MCSN Tuesday, 1-Nov-11

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Quiz #2 take 2

  • A pedagogical success. Nearly everyone did much better than before.
  • page 1 - bravo! (perhaps one or two arithmetic mistakes, or creating a semiwalk that was also a semipath...)
  • page 2, also bravo, mostly - but a few common misconceptions remain:
    • components can't overlap (because they're maximal)
    • cores: can't be determined from degree. For one thing, a vertex in the 4-core has to be connected to at least 4 others in the 4-core (by definition!). Therefore the smallest 4-core will have 5 vertices. Some people indicated a single node as belonging to the 4-core.
    • cliques: are defined to be maximal. So a triad isn't necessarily a clique, though if it's not a clique on its own it must be part of a larger clique. Note also that a square is not a clique unless it contains its diagonals.


  • How to define the flying teams?

Chapter 5: Affiliation networks


Basic ideas

  • People affiliate to groups (often defined by space, like the University of Alberta), and events (typically defined by space-time, like this class session), whether by choice or circumstance.
  • Such affiliations define bipartite networks comprising two kinds of vertex, which we can call actors and events (don't be confused - events could be more like groups).
  • In a bipartite network there are two kinds of vertex, type A and type B. All lines connect a type A vertex to a type B vertex - there are no direct connections between vertices of type A, nor are there direct connections between vertices of type B.
  • A bipartite network is also called "two mode", since there are two kinds of vertex, and is represented by a matrix rectangle rather than a square (see this in Excel)
  • Affiliations define social circles which overlap.
  • Network representation of identity as a model for social belonging:
    • Culture model (common in traditional ethnomusicology): each individual belongs to one "complex whole" as Tylor put it in 1847.
    • Identity model (more common in sociology and contemporary ethnomusicology): each individual associates with multiple "simple parts", each person in a slightly different way. These "parts" can be viewed as social circles whose intersection is the individual.
    • Note: social identity can't be captured in a single Pajek partition....why? The concept of partition is closer to the traditional "culture" model of exclusive all-encompassing identities.
  • Social circles may also imply power circles with critical implications for relationships among "events" (groups). Example: Interlocking directorates
  • Degree of a vertex indicates the scope of the corresponding social circle:
    • Degree of an event: size of the event
    • Degree of an actor: rate of participation of the actor

Typical assumptions about affiliation networks

  • Book states them as facts (see p. 101), but you should critique them in theory! test them in your projects!
  1. Affiliations are institutional or structural - less personal than friendships or sentiments. [What do you think? How could we test this?]
  2. "Although membership lists do not tell us exactly which people interact, communicate, and like each other, we may assume that there is a fair chance that they will." [what factors might impact the chances of actual dyadic interaction?]
  3. Actors at the intersection of multiple social circles...
    1. tend to interact even more
    2. enable indirect communication/control between the circles as a whole.
  4. "Joint membership in a social circle often entails similarities in other social domains." (i.e. homophily principle...Cause or effect?)

Matrix Representations

  • One-mode networks are naturally represented using
    • upper triangular matrix, no diagonal (undirected simple)
    • upper triangular matrix (undirected with loops)
    • square matrix (directed with loops)
  • Two-mode networks are naturally represented using rectangular matrices
    • Rows represent first mode (e.g. actors)
    • Columns represent second mode (e.g. events)
  • Deriving one-mode network from two-mode network.
    • Mapping the "hidden networks" implied by two-mode network (under assumptions above) can be highly significant
    • One-mode network derived from rows (e.g. actors)
    • One-mode network derived from columns (e.g. events)
  • Representing two-mode networks with lists of edges
    • Simply listing edges may violate condition that actors can't link to actors, or events to events
    • Thus we must also provide a means of identifying which vertices are rows (or, conversely, which vertices are columns)

Applications: creating and manipulating two mode networks

  • Two-mode network in Pajek
    • Vertex command is followed by two numbers: (a) the number of vertices; (b) the number of rows (whether actors or events)
    • When Pajek sees two numbers instead of one, it generates an affiliation partition to match.
  • Using txt2pajek to generate a sample two-mode network

Corporate interlocks in Scotland, 1904-5

  • Early 20th century: joint stock companies began to form
    • owned by shareholders
    • represented by boards of directors
  • Interlocking directorates linked the companies (and companies linked the directors)
  • Data: 136 multiple directors for 108 largest joint stock companies, of various types:
    • non-financial firms (64)
    • banks (8)
    • insurance companies (14)
    • investment and property companies (22)
  • Partition: indicates industry type
  1. oil & mining
  2. railway
  3. engineering & steel
  4. electricity & chemicals
  5. domestic products
  6. banks
  7. insurance
  8. investment
  • Vector: indicates total capital in 1,000 pounds sterling

Analyzing Scotland.paj

Two mode net

  • Info->network
    • Number of vertices
    • Number of lines
  • Affiliation partition separates firms and directors (examine)
  • Drawing and energizing. Note bipartite property.
  • Degree partition (size of events and rates of participation), can be displayed as vertex size (convert to vector)
  • Components

One mode nets

  • Derived networks: Each two-mode network induces two one-mode networks: (a) by events (groups), (b) by actors, as follows:
    • By events (groups): events are linked by one line per shared actor
    • By actors: actors are linked by one line per shared event (group)
    • Note: loops represent size of events, participation rates of actors:
      • each event (group) shares each actor with itself, so each actor induces a loop for every event in which it participates
      • each actor shares each event (group) with itself, so each event induces a loop for every actor participating in it
  • Derived networks are typically not simple, but one can replace multiple lines by a single line with value = number of lines replaced. This value is called line multiplicity and the resulting network is called a valued network.
  • We can convert into one-mode network of firms (no loops, no multiple lines).
    • View line values (info->network->line values)
    • Add degree information from the original network (create a degree partition, then extract using the affiliation partition)
    • m-slices
      • display 2-slice
      • valued core