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.

4.8

• How to define the flying teams?

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)
• 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

• 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
• 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 Scotland.net 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