1. Hierarchy Overview Background: Hierarchy surrounds us: what is it? Micro foundations of social stratification Ivan Chase: Structure from process Action --> Structure, not attributes David Krackhardt: Deliberate Structure w. in organizations Measures for the extent of hierarchy

2. Examples of Hierarchical Systems Linear Hierarchy (all triads transitive) Simple Hierarchy Branched Hierarchy Mixed Hierarchy

3. Examples of “Similar” Non-Hierarchical Systems Acyclic Cycle Line Graph

4. Chase’s Question: Where does hierarchy come from? Hierarchy surrounds us, in natural (animal and human) and controlled (laboratory, organizations) settings. How do we account for it? Most previous research focuses on the static structure of hierarchy Often consider the attributes of actors: strength, race, gender, education, size, etc.

5. Chase’s Question: Where does hierarchy come from? The “Correlational Model” Individual’s position in the hierarchy is due to their attributes (physical, social, etc.) Mathematically, for the correlational model to be true, the correspondence between attributes and rank in the hierarchy would have to be extremely high (Pearson correlation of >.9). (See Chase, 1974 for details)

6. Chase’s Question: Where does hierarchy come from? The “Pairwise interaction” model Pairwise differences in each dyad account for position in the hierarchy. “...it is assumed that each member of a group has a pairwise contest with each other member, that the winner of a contest dominates the loser in the group hierarchy, and that an individual has a particular probability of success in each contest.” Model implies that there be one individual with a.95 probability of beating every other individual, another with a.95 probability of beating everyone but the most dominant, and so forth down the line. The required conditions simply do not hold. As such, this explanation for where the hierarchy comes from cannot hold.

7. Chase’s Question: Where does hierarchy come from? Chase focuses on the simple mathematical fact: Every linear hierarchy must contain all transitive triads. That is, the triad census for the network must have only 3 T triads. A B C D E Number of Type triads ---------------------- 1 - 3 ----------------------- 2 - 012 0 3 - 102 0 4 - 021D 0 5 - 021U 0 6 - 021C 0 7 - 111D 0 8 - 111U 0 9 - 030T 10 10 - 030C 0 11 - 201 0 12 - 120D 0 13 - 120U 0 14 - 120C 0 15 - 210 0 16 - 300 0 --------------------------- Sum (2 - 16): 10 What process could generate all 030T triads?

8. Chase’s Question: Where does hierarchy come from? A CB Transitive (030T) triad A CB Intransitive (030C) triad The elements: Dominance relations must by asymmetric, thus, the set of possible triads is limited.

9. Why Chase Finds Linear Hierarchy: 003 012 021D 021U 021C 030C 030T p=1. p=.5 p=.25 p=.5 p=1 p=.5 P( 3 C) =.5*.5=.25 P( 3 T)= (.5*.5 +.25*1 +.25*1) =.75 Triad transitions (w/ Random Expectations) for Dominance Relations.

10. Dominance Strategies That ensure a transitive hierarchy 003 012 021D 030T The “Double Attack” Strategy: The first attacker quickly attacks the bystander. This means we arrive at 21D, and any action on the part of the other two chickens will lead to a transitive triad. The “Double Receive” Strategy: The first attacker dominates B, and then the bystander quickly dominates B as well, leading to 21U, and any dominance between the first and second attacker will lead to a transitive triple. 003 012 021U 030T

11. Dominance Strategies That may not lead to a transitive hierarchy “Attack the Attacker” The bystander attacks the first attacker. This could lead to a cyclic triad, and thus thwart hierarchy. 003 012 021C 030C 030T “Pass on the attack” The one who is attacked, attacks the bystander. Again, this could lead to a cycle, and thus thwart hierarchy. 003 012 021C 030C 030T

12. The evidence : 24 Chase Chicken Triads 003 012 021D 021U 021C 030C 030 T 23 17 1 (17 stay) 4 4 2 1 1 1 Domination Reversal New Domination (1 stays) (6 Fully Transitive) ( 0 stay) Most Common Path

13. Graph Theoretic Dimensions of Informal Organizations Moving beyond dominance relations in animals, what can SNA tell us about dominance in organizations? Krackhardt argues that an ‘Outree” is the archetype of hierarchy. (what are the allowed triad types for an out-tree?) Krackhardt focuses on 4 dimensions: 1) Connectedness 2) Digraph hierarchic 3) digraph efficiency 4) least upper bound

14. Graph Theoretic Dimensions of Informal Organizations Connectedness: The digraph is connected if the underlying graph is a component. We can measure the extent of connectedness through reachability. Where V is the number of pairs that are not reachable, and N is the number of people in the network.

15. Graph Theoretic Dimensions of Informal Organizations How to calculate Connectedness: 1 2 3 5 4 Digraph: 1 2 3 4 5 1 0 1 0 1 0 2 0 0 1 0 0 3 0 0 0 0 0 4 0 0 0 0 0 5 0 0 0 0 0 Graph: 1 2 3 4 5 1 0 1 0 1 0 2 1 0 1 0 0 3 0 1 0 0 0 4 1 0 0 0 0 5 0 0 0 0 0 Reach: 1 2 3 4 5 1 0 1 2 1 0 2 1 0 1 2 0 3 2 1 0 3 0 4 1 2 3 0 0 5 0 0 0 0 0 V = # of zeros in the upper diagonal of Reach: V = 4. C = 1 - [4/((5*4)/2)] = 1 - 4/1 =.6

16. Graph Theoretic Dimensions of Informal Organizations How to calculate Connectedness: 1 2 3 5 4 This is equivalent to the density of the reachability matrix. Reachable: 1 2 3 4 5 1 0 1 1 1 0 2 1 0 1 1 0 3 1 1 0 1 0 4 1 1 1 0 0 5 0 0 0 0 0  =  R/(N(N-1)) = 12 /(5*4) =.6 Reach: 1 2 3 4 5 1 0 1 2 1 0 2 1 0 1 2 0 3 2 1 0 3 0 4 1 2 3 0 0 5 0 0 0 0 0

17. Graph Theoretic Dimensions of Informal Organizations Graph Hierarchy: The extent to which people are asymmetrically reachable. Where V is the number of symmetrically reachable pairs in the network. Max(V) is the number of pairs where i can reach j or j can reach i.

18. Graph Theoretic Dimensions of Informal Organizations 1 2 3 5 4 Graph Hierarchy: An example Digraph: 1 2 3 4 5 1 0 1 0 1 0 2 0 0 1 0 0 3 0 1 0 0 0 4 0 0 0 0 0 5 0 0 0 0 0 D reach 1 2 3 4 5 1 0 1 2 1 0 2 0 0 1 0 0 3 0 1 0 0 0 4 0 0 0 0 0 5 0 0 0 0 0 V = 1 Max(V) = 4 H = 1/4 =.25 D reachable 1 2 3 4 5 1 0 1 2 1 0 2 0 0 1 0 0 3 0 1 0 0 0 4 0 0 0 0 0 5 0 0 0 0 0

19. Graph Theoretic Dimensions of Informal Organizations Graph Efficiency: The extent to which there are extra lines in the graph, given the number of components. Where v is the number of excess links and max(v) is the maximum possible number of excess links

20. 1 2 3 5 4 6 7 Graph Theoretic Dimensions of Informal Organizations Graph Efficiency: The minimum number of lines in a connected component is N-1 (assuming symmetry, only use the upper half of the adjacency matrix). In this example, the first component contains 4 nodes and thus the minimum required lines is 3. There are 4 lines, and thus V 1 = 4-3 = 1. The second component contains 3 nodes and thus minimum connectivity is = 2, there are 3 so V 2 = 1. Summed over all components V=2. The maximum number of lines would occur if every node was connected to every other, and equals N(N-1)/2. For the first component Max(V 1 ) = (6-3)=3. For the second, Max(V2) = (3-2)=1, so Max(V) = 4. Efficiency = (1- 2/4 ) =.5 1 2

21. Graph Theoretic Dimensions of Informal Organizations Graph Efficiency: Steps to calculate efficiency: a) identify all components in the graph b) for each component (i) do: i) calculate V i =  (G i )/2 - N i -1; ii) calculate Max(V i ) = N i (N i -1) - (N i -1) c) V =  (V i ), Max(V)=  (Max(V i ) d) efficiency = 1 = V/Max(V) Substantively, this must be a function of the average density of the components in the graph.

22. Graph Theoretic Dimensions of Informal Organizations Least Upper Boundedness: This condition looks at how many ‘roots’ there are in the tree. The LUB for any pair of actors is the closest person who can reach both of them. In a formal hierarchy, every pair should have at least one LUB. A D B C E In this case, E is the LUB for (A,D), B is the LUB for (F,G), H is the LUB for (D,C), etc. F G H

23. Graph Theoretic Dimensions of Informal Organizations Least Upper Boundedness: You get a violation of LUB if two people in the organization do not have an (eventual) common boss. Here, persons 4 and 7 do not have an LUB.

24. Graph Theoretic Dimensions of Informal Organizations Least Upper Boundedness: Calculate LUB by looking at reachability. Distance matrix 1 2 3 4 5 6 7 8 9 1 1 1 1 2 2 2 2 1 1 1 3 1 1 4 1 5 1 6 1 1 1 2 7 1 1 8 1 9 1 (Note that I set the diagonal = 1) Reachable matrix 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 2 1 1 1 3 1 1 4 1 5 1 6 1 1 1 1 7 1 1 8 1 9 1 A violation occurs whenever a pair is not reachable by at least one common node. We can get common reachability through matrix multiplication

25. Graph Theoretic Dimensions of Informal Organizations Least Upper Boundedness: Calculate LUB by looking at reachability. Reachable matrix 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 2 1 1 1 3 1 1 4 1 5 1 6 1 1 1 1 7 1 1 8 1 9 1 R`*R = CR Reachable Trans 1 2 3 4 5 6 7 8 9 1 2 1 1 3 1 1 4 1 1 1 5 1 1 1 6 1 7 1 1 8 1 1 9 1 1 1 1 1 X = Common Reach 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 2 1 2 1 2 2 1 3 1 1 2 1 1 2 4 1 2 1 3 2 1 5 1 2 1 2 3 1 6 1 1 1 1 7 1 2 1 2 8 1 1 2 1 9 1 1 2 1 1 1 2 1 5 Any place with a zero indicates a pair that does not have a LUB. (R by S)(S by R) (R by R)

26. Graph Theoretic Dimensions of Informal Organizations Least Upper Boundedness: Calculate LUB by looking at reachability. Where V = number of pairs that have no LUB, summed over all components, and:

27. Other characteristics of Hierarchy: DAG: Directed, Acyclic, Graph Graph that: contains no cycles at least one node has in-degree Rank Cluster Graph in which some number of nodes are mutually reachable, but asymmetrically reachable between groups. Tree A DAG with only one root Centralization We’ll return to this when we get to centralization

28. Another method: Approximation based on permutation One characteristic of a hierarchy is that most of the ties fall on the upper triangle of the adjacency matrix. Thus, one way to get an order is by juggling the rows and columns until most of the ties are in the upper triangle. 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 1 1 2 1 3 4 1 1 5 1 6 1 1 7 8 1 1 1 9 1 1 1 1 1 11 1 12 1 1 13 1 1 1 14 1 1 1 15 16 17 1 18 1

29. Another method: Approximation based on permutation 13 1 1 1 14 1 1 1 9 1 1 1 1 1 2 1 11 1 5 1 1 8 1 1 1 18 1 17 1 4 1 1 6 1 1 12 1 1 3 7 15 16 Re-ordered matrix