6 Block Modeling
maksim2042
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34 slides
Feb 19, 2009
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About This Presentation
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Slide Content
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
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
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
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.
•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
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
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
A single relation
Structural Equivalence
Graph reduced to positions
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
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 equivalence
2) Measure the degree to which pairs of actors are equivalent
3) Develop a representation of the equivalencies
4) Assess the adequacy of the representation
5) Repeat and refine
Basic Steps: Blockmodeling
1) Identify a definition of equivalence
2) Measure the degree to which pairs of actors are equivalent
3) Develop a representation of the equivalencies
4) Assess the adequacy of the representation
5) 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.
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.)
•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
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.
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
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 0
1 0 0 0 1 1 0 0 0 0 0 0 0 0
1 0 0 1 0 0 1 1 1 1 0 0 0 0
1 0 1 0 0 0 1 1 1 1 0 0 0 0
0 1 0 0 0 1 0 0 0 0 1 1 1 1
0 1 0 0 1 0 0 0 0 0 1 1 1 1
0 0 1 1 0 0 0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 0 0 0 0 0 0 0 0
0 0 0 0 1 1 0 0 0 0 0 0 0 0
0 0 0 0 1 1 0 0 0 0 0 0 0 0
0 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 0 0 0 1 1 0 0 0 0 0 0 0 0
1 0 0 1 0 0 1 1 1 1 0 0 0 0
1 0 1 0 0 0 1 1 1 1 0 0 0 0
0 1 0 0 0 1 0 0 0 0 1 1 1 1
0 1 0 0 1 0 0 0 0 0 1 1 1 1
0 0 1 1 0 0 0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 0 0 0 0 0 0 0 0
0 0 0 0 1 1 0 0 0 0 0 0 0 0
0 0 0 0 1 1 0 0 0 0 0 0 0 0
0 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 0
1 . 0 0 1 1 0 0 0 0 0 0 0 0
1 0 . 1 0 0 1 1 1 1 0 0 0 0
1 0 1 . 0 0 1 1 1 1 0 0 0 0
0 1 0 0 . 1 0 0 0 0 1 1 1 1
0 1 0 0 1 . 0 0 0 0 1 1 1 1
0 0 1 1 0 0 . 0 0 0 0 0 0 0
0 0 1 1 0 0 0 . 0 0 0 0 0 0
0 0 1 1 0 0 0 0 . 0 0 0 0 0
0 0 1 1 0 0 0 0 0 . 0 0 0 0
0 0 0 0 1 1 0 0 0 0 . 0 0 0
0 0 0 0 1 1 0 0 0 0 0 . 0 0
0 0 0 0 1 1 0 0 0 0 0 0 . 0
0 0 0 0 1 1 0 0 0 0 0 0 0 .
1 2 3 4 5 6
1 0 1 1 0 0 0
2 1 0 0 1 0 0
3 1 0 1 0 1 0
4 0 1 0 1 0 1
5 0 0 1 0 0 0
6 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
1234 5 6
1
2
3
4
5
6
1 . 0 0 1 1 0 0 0 0 0 0 0 0
1 0 . 1 0 0 1 1 1 1 0 0 0 0
1 0 1 . 0 0 1 1 1 1 0 0 0 0
0 1 0 0 . 1 0 0 0 0 1 1 1 1
0 1 0 0 1 . 0 0 0 0 1 1 1 1
0 0 1 1 0 0 . 0 0 0 0 0 0 0
0 0 1 1 0 0 0 . 0 0 0 0 0 0
0 0 1 1 0 0 0 0 . 0 0 0 0 0
0 0 1 1 0 0 0 0 0 . 0 0 0 0
0 0 0 0 1 1 0 0 0 0 . 0 0 0
0 0 0 0 1 1 0 0 0 0 0 . 0 0
0 0 0 0 1 1 0 0 0 0 0 0 . 0
0 0 0 0 1 1 0 0 0 0 0 0 0 .
1 2 3 4 5 6
1 0 1 1 0 0 0
2 1 0 0 1 0 0
3 1 0 1 0 1 0
4 0 1 0 1 0 1
5 0 0 1 0 0 0
6 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
1234 5 6
1
2
3
4
5
6
. 1 1 1 0 0 0 0 0 0 0 0 0 0
1 . 0 0 1 1 0 0 0 0 0 0 0 0
1 0 . 1 0 0 1 1 1 1 0 0 0 0
1 0 1 . 0 0 1 1 1 1 0 0 0 0
0 1 0 0 . 1 0 0 0 0 1 1 1 1
0 1 0 0 1 . 0 0 0 0 1 1 1 1
0 0 1 1 0 0 . 0 0 0 0 0 0 0
0 0 1 1 0 0 0 . 0 0 0 0 0 0
0 0 1 1 0 0 0 0 . 0 0 0 0 0
0 0 1 1 0 0 0 0 0 . 0 0 0 0
0 0 0 0 1 1 0 0 0 0 . 0 0 0
0 0 0 0 1 1 0 0 0 0 0 . 0 0
0 0 0 0 1 1 0 0 0 0 0 0 . 0
0 0 0 0 1 1 0 0 0 0 0 0 0 .
1 2 3
1 1 1 0
2 1 1 1
3 0 1 0
Once you partition the matrix, reduce it:
Regular equivalence
1 2
3
1 . 0 0 1 1 0 0 0 0 0 0 0 0
1 0 . 1 0 0 1 1 1 1 0 0 0 0
1 0 1 . 0 0 1 1 1 1 0 0 0 0
0 1 0 0 . 1 0 0 0 0 1 1 1 1
0 1 0 0 1 . 0 0 0 0 1 1 1 1
0 0 1 1 0 0 . 0 0 0 0 0 0 0
0 0 1 1 0 0 0 . 0 0 0 0 0 0
0 0 1 1 0 0 0 0 . 0 0 0 0 0
0 0 1 1 0 0 0 0 0 . 0 0 0 0
0 0 0 0 1 1 0 0 0 0 . 0 0 0
0 0 0 0 1 1 0 0 0 0 0 . 0 0
0 0 0 0 1 1 0 0 0 0 0 0 . 0
0 0 0 0 1 1 0 0 0 0 0 0 0 .
1 2 3
1 1 1 0
2 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 0
1 . 0 0 1 1 0 0 0 0 0 0 0 0
1 0 . 1 0 0 1 1 1 1 0 0 0 0
1 0 1 . 0 0 1 1 1 1 0 0 0 0
0 1 0 0 . 1 0 0 0 0 1 1 1 1
0 1 0 0 1 . 0 0 0 0 1 1 1 1
0 0 1 1 0 0 . 0 0 0 0 0 0 0
0 0 1 1 0 0 0 . 0 0 0 0 0 0
0 0 1 1 0 0 0 0 . 0 0 0 0 0
0 0 1 1 0 0 0 0 0 . 0 0 0 0
0 0 0 0 1 1 0 0 0 0 . 0 0 0
0 0 0 0 1 1 0 0 0 0 0 . 0 0
0 0 0 0 1 1 0 0 0 0 0 0 . 0
0 0 0 0 1 1 0 0 0 0 0 0 0 .
1234 5 6
1
2
3
4
5
6
C D Match
1 1 1
0 0 1
. 1 0
1 . 0
0 0 1
0 0 1
1 1 1
1 1 1
1 1 1
1 1 1
0 0 1
0 0 1
0 0 1
0 0 1
Sum: 12
C and D match
on 12 other
people
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 0
1 . 0 0 1 1 0 0 0 0 0 0 0 0
1 0 . 1 0 0 1 1 1 1 0 0 0 0
1 0 1 . 0 0 1 1 1 1 0 0 0 0
0 1 0 0 . 1 0 0 0 0 1 1 1 1
0 1 0 0 1 . 0 0 0 0 1 1 1 1
0 0 1 1 0 0 . 0 0 0 0 0 0 0
0 0 1 1 0 0 0 . 0 0 0 0 0 0
0 0 1 1 0 0 0 0 . 0 0 0 0 0
0 0 1 1 0 0 0 0 0 . 0 0 0 0
0 0 0 0 1 1 0 0 0 0 . 0 0 0
0 0 0 0 1 1 0 0 0 0 0 . 0 0
0 0 0 0 1 1 0 0 0 0 0 0 . 0
0 0 0 0 1 1 0 0 0 0 0 0 0 .
1234 5 6
1
2
3
4
5
6
C D Match
1 1 1
0 0 1
. 1 0
1 . 0
0 0 1
0 0 1
1 1 1
1 1 1
1 1 1
1 1 1
0 0 1
0 0 1
0 0 1
0 0 1
Sum: 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
C
C
C
Provides food for
Romantic Love
Bickers with
Romance
0 1 0 0 0
1 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
Feeds
0 0 1 1 1
0 0 1 1 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
Bicker
0 0 0 0 0
0 0 0 0 0
0 0 0 1 1
0 0 1 0 0
0 0 1 1 0
0 1 0 0 0
1 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 1 1 1
0 0 1 1 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
1 1 0 0 0
1 1 0 0 0
1 1 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 1 1
0 0 1 0 1
0 0 1 1 0
Stacked
relations, you simply stack the various relations:
H W
C
C
C
Provides food for
Romantic Love
Bickers with
Romance
0 1 0 0 0
1 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
Feeds
0 0 1 1 1
0 0 1 1 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
Bicker
0 0 0 0 0
0 0 0 0 0
0 0 0 1 1
0 0 1 0 0
0 0 1 1 0
0 1 0 0 0
1 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 1 1 1
0 0 1 1 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
1 1 0 0 0
1 1 0 0 0
1 1 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 1 1
0 0 1 0 1
0 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 8
11 7 8 8 4 4 0 12 12 12 8 8 8 8
11 7 8 8 4 4 12 0 12 12 8 8 8 8
11 7 8 8 4 4 12 12 0 12 8 8 8 8
11 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)
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 8
11 7 8 8 4 4 0 12 12 12 8 8 8 8
11 7 8 8 4 4 12 0 12 12 8 8 8 8
11 7 8 8 4 4 12 12 0 12 8 8 8 8
11 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
-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 0.95
0.55 -.57 1.00 1.00 -1.0 -1.0 0.73 0.73 0.73 0.73 -.75 -.75 -.75 -.75
0.55 -.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 0.73
0.95 -.75 0.73 0.73 -.75 -.75 1.00 1.00 1.00 1.00 -.77 -.77 -.77 -.77
0.95 -.75 0.73 0.73 -.75 -.75 1.00 1.00 1.00 1.00 -.77 -.77 -.77 -.77
0.95 -.75 0.73 0.73 -.75 -.75 1.00 1.00 1.00 1.00 -.77 -.77 -.77 -.77
0.95 -.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:
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 0.95
0.55 -.57 1.00 1.00 -1.0 -1.0 0.73 0.73 0.73 0.73 -.75 -.75 -.75 -.75
0.55 -.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 0.73
0.95 -.75 0.73 0.73 -.75 -.75 1.00 1.00 1.00 1.00 -.77 -.77 -.77 -.77
0.95 -.75 0.73 0.73 -.75 -.75 1.00 1.00 1.00 1.00 -.77 -.77 -.77 -.77
0.95 -.75 0.73 0.73 -.75 -.75 1.00 1.00 1.00 1.00 -.77 -.77 -.77 -.77
0.95 -.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 0.99
0.94 -.94 1.00 1.00 -1.0 -1.0 0.97 0.97 0.97 0.97 -.97 -.97 -.97 -.97
0.94 -.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 0.97
0.99 -.99 0.97 0.97 -.97 -.97 1.00 1.00 1.00 1.00 -.99 -.99 -.99 -.99
0.99 -.99 0.97 0.97 -.97 -.97 1.00 1.00 1.00 1.00 -.99 -.99 -.99 -.99
0.99 -.99 0.97 0.97 -.97 -.97 1.00 1.00 1.00 1.00 -.99 -.99 -.99 -.99
0.99 -.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.
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 0.99
0.94 -.94 1.00 1.00 -1.0 -1.0 0.97 0.97 0.97 0.97 -.97 -.97 -.97 -.97
0.94 -.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 0.97
0.99 -.99 0.97 0.97 -.97 -.97 1.00 1.00 1.00 1.00 -.99 -.99 -.99 -.99
0.99 -.99 0.97 0.97 -.97 -.97 1.00 1.00 1.00 1.00 -.99 -.99 -.99 -.99
0.99 -.99 0.97 0.97 -.97 -.97 1.00 1.00 1.00 1.00 -.99 -.99 -.99 -.99
0.99 -.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:
“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
“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.
“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.
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
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:
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.
Sibling
Child of
Si bl i ng
0 1 0 0 0
1 0 0 0 0
0 0 0 1 0
0 0 1 0 0
0 0 0 0 0
Chi l d of
0 0 1 1 0
0 0 0 0 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
x =
Nephew/Ni ece
0 0 0 0 1
0 0 1 1 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
SC = SC
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.
Sibling
Child of
Si bl i ng
0 1 0 0 0
1 0 0 0 0
0 0 0 1 0
0 0 1 0 0
0 0 0 0 0
Chi l d of
0 0 1 1 0
0 0 0 0 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
x =
Nephew/Ni ece
0 0 0 0 1
0 0 1 1 0
0 0 0 0 0
0 0 0 0 0
0 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”
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”
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