MCSN Tuesday, 8-Nov-11

Schedule

• office hrs Wed Nov 9 from 2:30 to 3:15 (I can stay until just before 4. No office hours next week however...)
• no class Thursday Nov 10 (Remembrance Day)
• short class Tuesday Nov 15 (new drafts of proposals due; intro to chapter 6 and network game)
• self-guided class Thurs Nov 17
  • construct class' affiliation network and analyze it
  • play the network message passing game and estimate centrality.
• course evaluation on Tuesday Nov 22
• quiz on Thursday November 24 (to cover material up to that point)
• presentations on Tuesday November 29 (presentations: 10 minutes each)
• self-guided class Dec 1 (polish your music compositions for possible performance)
• class on Dec 6 (last class - more presentations)

Project discussions

for those who didn't talk about their projects last time, and would like class feedback, suggestions, discussion...(but briefly so we can wrap up chapter 5 today)

Chapter 5: Affiliation networks

Concepts

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).
• Affiliation networks are bipartite (or two-mode), but the reverse doesn't hold (e.g. links between keys and locks is bipartite, but not an affiliation network in the usual sense). An affiliation network is a special interpretation of a bipartite network.
• Affiliations define social circles (the term comes from sociologist George Simmel) which overlap.
• Network representation of identity as a model for social belonging:
  • Culture model (common in traditional ethnomusicology, and applicable small-scale societies): each individual belongs to one "complex whole" as Tylor put it in 1847. This "complex whole" might be identified as the sum of many social circles, but they heavily overlap: everyone in a small community belongs to many or most of them (work circle, religious circle, etc.) with diversions primarily on the basis of age and gender.
  • Identity model (more common in sociology and contemporary ethnomusicology, and more applicable to large-scale societies): individuality is the sum total of multiple "simple parts", each person summing them 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 (group): size of the event (group)
  • 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. become bridges enabling indirect communication/control between the circles as a whole.
4. "Joint membership in a social circle often comes with similarities in other social domains." (i.e. homophily principle..."birds of a feather flock together". Is this cause of common affiliation, or effect? Understanding the difference might require different methods: (a) temporal network analysis, or (b) qualitative (interview, observation) analysis.)

Representations via matrix or edge list

Matrices are useful conceptual tools, but Pajek relies on lists of edges.

• Matrices represent one or two mode networks very naturally
• 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)
• One can also represent two-mode networks with lists of edges
  • Actors and events must be clearly differentiated
  • Simply listing edges may violate condition that actors can't link to actors, or events to events
  • Thus it's necessary to provide a simple means of identifying which vertices are rows (or, conversely, which vertices are columns)
  • If we number the vertices, it's easiest to separate rows and columns by assuming that the first N vertices are rows (and so the rest are columns). This is the approach taken in Pajek...

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

• 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

Deriving one mode nets from affiliation 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 Scotland.net into one-mode network of firms (settings: no loops, no multiple lines).
  • Lines between firms now represent the number of shared directors.
  • 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
    • m-slice is derived by deleting all lines of multiplicity less than m, and then deleting all isolated vertices. Detect isolated vertices by running component analysis with minimum component size = 2, then deleting the zero cluster.
    • vertices of the m-slice are precisely those that are attached to at least one line of multiplicity m or more.
    • m-slices are therefore nested (like k-cores): a vertex in the 3-slice (m=3) is automatically in the 2-slice (m=2), and a line in the 3-slice is automatically in the 2-slice.
      • 1-slice contains the 2-slice
      • 2-slice contains the 3-slice
      • 3-slice contains the 4-slice
      • etc.
    • Note: m-slices need not be connected - so after finding an m-slice, run component analysis to find its "pieces"
  • valued core
    • useful when we want to derive a partition from values on incident lines
    • first we decide if we want to derive this value from the maximum or sum of incident lines
    • then we sort this maximum or sum into a single "bin" representing a range of values
    • finally we have our vertex partition
  • valued core for m-slices
    • note that the highest m-slice to which a vertex belongs is simply the highest incident line multiplicity
    • so we can use valued core on the "max" setting, with bins defined as one per multiplicity (this is the default)
    • vertices are now partitioned by m-slice, but in order to find a particular m-slice we need to both (a) extract clusters, and (b) delete lines with value under m.

Analyzing Scotland.paj

• Derive network of firms
  • Energize with line values = similarities
  • Derive the 2-slice directly, and look at its components
• Run valued core and examine results with "info" and "draw"
  • Derive the 2-slice again from valued core result
  • Use valued core partition to derive Industrial Categories partition and Capital vector.
  • Examine resulting sociogram
• 3D views
  • Note: SVG is no longer supported; VRML may work on PCs
  • Two techniques will work:
    • Energize in 2D, then apply Layers command in Draw menu
      • Draw network of firms, with valued core partition selected
      • Select Layers>Type of layout>3D
      • Run Layers>In Z direction
      • Energize in 3D and rotate or spin
    • Energize with Fruchterman Reingold 3D