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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
Elements of a Role ,[object Object],[object Object],[object Object],[object Object],[object Object]
Coherence of Role Systems 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 .
Empirical social structures ,[object Object],[object Object],[object Object],[object Object]
Family Structure 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 (and there are, of course, many other relations inside the family) Romantic Love Bickers with
Generalization White et al :  From logical role systems to empirical social structures ,[object Object],[object Object],[object Object],.
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
Basic Steps: Blockmodeling 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
1) Identify a definition of equivalence ,[object Object],[object Object]
AutoMorphic Equivalence ,[object Object],[object Object],[object Object]
Automorphic Equivalence:
[object Object],[object Object],[object Object],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.
Practicality ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
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 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 1 2 3 4 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
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 . 1 2 3 4 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
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)
Measuring similarity 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:
The initial method for finding structurally equivalent positions was CONCOR, the CONvergence of iterated  COR relations.  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  COR relations.
Padget and Ansell: “ Robust Action and the Rise of the Medici” ,[object Object],[object Object],[object Object]
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.  Sibling Child of Sibling 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 Child 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/Niece 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 S  C = 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”

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6 Block Modeling

  • 1. 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
  • 2.
  • 3. Coherence of Role Systems 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 .
  • 4.
  • 5. Family Structure 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 (and there are, of course, many other relations inside the family) Romantic Love Bickers with
  • 6.
  • 7. Structural Equivalence A single relation
  • 8. Structural Equivalence Graph reduced to positions
  • 9. 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
  • 10. Basic Steps: Blockmodeling 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
  • 11.
  • 12.
  • 14.
  • 15. 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.
  • 16.
  • 17. 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
  • 18. . 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 1 2 3 4 5 6 1 2 3 4 5 6
  • 19. . 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
  • 20. 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 . 1 2 3 4 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
  • 21. 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
  • 22. 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)
  • 23. Measuring similarity 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:
  • 24. The initial method for finding structurally equivalent positions was CONCOR, the CONvergence of iterated COR relations. 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:
  • 25. 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 COR relations.
  • 26.
  • 27. Padget and Ansell: “ Robust Action and the Rise of the Medici” Medici Takeover
  • 28. 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.
  • 29. 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.
  • 30. An example: Relations among Italian families. Political and friendship ties
  • 31. 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:
  • 33. 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 Sibling 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 Child 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/Niece 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 S  C = SC
  • 34. 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”