MCSN Tuesday, 8-Nov-11
Contents
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!
- Affiliations are institutional or structural - less personal than friendships or sentiments. [What do you think? How could we test this?]
- "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?]
- Actors at the intersection of multiple social circles...
- tend to interact even more
- become bridges enabling indirect communication/control between the circles as a whole.
- "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
- oil & mining
- railway
- engineering & steel
- electricity & chemicals
- domestic products
- banks
- insurance
- 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.
- 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 creating a degree partition, 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.
- The slices can thus be represented with contour lines (as in cartography)
- 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
- 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)
- Energize with line values = similarities
- Let's add degree information from the original network
- create a degree partition for the full 2-mode network
- extract a degree partition for firms by setting the degree partition as #1, and using the affiliation partition as #2
- Derive the 2-slice directly
- delete lines of multiplicity under 2
- run degree partition
- delete class 0
- examine
- construct components and reexamine
- 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
- Energize in 2D, then apply Layers command in Draw menu