Social Balance Hierarchy Overview Background Basic Balance Theory

Social Balance & Hierarchy Overview Background: • Basic Balance Theory • Extensions to directed graphs • Triads Theoretical Implications: • Micro foundations of macro structure • Implications for networks dynamics

How do we capture hierarchy in a network? The shape of the popularity distribution is essentially constant over time and across each state/cohort combination. Males differ from females in having greater numbers of people receiving zero nominations. In all cases (and within each network), we get a fairly skewed distribution of popularity, reflecting a small number of very popular students.

Popularity Structure: Macro level stability The shape of the popularity distribution is essentially constant over time and across each state/cohort combination. Males differ from females in having greater numbers of people receiving zero nominations. In all cases (and within each network), we get a fairly skewed distribution of popularity, reflecting a small number of very popular students.

Popularity Structure: Macro level stability The Skewness coefficient captures the length of the tail of the popularity distribution, and is essentially constant across networks of more than ~100 students.

Popularity Structure: Macro level stability Add Health comparison Distribution of Popularity By size and city type

Social Balance & Transitivity Heider’s work on cognition of social situations, which can be boiled down to the relations among three ‘actors’: Object X P Person O Other Heider was interested in the correspondence of P and O, given their beliefs about X

Social Balance & Transitivity Each dyad (PO, PX, OX) can take on one of two values: + or 8 POX triples: Two Relations: Like: + Dislike - x + p + x + - o p + o p p o p - - + p x - - o x - - + + x x + o x + - o p + + o - - p - o

Social Balance & Transitivity The 8 triples can be reduced if we ignore the distinction between POX: x + p + x + - o p + p o p - p + x p + + + - + o - o x + - o p - - p - + o - - x - - + + x x + + o + + - - -

Social Balance & Transitivity We determine balance based on the product of the edges: + + (+)(+)(+) = (+) Balanced “A friend of a friend is a friend” - (-)(+)(-) = (-) Balanced “An enemy of my enemy is a friend” - (-)(-)(-) = (-) Unbalanced “An enemy of my enemy is an enemy” + (+)(-)(+) = (-) + + + - Unbalanced “A Friend of a Friend is an enemy”

Social Balance & Transitivity Heider argued that unbalanced triads would be unstable: They should transform toward balance + + Become Friends - Become Enemies + + + + - - -

Social Balance & Transitivity IF such a balancing process were active throughout the graph, all intransitive triads would be eliminated from the network. This would result in one of two possible graphs (Balance Theorem): Complete Clique Balanced Opposition Friends with Enemies with

Social Balance & Hierarchy Empirically, we often find that graphs break up into more than two groups. What does this imply for balance theory? It turns out, that if you allow all negative triads, you can get a graph with many clusters. That is, instead of treating (-)(-)(-) as an forbidden triad, treat it as allowed. This implies that the micro rule is different: negative ties among enemies are not as motivating as positive ties.

Social Balance & Transitivity Empirically, we also rarely have symmetric relations (at least on affect) thus we need to identify balance in undirected relations. Directed dyads can be in one of three states: 1) Mutual 2) Asymmetric 3) Null Every triad is composed of 3 dyads, and we can identify triads based on the number of each type, called the MAN label system

Social Balance & Transitivity Balance in directed relations Actors seek out transitive relations, and avoid intransitive relations. A triple is transitive If: i j & j k then: i k • A property of triples within triads • Assumes directed relations • The saliency of a triad may differ for each actor, depending on their position within the triad.

Social Balance & Transitivity Once we admit directed relations, we need to decompose triads into their constituent triples. Ordered Triples: b a c 120 C a a b b c c b c a b c; b; c; a; b; a; a a b b c c c b c a b a Transitive Vacuous Intransitive Vacuous

Network Sub-Structure: Triads (0) (1) (2) (3) (4) (5) 003 012 102 111 D 201 210 021 D 111 U 120 D (6) 300 Intransitive Transitive 021 U 030 T 120 U 021 C 030 C 120 C Mixed

An Example of the triad census Type Number of triads -------------------1 - 003 21 -------------------2 - 012 26 3 - 102 11 4 - 021 D 1 5 - 021 U 5 6 - 021 C 3 7 - 111 D 2 8 - 111 U 5 9 - 030 T 3 10 - 030 C 1 11 - 201 1 12 - 120 D 1 13 - 120 U 1 14 - 120 C 1 15 - 210 1 16 - 300 1 -------------------Sum (2 - 16): 63

Social Balance & Transitivity As with undirected graphs, you can use the type of triads allowed to characterize the total graph. But now the potential patterns are much more diverse 1) All triads are 030 T: A perfect linear hierarchy.

Social Balance & Transitivity Triads allowed: {300, 102} N* M M 1 0 0 1

Social Balance & Transitivity Cluster Structure, allows triads: {003, 300, 102} N* Eugene Johnsen (1985, 1986) specifies a number of structures that result from various triad configurations M M N* N* 1 M 1 1 M

Social Balance & Transitivity PRC{300, 102, 003, 120 D, 120 U, 030 T, 021 D, 021 U} Ranked Cluster: M A* A* N* M M A* A* 1 1 1 0 0 1 1 1 0 0 0 0 0 1 A* A* N* M M And many more. . .

Social Balance & Transitivity Substantively, specifying a set of triads defines a behavioral mechanism, and we can use the distribution of triads in a network to test whether the hypothesized mechanism is active. We do this by (1) counting the number of each triad type in a given network and (2) comparing it to the expected number, given some random distribution of ties in the network. See Wasserman and Faust, Chapter 14 for computation details, and the SPAN manual for SAS code that will generate these distributions, if you so choose.

Social Balance & Transitivity Triad: BA CL RC R 2 C TR HC 39+ p 1 p 2 p 3 003 012 102 021 D 021 U 021 C 111 D 111 U 030 T 030 C 201 120 D 120 U 120 C 210 300 Triad Micro-Models: BA: Ballance (Cartwright and Harary, ‘ 56) CL: Clustering Model (Davis. ‘ 67) RC: Ranked Cluster (Davis & Leinhardt, ‘ 72) R 2 C: Ranked 2 -Clusters (Johnsen, ‘ 85) TR: Transitivity (Davis and Leinhardt, ‘ 71) HC: Hierarchical Cliques (Johnsen, ‘ 85) 39+: Model that fits D&L’s 742 mats N : 39 -72 p 1 -p 4: Johnsen, 1986. Process Agreement Models. p 4

Social Balance & Transitivity Structural Indices based on the distribution of triads The observed distribution of triads can be fit to the hypothesized structures using weighting vectors for each type of triad. Where: l = 16 element weighting vector for the triad types T = the observed triad census m. T= the expected value of T ST = the variance-covariance matrix for T

Triad Census Distributions Standardized Difference from Expected Data from Add Health 400 t-value 300 200 100 0 -100 012 102 021 D 021 U 021 C 111 D 111 U 030 T 030 C 201 120 D 120 U 120 C 210 300

Social Balance & Transitivity For the Add Health data, the observed distribution of the tau statistic for various models was: Indicating that a ranked-cluster model fits the best.

Popularity Structure: Macro level stability

Testing Theories of Friendship Standardized Coefficients from an Exponential Random Graph Model 0. 8 0. 6 0. 4 0. 2 0 -0. 2 -0. 4 -0. 6 t gh Fi ng ki rin e ok e ac R Sm e x bs lu Se C de ra G ge le ol D C th Bo S PA G SE m e e e m m m Sa Sa ty vi iti ns ity v iti s an ity oc r ip ec tra In Tr R

Social Balance & Transitivity So far, we’ve focused on the graph ‘at equilibrium. ’ That is, we have hypothesized structures once people have made all the choices they are going to make. What we have not done, is really look closely at the implication of changing relations. That is, we might say that triad 030 C should not occur, but what would a change in this triad imply from the standpoint of the actor making a relational change?

Social Balance & Transitivity 102 030 C 120 C 111 U 021 C 201 003 012 111 D 021 D 210 300 120 U vacuous transition Increases # transitive 030 T 021 U Decreases # intransitive 120 D Decreases # transitive Increases # intransitive Vacuous triad Intransitive triad Transitive triad

Social Balance & Transitivity

Social Balance & Transitivity Random Walk TRIAD 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 100 200 300

Social Balance & Transitivity Transitive =. 5, Intran & Pos = 0. TRIAD 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 100 200 ITTER 300

Social Balance & Transitivity Trans=1. 0 TRIAD 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 100 200 ITTER 300

Social Balance & Transitivity Strong Negative (INT=-2) TRIAD 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 100 200 ITTER 300

Social Balance & Transitivity Observed triad transition patterns, from Hallinan’s data. 030 C 120 C 102 021 C 111 U 201 111 D 003 012 210 021 D 021 U 120 U 030 T 120 D 300

Social Balance & Transitivity Strong Transitivity Simulation Density Transitivity

Social Balance & Transitivity Strong Intransitivity Simulation

Social Balance & Transitivity Strong Intransitivity Simulation Ideal-typical results

Social Balance & Transitivity Moderate values on both Simulation

Social Balance & Transitivity

Social Balance & Transitivity A Brief History of Balance Through Time Newcomb, PI layout

Social Balance & Transitivity A Brief History of Balance Through Time

PNAS

Homophily as social balance: Variable attributes

Homophily as social balance: Fixed attributes

Micro mobility within settings Here we see (normalized) mobility tables, looking at movement between quintiles of the popularity distribution. While most movement is shortdistances, there is a good deal of movement in overall status. Only half of the most/least popular kids remain so a year later, dropping to between 30 and 40 across wider time-spans. Tables on popularity quintiles, correlations on percentile-rank between each wave.

Micro mobility within settings An individual-based perspective: chances of being in the top quintile x times given time in the bottom quintile. For example only 3. 9% of kids are in the most popular quintile all 5 waves, and a full 50% are in the top quintile at least once over the observation period. Similarly, only 1. 9% of kids are least popular all 5 waves, but 43% are least popular at least once.

Micro mobility within settings Another view: Tracking trajectories over time. Here we have a random sample of 15 kids’ trajectories, with increasing popularity in red, decreasing in blue. Note the many different patterns…

Micro mobility within settings (Add Health) An individual’s position in the status hierarchy is also not stable: Jefferson Sunshine

Capturing Trajectories Tracking Trajectories: How do we characterize a popularity trajectory? - Cluster analysis: - Goal is to find similar patterns across the set of 5 -wave trajectories. - Advantage: you have great flexibility in the actual pattern - Disadvantages: - Exploratory & can capitalize on randomness - Time/Data intensive - Need to decide on numbers-of-clusters -In the end this was not convincing: no clear separation in the clusters nor interpretable clustering tree. -Instead, what jumps out is the sheer variability in experiences across cases.

Capturing Trajectories Tracking Trajectories: Smooth “field of experiences” approach: - Fit a simple linear model to change over time for each student. : the combination of intercept and slope then describe the *general* trajectory of popularity each student experiences…. …but, the trend removes all the variability in movement around the trend; which is likely important for one’s experience in the setting: we want to distinguish a steady trajectory from a wildly swinging one. So we add an additional indicator of the standard deviation of popularity to the model. Combined, these two features (regression parameters & variance) capture the direction and “width” of trajectory experiences.

Capturing Trajectories The regression estimates define a simple 2 -d space of slope (Y) and intercept (x). This figure represents the distribution of cases across that space, with key points labeled.

Capturing Trajectories The regression estimates define a simple 2 -d space of slope (Y) and intercept (x). This region represents steady & sharp increases in popularity, from a low starting point to a high ending point. These kids are upwardly mobile

Capturing Trajectories The regression estimates define a simple 2 -d space of slope (Y) and intercept (x). In contrast, here we have steady but sharp decline – these kids are downwardly mobile.

Capturing Trajectories The regression estimates define a simple 2 -d space of slope (Y) and intercept (x). …and three “steady” states: kids with stable popularity. Note there is no clumping in the space, but a true continuous field, and also note that variability is correlated with slope, so we need to disentangle that in the models.

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

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

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.

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)

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.

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 030 T triads. A B C D E Number of Type triads -----------1 - 003 0 -----------2 - 012 0 3 - 102 0 4 - 021 D 0 5 - 021 U 0 6 - 021 C 0 7 - 111 D 0 8 - 111 U 0 9 - 030 T 10 10 - 030 C 0 11 - 201 0 12 - 120 D 0 13 - 120 U 0 14 - 120 C 0 15 - 210 0 16 - 300 0 -------------Sum (2 - 16): 10 What process could generate all 030 T triads?

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

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

Dominance Strategies That ensure a transitive hierarchy The “Double Attack” Strategy: The first attacker quickly attacks the bystander. This means we arrive at 21 D, 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 21 U, and any dominance between the first and second attacker will lead to a transitive triple. 003 012 030 T 021 D 003 012 030 T 021 U

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

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

The Origins of Status Hierarchies: A formal model and empirical test Hierarchy & Inequality appear virtually universal. How do we account for it? Two alternative explanations: • Individualist – that people vary in qualities that are locally salient • Structuralist – that differentiation results from the quality of social positions individuals occupy. • Third option: Hierarchy is explained as the product of an emergent social process without presupposing that the resulting assignment of actors to positions is a reflection of underlying qualities. The key is that: “social hierarchies are understood to emerge and persists spontaneously rather than by conscious creation, but at the same time without ensuring that rewards exactly reflect differences in individual qualities” (p. 1146)

The Origins of Status Hierarchies: A formal model and empirical test Theoretical claim: “the reason positions with greater and lesser advantage exist is that judgments about relative quality are socially influenced. Socially influenced judgments amplify underlying differences, so that actors who objectively rank above the mean on some abstract quality dimension are overvalued with those ranking below the mean are undervalued…. Amplification occurs because observable interactions expressing judgments of quality are also cues to other actors seeking guidance for their own judgments. ” (p. 1147) Examples include the “Mathew effect” in science

The Origins of Status Hierarchies: A formal model and empirical test This claim is theoretically consistent with a Nash Equilibrium, in which everyone’s current choice of action is preferable to (or as good as) the alternatives so long as everyone else’s choice of action remains constant. IF attribution builds on others’ attributions, then, “the patterns should tend toward a stable state in which collective attributions confirm themselves in each time period. ” The theory implies that “the only equilibria possible when there absolutely no underlying differences across individuals are one in which everyone is ranked equally and one in which one actor receives attention while all others receive none. ” (p. 1149) This follows because of the ‘cascade’ effect of social influence.

The Origins of Status Hierarchies: A formal model and empirical test Since most observed structures fall between these two poles, something else must be going on as well. The mechanism employed rests on the returns to asymmetric admiration. “It is painful to pay attention to another person if the favor is not repaid. By the same token, it is particularly pleasant to receive attention when it is not solicited. ” “Individuals should be less willing to demonstrate esteem toward those who do not return the favor and conversely may prefer to receive such demonstrations without reciprocating. ”

The Origins of Status Hierarchies: A formal model and empirical test The theory then makes three predictions: 1) Asymmetry in social relationships will be proportional to the difference in choice status (indegree) between pairs of actors. A high-status actor will be more weakly connected to any low-status actor that the latter is to her or him. This has to exceed the chance/tautology levels. 2) All else equal, pairs of actors who are similar in choice status will also be similar in the patterns of attachments they make to others. 3) Across all actors, the sum of attachments directed to others will be proportional to, but more evenly distributed than, the sum of attachments received.

The Origins of Status Hierarchies: A formal model and empirical test Formally, the model: 1) Assumes a closed, finite population 2) The quantity of attachments varies across individuals 3) Each actor cares about: a) The quality of each potential alter b) The gap between their and the alter’s attachment to each other. Ui = utility for person i aij = attachment of person i to person j qj = quality of person j S = a weight of symmetry considerations relative to the quality of i’s alters in determining i’s welfare. In this model, q is determined exogenously

The Origins of Status Hierarchies: A formal model and empirical test To extend the model for social influence, assume that quality judgments are a function of peer influence: qij= i’s assessment of j’s quality Qj = exogenous quality of j W = relative weight of social influence on I’s judgment of j.

The Origins of Status Hierarchies: A formal model and empirical test Prediction from equation 6 (aggregate quality centered at 0)

The Origins of Status Hierarchies: A formal model and empirical test (Close-up of threshold region)

The Origins of Status Hierarchies: A formal model and empirical test Analysis of the model results is a set of testable propositions: 1) Asymmetry in attachments between any two actors is proportional to their differences in choice status 2) The relationship between choice status and asymmetry declines with group size 3) Any pair of actors i, j will be similar in the attachments they direct toward others in proportion as they are similar in choice status. 4) If sum(choice kj) – sum(choice ki) = 0, then aik – aij = 0. 5) Actors direct attachments to others in proportion to the quantity of attachments received 6) The slope of the function that transforms choice status into attachments directed outward is always less than unity. Consequently, the distribution of choice status (popularity) is more unequal than the distribution of out degree.

The Origins of Status Hierarchies: A formal model and empirical test . 80

The Origins of Status Hierarchies: A formal model and empirical test

The Origins of Status Hierarchies: A formal model and empirical test So the model seems to be well supported by the data. -I’ve played a little with these models and Add Health, and they don’t perform as well, but the data don’t fit the assumptions either….

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

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.

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

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

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.

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

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

Graph Theoretic Dimensions of Informal Organizations Graph Efficiency: 1 4 2 1 6 3 5 7 2 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(V 2) = (3 -2)=1, so Max(V) = 4. Efficiency = (1 - 2/4 ) =. 5

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 Vi = S(Gi)/2 - Ni-1; ii) calculate Max(Vi) = Ni(Ni-1) - (Ni-1) c) V = S(Vi), Max(V)= S(Max(Vi) d) efficiency = 1 - V/Max(V) Substantively, this must be a function of the average density of the components in the graph.

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. E H B F A G C D 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.

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.

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

Graph Theoretic Dimensions of Informal Organizations Least Upper Boundedness: Calculate LUB by looking at reachability. 1 2 3 4 5 6 7 8 9 Reachable Trans 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 X 1 2 3 4 5 6 7 8 9 Reachable matrix 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 (R by S) (S by R) R`*R = CR = 1 2 3 4 5 6 7 8 9 1 1 1 0 0 0 1 Common Reach 2 3 4 5 6 7 8 1 1 0 0 0 2 1 2 2 0 0 0 1 2 1 1 0 0 0 2 1 3 2 0 0 0 2 1 2 3 0 0 0 0 1 1 1 0 0 1 2 1 0 0 1 1 2 1 9 1 1 2 1 5 (R by R) Any place with a zero indicates a pair that does not have a LUB.

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:

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

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 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 1 1 1 1 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 1 1 1 1 1 1 1

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

Group Hierarchy We wanted to examine hierarchy variability across all the add health schools. Mc. Farland et al (2014) We have a continuous measure based on the triad distribution, but also created this as a way to illustrate: a) b) c) d) e) Blockmodel the network to identify positions. Use Becker’s algorithm to sort positing mixing matrix in hierarchical order. Network level hierarchy is ratio of ties above diagonal to ties below Position level hierarchy (y-axis) is ratio of ties received to ties sent For clarity, only ploted ties that are at least 75% of the mean value Hierarchy here is the coefficient from the GWESP term in our models

Group Hierarchy We wanted to examine hierarchy variability across all the add health schools. We have a continuous measure based on the triad distribution, but also created this as a way to illustrate: a) b) c) d) e) Blockmodel the network to identify positions. Use Becker’s algorithm to sort positing mixing matrix in hierarchical order. Network level hierarchy is ratio of ties above diagonal to ties below Position level hierarchy (y-axis) is ratio of ties received to ties sent For clarity, only ploted ties that are at least 75% of the mean value Mc. Farland et al (2014)

Martin: Social Structures Martin’s project is directly related to the early balance models: Can we find necessary/likely/expected structures created by the constraints forced by the type of tie. Mutual (a. Rb b. Ra, a. Ra, (a. Rb & b. Rc, a. Rc) Equality Archetype is the clique Asymmetric (a. Rb has no implication for b. Ra) Archetype is the gift/donation Question’s the emergence of equality & stability given tendencies toward “Matthew effects” Anti-symmetric (a. Rb not(b. Ra), transitivity) Pecking Orders & Patronage

Martin: Social Structures Mutual (a. Rb b. Ra, a. Ra, (a. Rb & b. Rc, a. Rc) Equality Archetype is the clique Problem is that the structure collapses under its own weight, since we can’t have clusters of N*N-1 people. So we differentiate: By types of tie/person:

Martin: Social Structures Mutual (a. Rb b. Ra, a. Ra, (a. Rb & b. Rc, a. Rc) Equality Archetype is the clique Problem is that the structure collapses under its own weight, since we can’t have clusters of N*N-1 people. So we differentiate: Social distance OR geographic distance

Martin: Social Structures Mutual (a. Rb b. Ra, a. Ra, (a. Rb & b. Rc, a. Rc) Equality Archetype is the clique Problem is that the structure collapses under its own weight, since we can’t have clusters of N*N-1 people. So we differentiate (creating weighted ties): Then he moves on to discuss alternative social rules to balance, which never quite works leading to claims about rules across equivalents (which we will cover later)

Martin: Social Structures Mutual (a. Rb b. Ra, a. Ra, (a. Rb & b. Rc, a. Rc) Equality Archetype is the clique A social constraint space for network structure

Martin: Social Structures Asymmetric (a. Rb has no implication for b. Ra) How do we make sense of our options? individual models (p 1) or generalized exchange (later)

Martin: Social Structures Anti-symmetric (a. Rb not(b. Ra), transitivity) Pecking Orders & Patronage