Block Modeling

Overview

Social life can be described (at least in part) through social roles.

To the extent that roles can be characterized by regular interaction patterns, we can summarize roles through common relational patterns.

Social life as interconnected system of roles

Important feature: thinking of roles as connected in a role system = social structure

•Rights and obligations with respect to other people or classes of people

•Roles require a ‘role compliment’ another person who the role-occupant acts with respect to

Examples:Parent - child, Teacher - student, Lover - lover, Friend - Friend,

Husband - Wife, etc.

Nadel (Following functional anthropologists and sociologists) defines ‘logical’ types of roles, and then examines how they can be linked together.

Elements of a Role

Necessary:Some roles fit together necessarily. For example, the expected

interaction patterns of “son-in-law” are implied through the joint roles of “Husband” and “Spouse-Parent”

Coincidental:Some roles tend to go together empirically, but they need not

(businessman & club member, for example).

Distinguishing the two is a matter of usefulness and judgement, but relates to social substitutability. The distinction reverts to how the system as a whole will be held together in the face of changes in role occupants.

Coherence of Role Systems

Empirical social structures

• With the fall of functionalism in the late 60s, many of the ideas about social structure and system were also tossed.

• White et al demonstrate how we can understand social structure as the intercalation of roles, without the a priori logical categories.

• Empirical role is: • A set of relations signifying exchange of something (support, ideas,

commands, etc) between actors.

Start with some basic ideas of what a role is: An exchange of something (support, ideas, commands, etc) between actors. Thus, we might represent a family as:

H W

C

C

C

Provides food for

Romantic Love

Bickers with(and there are, of course, many other relations inside the family)

Family Structure

White et al: From logical role systems to empirical social structures

The key idea, is that we can express a role through a relation (or set of relations) and thus a social system by the inventory of roles. If roles equate to positions in an exchange system, then we need only identify particular aspects of a position. But what aspect?

Structural Equivalence•Two actors are structurally equivalent if they have the same types of ties to the same people.

Generalization

Structural Equivalence

A single relation

Structural Equivalence

Graph reduced to positions

Alternative notions of equivalence

Instead of exact same ties to exact same alters, you look for nodes with similar ties to similar types of alters

In any positional analysis, there are 4 basic steps:

1) Identify a definition of equivalence2) Measure the degree to which pairs of actors are equivalent3) Develop a representation of the equivalencies4) Assess the adequacy of the representation5) Repeat and refine

Basic Steps: Blockmodeling

1) Identify a definition of equivalence

Structural Equivalence: Two actors are equivalent if they have the same type of ties to the same people.

AutoMorphic Equivalence

• Actors occupy indistinguishable structural locations in the network. That is, that they are in isomorphic positions in the network.

• Two graphs are isomorphic if there is some mapping of nodes to positions that equates the two.

• In general, automorphically equivalent nodes are equivalent with respect to all graph theoretic properties (I.e. degree, number of people reachable, centrality, etc.)

Automorphic Equivalence:

Regular equivalence does not require actors to have identical ties to identical actors or to be structurally indistinguishable.

Actors who are regularly equivalent have identical ties to and from equivalent actors.

If actors i and j are regularly equivalent, then for all relations and for all actors, if i --> k, then there exists some actor l such that j--> l and k is regularly equivalent to l.

Regular Equivalence

i j

k l

Regular Equivalence:

There may be multiple regular equivalence partitions in a network, and thus we tend to want to find the maximal regular equivalence position, the one with the fewest positions.

Note that:Structurally equivalent actors are automorphically equivalent,Automorphically equivalent actors are regularly equivalent.

Structurally equivalent and automorphically equivalent actors are role equivalent

In practice, we tend to ignore some of these fine distinctions, as they get blurred quickly once we have to operationalize them in real graphs. It turns out that few people are ever exactly equivalent, and thus we approximate the links between the types.

In all cases, the procedure can work over multiple relations simultaneously.

The process of identifying positions is called blockmodeling, and requires identifying a measure of similarity among nodes.

Practicality

0 1 1 1 0 0 0 0 0 0 0 0 0 01 0 0 0 1 1 0 0 0 0 0 0 0 01 0 0 1 0 0 1 1 1 1 0 0 0 01 0 1 0 0 0 1 1 1 1 0 0 0 00 1 0 0 0 1 0 0 0 0 1 1 1 10 1 0 0 1 0 0 0 0 0 1 1 1 10 0 1 1 0 0 0 0 0 0 0 0 0 00 0 1 1 0 0 0 0 0 0 0 0 0 00 0 1 1 0 0 0 0 0 0 0 0 0 00 0 1 1 0 0 0 0 0 0 0 0 0 00 0 0 0 1 1 0 0 0 0 0 0 0 00 0 0 0 1 1 0 0 0 0 0 0 0 00 0 0 0 1 1 0 0 0 0 0 0 0 00 0 0 0 1 1 0 0 0 0 0 0 0 0

Blockmodeling is the process of identifying these types of positions. A block is a section of the adjacency matrix - a “group” of people.

Here I have blocked structurally equivalent actors

. 1 1 1 0 0 0 0 0 0 0 0 0 01 . 0 0 1 1 0 0 0 0 0 0 0 01 0 . 1 0 0 1 1 1 1 0 0 0 01 0 1 . 0 0 1 1 1 1 0 0 0 00 1 0 0 . 1 0 0 0 0 1 1 1 10 1 0 0 1 . 0 0 0 0 1 1 1 10 0 1 1 0 0 . 0 0 0 0 0 0 00 0 1 1 0 0 0 . 0 0 0 0 0 00 0 1 1 0 0 0 0 . 0 0 0 0 00 0 1 1 0 0 0 0 0 . 0 0 0 00 0 0 0 1 1 0 0 0 0 . 0 0 00 0 0 0 1 1 0 0 0 0 0 . 0 00 0 0 0 1 1 0 0 0 0 0 0 . 00 0 0 0 1 1 0 0 0 0 0 0 0 .

1 2 3 4 5 61 0 1 1 0 0 02 1 0 0 1 0 03 1 0 1 0 1 04 0 1 0 1 0 1 5 0 0 1 0 0 06 0 0 0 1 0 0

Once you block the matrix, reduce it, based on the number of ties in the cell of interest. The key values are a zero block (no ties) and a one-block (all ties present):

Structural equivalence thus generates 6 positions in the network

1 2 3 4 5 6

12

3

4

5

6

. 1 1 1 0 0 0 0 0 0 0 0 0 01 . 0 0 1 1 0 0 0 0 0 0 0 01 0 . 1 0 0 1 1 1 1 0 0 0 01 0 1 . 0 0 1 1 1 1 0 0 0 00 1 0 0 . 1 0 0 0 0 1 1 1 10 1 0 0 1 . 0 0 0 0 1 1 1 10 0 1 1 0 0 . 0 0 0 0 0 0 00 0 1 1 0 0 0 . 0 0 0 0 0 00 0 1 1 0 0 0 0 . 0 0 0 0 00 0 1 1 0 0 0 0 0 . 0 0 0 00 0 0 0 1 1 0 0 0 0 . 0 0 00 0 0 0 1 1 0 0 0 0 0 . 0 00 0 0 0 1 1 0 0 0 0 0 0 . 00 0 0 0 1 1 0 0 0 0 0 0 0 .

1 2 31 1 1 02 1 1 1 3 0 1 0

Once you partition the matrix, reduce it:

Regular equivalence

1 2

3

To get a block model, you have to measure the similarity between each pair. If two actors are structurally equivalent, then they will have exactly similar patterns of ties to other people. Consider the example again:

. 1 1 1 0 0 0 0 0 0 0 0 0 01 . 0 0 1 1 0 0 0 0 0 0 0 01 0 . 1 0 0 1 1 1 1 0 0 0 01 0 1 . 0 0 1 1 1 1 0 0 0 00 1 0 0 . 1 0 0 0 0 1 1 1 10 1 0 0 1 . 0 0 0 0 1 1 1 10 0 1 1 0 0 . 0 0 0 0 0 0 00 0 1 1 0 0 0 . 0 0 0 0 0 00 0 1 1 0 0 0 0 . 0 0 0 0 00 0 1 1 0 0 0 0 0 . 0 0 0 00 0 0 0 1 1 0 0 0 0 . 0 0 00 0 0 0 1 1 0 0 0 0 0 . 0 00 0 0 0 1 1 0 0 0 0 0 0 . 00 0 0 0 1 1 0 0 0 0 0 0 0 .

1 2 3 4 5 6

12

3

4

5

6

C D Match1 1 10 0 1. 1 01 . 00 0 10 0 11 1 1 1 1 1 1 1 11 1 10 0 10 0 10 0 10 0 1Sum: 12

C and D match on 12 other people

If the model is going to be based on asymmetric or multiple relations, you simply stack the various relations:

H W

CC

C

Provides food for

Romantic Love

Bickers with

Romance0 1 0 0 01 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

Feeds0 0 1 1 10 0 1 1 10 0 0 0 00 0 0 0 00 0 0 0 0

Bicker0 0 0 0 00 0 0 0 00 0 0 1 10 0 1 0 00 0 1 1 0

0 1 0 0 01 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 1 1 10 0 1 1 10 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 01 1 0 0 01 1 0 0 01 1 0 0 00 0 0 0 00 0 0 0 00 0 0 1 10 0 1 0 10 0 1 1 0

Stacked

0 8 7 7 5 5 11 11 11 11 7 7 7 7 8 0 5 5 7 7 7 7 7 7 11 11 11 11 7 5 0 12 0 0 8 8 8 8 4 4 4 4 7 5 12 0 0 0 8 8 8 8 4 4 4 4 5 7 0 0 0 12 4 4 4 4 8 8 8 8 5 7 0 0 12 0 4 4 4 4 8 8 8 811 7 8 8 4 4 0 12 12 12 8 8 8 811 7 8 8 4 4 12 0 12 12 8 8 8 811 7 8 8 4 4 12 12 0 12 8 8 8 811 7 8 8 4 4 12 12 12 0 8 8 8 8 7 11 4 4 8 8 8 8 8 8 0 12 12 12 7 11 4 4 8 8 8 8 8 8 12 0 12 12 7 11 4 4 8 8 8 8 8 8 12 12 0 12 7 11 4 4 8 8 8 8 8 8 12 12 12 0

For the entire matrix, we get:

(number of agreements for each ij pair)

1.00 -0.20 0.08 0.08 -0.19 -0.19 0.77 0.77 0.77 0.77 -0.26 -0.26 -0.26 -0.26-0.20 1.00 -0.19 -0.19 0.08 0.08 -0.26 -0.26 -0.26 -0.26 0.77 0.77 0.77 0.77 0.08 -0.19 1.00 1.00 -1.00 -1.00 0.36 0.36 0.36 0.36 -0.45 -0.45 -0.45 -0.45 0.08 -0.19 1.00 1.00 -1.00 -1.00 0.36 0.36 0.36 0.36 -0.45 -0.45 -0.45 -0.45-0.19 0.08 -1.00 -1.00 1.00 1.00 -0.45 -0.45 -0.45 -0.45 0.36 0.36 0.36 0.36-0.19 0.08 -1.00 -1.00 1.00 1.00 -0.45 -0.45 -0.45 -0.45 0.36 0.36 0.36 0.36 0.77 -0.26 0.36 0.36 -0.45 -0.45 1.00 1.00 1.00 1.00 -0.20 -0.20 -0.20 -0.20 0.77 -0.26 0.36 0.36 -0.45 -0.45 1.00 1.00 1.00 1.00 -0.20 -0.20 -0.20 -0.20 0.77 -0.26 0.36 0.36 -0.45 -0.45 1.00 1.00 1.00 1.00 -0.20 -0.20 -0.20 -0.20 0.77 -0.26 0.36 0.36 -0.45 -0.45 1.00 1.00 1.00 1.00 -0.20 -0.20 -0.20 -0.20-0.26 0.77 -0.45 -0.45 0.36 0.36 -0.20 -0.20 -0.20 -0.20 1.00 1.00 1.00 1.00-0.26 0.77 -0.45 -0.45 0.36 0.36 -0.20 -0.20 -0.20 -0.20 1.00 1.00 1.00 1.00-0.26 0.77 -0.45 -0.45 0.36 0.36 -0.20 -0.20 -0.20 -0.20 1.00 1.00 1.00 1.00-0.26 0.77 -0.45 -0.45 0.36 0.36 -0.20 -0.20 -0.20 -0.20 1.00 1.00 1.00 1.00

Correlation between each node’s set of ties.

For the example, this would be:

Measuring similarity

The initial method for finding structurally equivalent positions was CONCOR, the CONvergence of iterated CORrelations.

1.00 -.77 0.55 0.55 -.57 -.57 0.95 0.95 0.95 0.95 -.75 -.75 -.75 -.75-.77 1.00 -.57 -.57 0.55 0.55 -.75 -.75 -.75 -.75 0.95 0.95 0.95 0.950.55 -.57 1.00 1.00 -1.0 -1.0 0.73 0.73 0.73 0.73 -.75 -.75 -.75 -.750.55 -.57 1.00 1.00 -1.0 -1.0 0.73 0.73 0.73 0.73 -.75 -.75 -.75 -.75-.57 0.55 -1.0 -1.0 1.00 1.00 -.75 -.75 -.75 -.75 0.73 0.73 0.73 0.73-.57 0.55 -1.0 -1.0 1.00 1.00 -.75 -.75 -.75 -.75 0.73 0.73 0.73 0.730.95 -.75 0.73 0.73 -.75 -.75 1.00 1.00 1.00 1.00 -.77 -.77 -.77 -.770.95 -.75 0.73 0.73 -.75 -.75 1.00 1.00 1.00 1.00 -.77 -.77 -.77 -.770.95 -.75 0.73 0.73 -.75 -.75 1.00 1.00 1.00 1.00 -.77 -.77 -.77 -.770.95 -.75 0.73 0.73 -.75 -.75 1.00 1.00 1.00 1.00 -.77 -.77 -.77 -.77-.75 0.95 -.75 -.75 0.73 0.73 -.77 -.77 -.77 -.77 1.00 1.00 1.00 1.00-.75 0.95 -.75 -.75 0.73 0.73 -.77 -.77 -.77 -.77 1.00 1.00 1.00 1.00-.75 0.95 -.75 -.75 0.73 0.73 -.77 -.77 -.77 -.77 1.00 1.00 1.00 1.00-.75 0.95 -.75 -.75 0.73 0.73 -.77 -.77 -.77 -.77 1.00 1.00 1.00 1.00

Concor iteration 1:

Concor iteration 2:1.00 -.99 0.94 0.94 -.94 -.94 0.99 0.99 0.99 0.99 -.99 -.99 -.99 -.99-.99 1.00 -.94 -.94 0.94 0.94 -.99 -.99 -.99 -.99 0.99 0.99 0.99 0.990.94 -.94 1.00 1.00 -1.0 -1.0 0.97 0.97 0.97 0.97 -.97 -.97 -.97 -.970.94 -.94 1.00 1.00 -1.0 -1.0 0.97 0.97 0.97 0.97 -.97 -.97 -.97 -.97-.94 0.94 -1.0 -1.0 1.00 1.00 -.97 -.97 -.97 -.97 0.97 0.97 0.97 0.97-.94 0.94 -1.0 -1.0 1.00 1.00 -.97 -.97 -.97 -.97 0.97 0.97 0.97 0.970.99 -.99 0.97 0.97 -.97 -.97 1.00 1.00 1.00 1.00 -.99 -.99 -.99 -.990.99 -.99 0.97 0.97 -.97 -.97 1.00 1.00 1.00 1.00 -.99 -.99 -.99 -.990.99 -.99 0.97 0.97 -.97 -.97 1.00 1.00 1.00 1.00 -.99 -.99 -.99 -.990.99 -.99 0.97 0.97 -.97 -.97 1.00 1.00 1.00 1.00 -.99 -.99 -.99 -.99-.99 0.99 -.97 -.97 0.97 0.97 -.99 -.99 -.99 -.99 1.00 1.00 1.00 1.00-.99 0.99 -.97 -.97 0.97 0.97 -.99 -.99 -.99 -.99 1.00 1.00 1.00 1.00-.99 0.99 -.97 -.97 0.97 0.97 -.99 -.99 -.99 -.99 1.00 1.00 1.00 1.00-.99 0.99 -.97 -.97 0.97 0.97 -.99 -.99 -.99 -.99 1.00 1.00 1.00 1.00

The initial method for finding structurally equivalent positions was CONCOR, the CONvergence of iterated CORrelations.

Padget and Ansell:“Robust Action and the Rise of the Medici”

•Substantive question relates to effective state-building: there is a tension between the need to control and organization and the ability to build the legitimacy and recognition required for reproduction. The distinction between “boss” and “judge”

•They use the marriage, economic and patronage networks

•Empirically, we know that the state oligarchy structure of Florence stabilized after the rise of the medici:

Padget and Ansell:“Robust Action and the Rise of the Medici”

Medici Takeover

Padget and Ansell:“Robust Action and the Rise of the Medici”

The story they tell revolves around how Cosimo de’Medici was able to found a system that lasted nearly 300 years, uniting a fractured political structure.

The paradox of Cosimo is that he didn’t seem to fit the role of a Machiavellian leader as decisive and goal oriented.

The answer lies in the power resulting from ‘robust action’ embedded in a network of relations that gives rise to no clear meaning and obligation, but instead allows for multiple meanings and obligations.

A real example:Padget and Ansell:“Robust Action and the Rise of the Medici”

“Political Groups” in the attribute sense do not seem to exist, so P&A turn to the pattern of network relations among families.

This is the BLOCK reduction of the full 92 family network.

An example:Relations among Italian families.

Political and friendship ties

Generalized Block Models

The recent work on generalization focuses on the patterns that determine a block.

Instead of focusing on just the density of a block, you can identify a block as any set that has a particular pattern of ties to any other set. Examples include:

Generalized Block Models

Compound Relations.

One of the most powerful tools in role analysis involves looking at role systems through compound relations.

A compound relation is formed by combining relations in single dimensions. The best example of compound relations come from kinship.

SiblingChild of

Sibling0 1 0 0 01 0 0 0 00 0 0 1 00 0 1 0 00 0 0 0 0

Child of0 0 1 1 00 0 0 0 10 0 0 0 00 0 0 0 00 0 0 0 0

x =

Nephew/Niece0 0 0 0 10 0 1 1 00 0 0 0 00 0 0 0 00 0 0 0 0

SC = SC

An example of compound relations can be found in W&F. This role table catalogues the compounds for two relations “Is boss of” and “Is on the same level as”