational groups serve as
  mechanisms for unification
• The existence of national network suggests a
  vehicle for achieving consensus of their
  interests.
• Directors who are members of important policy
  planning organizations are more likely to
  maintain the types of interlock patterns which
  generate components than their non-policy group
  colleague.
• Directors with multiple positions or bank board
  members are the most likely candidates for
  component membership.

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Major Findings

• Parallel structures between the corporate and
  director networks.
• Bank board membership is overlapped between
  corporate network and private networks, and it
  was identified as the location in which class and
  international interests are most intertwined.
• Analyses of the corporate and director networks
  have provided complementary information in
  addressing the relationship between class and
  organization in the business world.
• Within the network of relations of directors the
  overlap between institutional and class interest
  were supported.
• There are two routes mixed together toward the
  highest level in the U.S. capitalism either
  membership of the largest corporations or social
  elite credentials combined with a somewhat
  smaller firm.

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The Duality of Persons and Groups

• Ronald L. Breiger

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Basic Assumption Concept

• Basic assumption ...the subsistence of any
  aspect of human life depends on the coexistence
  of diametrically opposed elements... 
• Basic Concept
• The value of tie between any two individuals is
  defined as the number of groups of which they
  both members
• The value of tie between any two groups is
  defined as the number of persons who belong to
  both.

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Equation

• In A(i x j)
• if we intersect any rows i and j
  person-to-person relation (P)
• if we intersect any column i and j
  group-to-group relations (G).

Eq. 3
Eq. 4
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Reflexivity

• The main diagonal in A(p x q) In P, the number
  of ties between person and himself the number
  of groups to which he belong In G, the number of
  ties between a group and itself the number of
  members
• ? of main diagonal P ? of main diagonal G
• The vector of row-marginals of A main diagonal
  of P the vector column-marginal of A main
  diagonal of G sum of row-marginal the sum of
  column-marginal
• If all the persons belonging to all groups are
  found to belong to any single group, then the
  lower bound (largest value cell on the main
  diagonal of G) is the actual number of persons
  if no group overlaps, the upper bound (sum of
  main diagonal of G) is the actual number of
  persons.

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Application for clique

• study by Davis et.al (1941)

• transpose unpermuted matrix A,
• by equation AT(A) create matrix G,
• delete column which contains no zero value
• re-create the modified translation of matrix
  person-to-group,
• using equation A(AT )create matrix P
  (person-to-person)

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Reachability

• Number of p x p ties of length n between every
  two person binarized P matrix raise to the nth
  power (Pn)
• number of n-paths between two groups is contained
  in the matrix G raise to the nth power (Gn)
• Thus
• Substituting with equation 3 and 4, we got proof
  of

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Reachability (cont.)

• From this we can determine the number of groups
  that a person can reach (and conversely the
  number of persons that a group can reach)
• However, this lengths of paths has natural limits

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Building Asymmetry into the Basic Approach

• Consider F(pxq) primary affiliation, A(pxq)
  all affiliation, S(pxq) secondary
  affiliation S A n F
• asymmetric ties exist from person i to person j
  if a group which is i primary affiliation is a
  secondary affiliation for j.

So
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Simultaneous Group and Individual Centralities

• Phillip Bonacich

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Basic assumption based on Breiger

• A central firm gets its central position from the
  membership patterns of its members. Dually, a
  central individual should be one who belongs to a
  variety of important firms.
• Let gj be the centrality of group j, then
• Vector centrality scores g is an eigenvector of S
  and ? is the largest eigenvalue of S.

(eq. 1) Or
(eq. 2)
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• Considering basic assumption of duality and P
  A(AT) and G (AT)A, hence
• Ag ?p and ATp ?g (eq. 34)
• AAT ?2p and ATA ?2g (eq. 5)
• Application to Bipartite Graph
• Same approach can be used in Bipartite graph,
  since bipartite has n m vertices and line
  aij1
• Since the vector of individual and group
  centrality has been standardize (the length n
  m) if all individual were equal in the
  centrality, all centrality scores 1. Thus, the
  value of 1.00 serves as a base line comparison.

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Effect of Size

• Problem strongly affected by size of groups and
  number of groups to which individual belongs.
• How to control conventional way is to modify ATA
  before using equation (5), by standardized it by
  dividing it with the geometric mean of their
  size.

So Centrality is
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Correspondence Analysis

• The goal
• create multidimensional space for vector
  coresponding to the row and column categories
  that describe the geometric distance between
  categories.
• Important output
• The most powerful and useful dimension for
  correspondence analysis describe similarities in
  row and column pattern.
• However, within correspondence analysis we cannot
  detect which dimension correspond to centrality.

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Correspondence Analysis (exp.)
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Centrality in Affiliation Networks

• Katherine Faust

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Centrality Motivations for Affiliation Networks

• Acknowledge centrality indices for both actors
  and events, dually.
• Relates each event to a subset of actors and
  relates each actors to a subset of events
• Actors create linkages between events and events
  create linkages between actors. Consider the
  overlapping memberships
• Subset-superset inclusions of actors and events.
  The distinction between 'primary' and 'secondary'
  actors.

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Centrality Degree Centrality

• main diagonal of the actor co-membership matrix
  XN (for actors) or the event overlap matrix X M
  (for events), or to the row total (for actors) or
  the column total (for events) of the affiliation
  matrix A.
• Critics it does not consider the centrality of
  the actors (events) to which an actor (event) is
  adjacent.

OR
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Eigenvector Centrality

• the centrality of an actor should be proportional
  to the strength of the actor's ties to other
  network members and the centrality of these other
  actors.
• Vector centrality scores c is an eigenvector of X
  and ? is the largest eigenvalue of X.
• Considering basic assumption of duality and P
  A(AT) and G (AT)A, hence

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Eigenvalue Centrality

• Use in Bipartite graph
• Use in corresponding analysis assign scores to
  the rows and columns. The CA scores for rows
  (actors) and columns (events) are related to each
  other through the following equations

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Eigenvalue centrality

• Critics
• eigenvector analysis is affected by size.
• Two approach to mitigate this
• standardize the event overlap measure prior to
  analysis,
• remove the component of centrality due to paths
  of length 1.

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Closeness Centrality

• Closeness for bipartite graph.
• In bipartite graph, the distances if from an
  actor in the graph as a function of the distances
  from the events to which it belongs.
• Thus, the distance from node i (actor) to any
  node j (either actor or event) is d(i,j) 1
  mink d(k,j), for event nodes k adjacent to i.

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Closeness centrality

• Closeness centrality of an actor in bipartite
  graph
• Closeness centrality of an actor as a function of
  the geodesic distances from its events to other
  actors and events

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Betweenness Centrality

• Concentrate in betweenness for bipartite graph.
• In affiliation network, linkages between pairs of
  actors are always through participation in events
  (events always on geodesics between actors, vice
  versa).
• A portion of the betweenness centrality of event
  mk can be expressed in terms of the number of
  co-memberships of pairs of its member
• An event gains betweenness centrality if
  contains non-central actors pairs of actors
  share only that event in common.

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Flow Betweenness Centrality

• acknowledging the value of the relation
• Flow betweenness extent betweenness centrality
  1) it consider all paths between nodes, 2)
  appropriate for both graph and valued graph.
• flow betweenness centrality of pi is defined as
  the total flow between all pairs of nodes that
  depends on node pi.
• Limitation 1) actor with single event can have
  non-zero flow betweenness. 2) actor with more
  events can have lower centrality than actor with
  less events.

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Galois Lattice
A Galois lattice represents the relation between
the sets N and M in terms of the ? and ? mappings
between subsets of entities from each set. A
Galois lattice is presented in a diagram in which
points represent entities or subsets of entities
from the two sets, and lines represent
subset-superset relations.

• Actors that are relatively low in the diagram are
  relatively more central (n5, followed by n1, n3,
  n6) . Similarly, events that are relatively
  high in the diagram are relatively more central
  (all events are closely related / equal
  centrally)

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Graph Cover

• The importance of an actor (or a subset of
  actors) in the kind and extent of information
  about affiliation that is available to an actor
  or to a subset of actors.
• Actors that covers a large number of other
  actors or a large portion of co-membership has
  more access.
• Wide cover if
• Strict cover (more general than wide cover) if

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Using Galois Lattice to Represent Network Data

• Linton C. Freeman
• Douglas R. White

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Objectives

• To present the graph visualization for two-mode
  network data.
• Since two mode represent duality structure, its
  visualization should facilitate
• actor-event structure,
• actor-actor structure,
• event-event structure.

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What is lattice

• When power set is partially ordered and every
  pair of elements has both meet and join
• Consider set of X 1,2,3, power set P(X)
  consist of Xi ? X, in this case (1,2,3), (1,2),
  (1,3), (2,3), (1), (2), (3), this order (Xi) X
  ?(Xi ? X), such that elements (1,2) and (2,3)
  has 2 meet, and 1,2,3 as their join.

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• Galois lattice is based on
• triple (A, E, I) defined by
• two-mode network
• A is actor, E is event, and
• I is binary relation
• The I relation can be used to define a mapping
• And Define a mapping
• The ?? are both constructed from the same pairs
  in relation I
• This mapping can be drawn into a graph

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• subset actor 1 mapped to subset events A, C, D,
  subset events A, C, D mapped to actor 1, thus

Move up is larger collection of actors, and move
down is larger collection of events In practice
the graph usually
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• Interpretation
• Event D contain event C
• any actor present in event D will
• always present in event C
• Actor 5 never attend at any event
• unless actor 6 and 4 were both there
• The attendance of actor 6 require the attendancy
  of actor 4

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• Example Davis, Gardner Gardner Data