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Positional Analysis for

Social Networks & Health

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Outline

Introduction:

Positions vs. Connections

Types of positional models

“Centrality”
Block Models

Centrality Models
Block Models
Conclusions

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Connectionist:

Positional:

Networks as pipes

Networks as roles

Ego

Complete

Multiple

Structural Holes
Density
Mixing Models
Size

Community detection
Reachability
Homophily
Degree Distribution
Social Balance
ERGM

Multi-layer networks

Multi-level models of multiple networks

Local Roles

(Mandel 1983, Mandel & Winship 1984)

Relational Block Models
Motifs

Centrality

Cohesive blocking

2 ideas:

Patterns in networks
Patterns of networks

Connections & Positions: Network Problems

Positional Analysis

Introduction

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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.
Identifying these sets is the goal of block-model analyses.

Nadel: The Coherence of Role Systems

Background ideas for White, Boorman and Brieger. Social life as interconnected system of roles
Important feature: thinking of roles as connected in a role system = social structure

White, Boorman and Breiger: Social structure from Multiple Networks I. Blockmodels of Roles and Positions

The key article describing the theoretical and technical elements of block-modeling

Positional Analysis

Introduction

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Nadel: The Coherence of Role Systems

Elements of a Role:

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.

Positional Analysis

Introduction

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Nadel describes how various roles fit together to form a coherent whole. Roles are collected in people through the ‘summation of roles”

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.

Positional Analysis

Introduction

Nadel: The Coherence of Role Systems

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

Provides food for

Romantic Love

Bickers with

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

Positional Analysis

Introduction

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P

C

Provides food for

Romantic Love

Bickers with

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

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

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

Positional Analysis

Introduction

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Positional Analysis

Types of positional models

Informally we tend to think of “position” in two senses, a node-metric sense (“Centrality”) and a collective property/group sense (“blockmodel”).

The node metric sense uses some graph-theoretic property of the node to characterize a social position of interest. Generally treats positional features as scales rather than categories.

Examples:

“Loners” 🡪 operationalized as nodes with degree zero

“Leaders” 🡪 operationalized as nodes with lower network constraint

“Bridges” 🡪 operationalized as nodes with high betweenness

“Social butterfly” 🡪 operationalized as nodes that change relations frequently

These are not all strictly centrality scores, but often they are (which is why the connotations of being in the middle of thigs implied by the term “centrality” is really sort of not useful).

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Positional Analysis

Types of positional models

Informally we tend to think of “position” in two senses, a node-metric sense (“Centrality”) and a collective property/group sense (“blockmodel”).

The collective property sense identifies sets of nodes with equivalent* tie patterns to define a partition of nodes into classes. It makes the assumption that these are substantively discrete – classes rather than scales.

Examples:

set of all people who can hire faculty 🡪 “Deans”

set of all people descendant from the King 🡪 “Potential heir”

set of all female siblings of a father 🡪 “Aunts”

set of all allies of an enemy 🡪 “Enemy”

Positions differ from communities in two ways:

Adjacency– two members of the same community should have a very high probability of being adjacent; two members of the same position need not.
Ability to compound the relation – two-step links through a position is meaningful (i.e. “father's brother” is “uncle”), but is not for communities.

*Equivalence means something special in this context

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Positional Analysis

Types of positional models

Both approaches tend to be used in health as ways to create variables in a GLM.

Generally that simplifies things as the final modeling parts are not substantively different from standard health behavior models.

Does raise some obvious non-independence issues worth at least thinking through.

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Positional Analysis

Centrality models

The general approach is to model a health outcome as a function of a node’s network characteristic.

Self-rated health = network metric + <other stuff>

The logic behind this model is that we expect some particular feature of the node’s position in the network to be associated with health.

Metric Family	Example	Mechanism
Relational Volume	Isolate	Loneliness leads to self-doubt, depression
	Popularity	Approval of others boosts self esteem
Multiplexity	Proportion of coworkers who are friends	Integration of social worlds leads to consistency of expectations and better mental health
Bridging	Betweenness centrality	Access to different populations promotes diverse information
	Intransitivity of local friendships	Negotiating friends who dislike each other causes stress
Cohesion	Reciprocity ratio	Other’s recognition improves sense of self
	Proportion of friends that overlap	Tight-knit social units provide care

(and many others)

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Positional Analysis

Centrality models

And many others…is huge!

http://schochastics.net/sna/periodic.html

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Positional Analysis

Centrality models

Degree Centrality: Count of the number of adjacent nodes

Closeness Centrality: Inverse of average distance to all other nodes in the network

Betweenness Centrality: Sum of pairs who’s geodesic a node sits on

Eigenvector Centrality: Normalized recursive sum of adjacent nodes’ degrees.

🡪 eigenvector of the largest(1^st) eigenvalue

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Positional Analysis

Centrality models

Borgatti & Everett (2020) provide 3 perspectives to characterize centrality:

Walk structure participation perspective - organizes centrality by the pattern of walks & paths in the definition

Induced Centrality perspective – centralities matter because of what they do to the overall network structure – captures the extent to which a node contributes to a graph level property.

Flow outcomes perspective – focuses on the ways in which the node affects a propagation process outcome of the network.

Most^* common centralities can be discussed from each of these perspectives; they are ways to provide understanding for the network process of interest.

*Most because it is possible to define silly path or graph properties that’d technically count, but nobody uses these.

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Positional Analysis

Centrality models

Walk structure participation perspective : 3 dimensions

Kind of traversal: Geodesic, Path, Trail, Walk

Position on the traversal: Endpoint (radial) or interior (medial)

Property of the traversal: Frequency or Length?

For example:

Degree is a frequency, endpoint & geodesic

Closeness is length, endpoint, geodesic

betweenness is frequency, interior, geodesic

eigenvector is length*frequency, endpoint, walks

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Positional Analysis

Centrality models

1) Walk structure participation perspective: 3 dimensions

This is comprehensive, but often not clearly substantive.

i.e. doesn’t always provide a clear “which should I use” sort of guide.

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Positional Analysis

Centrality models

2) Induced Centrality perspective

A node is as important to the network as its removal would be consequential.

There are induced-interpretations of most standard graph metrics (though not all). Some of these are trivial (degree’s relation to density, say) others highly dependent on the path structure (betweenness and distance).

Particularly useful for thinking about mechanisms on the network. For example, if you calculate the speed with which a bit diffuses through a network then recalculate removing one node at a time, you get each node’s unique contribution to the total diffusion risk in the network.

Note as well you can apply this idea to groups of nodes – all nodes in some class, or all pairs of nodes, etc.

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Positional Analysis

Centrality models

3) Flow outcomes perspective

Metrics matter based on how they govern a particular kind of flow. Borgatti & Everett (2020) give a handful of archetypes:

Name	Traversal	Contagion	Example
Used Book	Trail	Move / Transfer	Read a book, pass it on. But if it returns to you, pass it to somebody new.
News/Gossip	Trail	Copy, directed	Pass a story to confidants, who pass it on, but not to same person repeatedly.
Itinerant	Path	Move/ Transfer	Live with somebody for a while, but outstay welcome so can’t come back
Virus	Path	Copy, directed	SIR model, highly infective reaches all neighbors quickly
Coin	Walk	Move/ Transfer	Coin moves through the economy – only in one place at a time.
Attitude	Walk	Copy, bidirected	All continuously affecting each other
Travel	Geodesic	Move	Search out fastest route.

Interestingly, most off-the-shelf centrality scores don’t map onto these common processes exactly

(mainly because most work w. geodesics rather than paths/walks)

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Positional Analysis

Centrality models

3) Flow outcomes perspective

Metrics matter based on how they govern a particular kind of flow. Borgatti (2005) version:

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A key element of the positional approach to networks is to consider nodes as combinations of metrics, not just a single dimension. This creates a linkage between the “metric” and “classes” version of positional models.

Low

Degree

Low

Closeness

Low

Betweenness

High Degree

Embedded in cluster that is far from the rest of the network

Ego's connections are redundant - communication bypasses him/her

High Closeness

Key player tied to important important/active alters

Probably multiple paths in the network, ego is near many people, but so are many others

High Betweenness

Ego's few ties are crucial for network flow

Very rare cell. Would mean that ego monopolizes the ties from a small number of people to many others.

Positional Analysis

Centrality models

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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:

P

C

Provides food for

Romantic Love

Bickers with

(and there are, of course, many other relations inside a family)

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

Positional Analysis

Block models

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Blockmodeling: basic steps

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

At the end of the day, this is community detection on a role-relevant similarity matrix rather than an adjacency matrix.

The “trick” is defining similar-with-respect-to-what

Positional Analysis

Block models

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If the model is going to be based on asymmetric or multiple relations, you simply stack the various relations, usually including both “directions” of asymmetric relations:

P

C

Provides food for

Romantic Love

Bickers with

Sim

1 1 0 0 0

0 0 1 1 1

Positional Analysis

Block models

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Positional Analysis

Block models

Traditional equivalence measures

Classic models build on a continuum between structural equivalence regular equivalence:

Structural equivalence: Same ties to the exact same people. Nodes are only distinguishable by their label. Example: Two pendants around the same hub.

Automorphic equivalence: Same pattern of ties to all others in the network. Nodes are indistinguishable on any summary metric. Example: Sports team positions

Regular Equivalence: Same types of ties to similar types of people. Idea is that nodes of one class relate similarly to nodes of another class, though they may differ in volume. Example: Nurses to Doctors; managers to vice-presidents, kids to parents.

In practice, it tends to be fairly difficult to distinguish the three forms as the operationalization rarely generates pure SE. So a poor operationalization of SE gives you AE, or something like a mix of AE and RE…

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Positional Analysis

Block models

Traditional equivalence approaches

ConCor: Convergence of Iterated Correlations (Boorman, Breiger & White)

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 0 0 1 1 0 0 0 0 0 0 0 0

Here I have blocked structurally equivalent actors

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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

-0.19 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

-0.26 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

Positional Analysis

Block models

Traditional equivalence approaches

ConCor: Convergence of Iterated Correlations (Boorman, Breiger & White)

Base similarity is correlation across the rows/columns of each pari to all other pairs

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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

-.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

-.75 0.95 -.75 -.75 0.73 0.73 -.77 -.77 -.77 -.77 1.00 1.00 1.00 1.00

Concor iteration 1:

Positional Analysis

Block models

Traditional equivalence approaches

ConCor: Convergence of Iterated Correlations (Boorman, Breiger & White)

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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

-.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

-.99 0.99 -.97 -.97 0.97 0.97 -.99 -.99 -.99 -.99 1.00 1.00 1.00 1.00

Positional Analysis

Block models

Traditional equivalence approaches

ConCor: Convergence of Iterated Correlations (Boorman, Breiger & White)

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1.00 -1.0 1.00 1.00 -1.0 -1.0 1.00 1.00 1.00 1.00 -1.0 -1.0 -1.0 -1.0

-1.0 1.00 -1.0 -1.0 1.00 1.00 -1.0 -1.0 -1.0 -1.0 1.00 1.00 1.00 1.00

1.00 -1.0 1.00 1.00 -1.0 -1.0 1.00 1.00 1.00 1.00 -1.0 -1.0 -1.0 -1.0

-1.0 1.00 -1.0 -1.0 1.00 1.00 -1.0 -1.0 -1.0 -1.0 1.00 1.00 1.00 1.00

1.00 -1.0 1.00 1.00 -1.0 -1.0 1.00 1.00 1.00 1.00 -1.0 -1.0 -1.0 -1.0

-1.0 1.00 -1.0 -1.0 1.00 1.00 -1.0 -1.0 -1.0 -1.0 1.00 1.00 1.00 1.00

Concor iteration 3:

Positional Analysis

Block models

Traditional equivalence approaches

ConCor: Convergence of Iterated Correlations (Boorman, Breiger & White)

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Concor iteration 3: Permuted

1.00 1.00 1.00 1.00 1.00 1.00 1.00 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0

-1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1

3

4

7

8

9

10

2

5

6

11

12

13

14

Positional Analysis

Block models

Traditional equivalence approaches

ConCor: Convergence of Iterated Correlations (Boorman, Breiger & White)

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Because CONCOR splits every sub-group into two groups, you get a partition tree that looks something like this:

Positional Analysis

Block models

Traditional equivalence approaches

ConCor: Convergence of Iterated Correlations (Boorman, Breiger & White)

The question is how to stop splitting – one level, each branch, etc. Very hands on and inductive. Original advice was to fit until you had all one/zero blocks, but that rarely works.

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CONCOR example:

Consider a simple senate voting network:

Network is dense, since every cell has some score and dynamic the pattern changes over time.

Color by structural equivalence…

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Network is dense, since every cell has some score and dynamic the pattern changes over time.

Adjust position to collapse SE positions.

CONCOR example:

Consider a simple senate voting network:

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Network is dense, since every cell has some score and dynamic the pattern changes over time.

And then adjust color, line width, etc. for clarity.

While we’ve gone some distance with identifying relevant information from the mass, how do we account for time?

CONCOR example:

Consider a simple senate voting network:

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CONCOR example:

Repeat at each wave, linking positions over time

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CONCOR example:

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Automorphic and Regular equivalence are more difficult to find, and require iteratively searching over possible class assignments for sets that have the same graph theoretic patterns. Usually start with a set of nodes defined as similar on a number of network measures, then look within these classes for automorphic equivalence classes.

The classic reference is REGE (White & Reitz 1985), which recursively defines the degree of equivalence between pairs and then adjusts for as many iterations as you specify. Slow and doesn’t always converge.

A theoretically appealing method for finding structures that are very similar to regular equivalence, role equivalence, uses the triad census. Each node is involved in (n-1)(n-2)/2 triads, and occupies a particular position in each of these triads.

Burt (1990) “Detecting Role Equivalence” Social Networks

Positional Analysis

Block models

Traditional equivalence approaches

Role equivalence

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003

(0)

012

(1)

102

021D

021U

021C

(2)

111D

111U

030T

030C

(3)

201

120D

120U

120C

(4)

210

(5)

300

(6)

16 directed triads

“A friend of a friend is a friend”

Triads also provide a tight coupling between behavior rules and (local) structure

Triad Census: The periodic table of social elements

Positional Analysis

Block models

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003

(0)

012

(1)

102

021D

021U

021C

(2)

111D

111U

030T

030C

(3)

201

120D

120U

120C

(4)

210

(5)

300

(6)

16 directed triads

“Hierarchical agreement”

Triads also provide a tight coupling between behavior rules and (local) structure

Triad Census: The periodic table of social elements

Positional Analysis

Block models

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003

(0)

012

(1)

102

021D

021U

021C

(2)

111D

111U

030T

030C

(3)

201

120D

120U

120C

(4)

210

(5)

300

(6)

16 directed triads

“Reciprocity”

Triads also provide a tight coupling between behavior rules and (local) structure

Triad Census: The periodic table of social elements

Positional Analysis

Block models

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An Example of the triad census

Type Number of triads

---------------------------------------

1 - 003 21

---------------------------------------

2 - 012 26

3 - 102 11

4 - 021D 1

5 - 021U 5

6 - 021C 3

7 - 111D 2

8 - 111U 5

9 - 030T 3

10 - 030C 1

11 - 201 1

12 - 120D 1

13 - 120U 1

14 - 120C 1

15 - 210 1

16 - 300 1

---------------------------------------

Sum (2 - 16): 63

Positional Analysis

Block models

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Limits on the set of triads constrains the global structure

1) All triads are 030T:

A perfect linear hierarchy.

030T

Positional Analysis

Block models

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Triads allowed: {300, 102}

M

N*

1

0

102

300

Positional Analysis

Block models

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Cluster Structure, allows triads: {003, 300, 102}

M

N*

M

N*

Eugene Johnsen (1985, 1986) specifies a number of structures that result from various triad configurations

1

003

102

300

Positional Analysis

Block models

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PRC{300,102, 003, 120D, 120U, 030T, 021D, 021U} Ranked Cluster:

M

N*

M

N*

M

A*

1

0

1

0

And many more...

Positional Analysis

Block models

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003

(0)

012

(1)

102

021D

021U

021C

(2)

111D

111U

030T

030C

(3)

201

120D

120U

120C

(4)

210

(5)

300

(6)

Intransitive

Transitive

Mixed

Positional Analysis

Block models

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003

012_S

012_E

012_I

102_D

102_I

021D_S

021D_E

021U_S

021U_E

021C_S

021C_B

021C_E

111D_S

111D_B

111D_E

111U_S

111U_B

111U_E

030T_S

030T_B

030T_E

030C

201_S

201_B

120D_S

120D_E

120U_S

120U_E

120C_S

120C_B

120C_E

210_S

210_B

300

Triadic Position Census: 36 Positions within 16 Directed Triads

Indicates the position.

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A B C D E F G H I J K L M N

36 36 10 10 10 10 43 43 43 43 43 43 43 43

0 0 0 0 0 0 0 0 0 0 0 0 0 0

20 20 41 41 41 41 14 14 14 14 14 14 14 14

9 9 11 11 11 11 12 12 12 12 12 12 12 12

0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 10 1 1 1 1 8 8 8 8 8 8 8 8

2 2 10 10 10 10 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 5 5 5 5 1 1 1 1 1 1 1 1

Triad position vectors for a simple example network with 3 positions:

003

102_D

102_I

201_S

201_B

300

Positional Analysis

Block models

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A 1.00 1.00 0.64 0.64 0.64 0.64 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98

B 1.00 1.00 0.64 0.64 0.64 0.64 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98

C 0.64 0.64 1.00 1.00 1.00 1.00 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50

D 0.64 0.64 1.00 1.00 1.00 1.00 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50

E 0.64 0.64 1.00 1.00 1.00 1.00 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50

F 0.64 0.64 1.00 1.00 1.00 1.00 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50

G 0.98 0.98 0.50 0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

H 0.98 0.98 0.50 0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

I 0.98 0.98 0.50 0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

J 0.98 0.98 0.50 0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

K 0.98 0.98 0.50 0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

L 0.98 0.98 0.50 0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

M 0.98 0.98 0.50 0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

N 0.98 0.98 0.50 0.50 0.50 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Triad position vectors for a simple example network with 3 positions:

Positional Analysis

Block models

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Triad position vectors for a simple example network with 3 positions:

How do we do it at scale?

Positional Analysis

Block models

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One Prosper School

(6^th grade)….of 368.

Positional Analysis

Block models

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Stage 1: Within settings:

Build triadic involvement distance matrix
Ward’s min Variance Clustering
Calculate modularity score for the partition applied to the similarity matrix at each cut level
Accept the cut with the highest modularity score
Units are students

Positional Analysis

Block models

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One Prosper School

(6^th grade)….each color is a position

Positional Analysis

Block models

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Example positions identified in a single school network

(role 7 is a “leading crowd” in the simplest sum-of-in-degree sense)

Stage 1: Within settings:

Build triadic involvement distance matrix
Ward’s min Variance Clustering to build dendrogram
Calculate modularity score for the partition applied to the similarity matrix
Accept the cut with the highest modularity score
🡪 2912 clusters

Thus far…standard single-network model.

But how do you compare blocks across networks when label values are meaningless?

Positional Analysis

Block models

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Stage 2: 2^nd-order clustering across settings

Calculate the triad position profile for each within-setting cluster
Identify similarity across the cluster profiles by clustering a 2^nd time
Units are clusters (of students)

Positional Analysis

Block models

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Popular

Loners

Uninvolved

Outsiders

Hangers-on

Aloofs

Leading

Crowd

Segmented

Peers

Lieutenants

Federated

Friends

Core

Peripheral

1391

1521

928

463

High(er) out degree

Low out-degree

740

409

534

206

Asym

Transitivity

372

1149

Leading Crowd

Secondary Core

165

207

T300

Asym

165

763

Higher indegree

Lower indegree

2^nd Order Clustering Dendrogram

2912 within-school clusters

Positional Analysis

Block models

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Uninvolved

outsiders

Popular Loners

Hangers-on

Power Centrality

Closeness Centrality

Total Degree

Ego Density

Ego Transitivity

In-Degree

Information Centrality

Out-Degree

Two-step Reach

Reciprocity

Betweenness

Power Centrality

Closeness Centrality

Total Degree

Ego Density

Ego Transitivity

In-Degree

Information Centrality

Out-Degree

Two-step Reach

Reciprocity

Betweenness

Power Centrality

Closeness Centrality

Total Degree

Ego Density

Ego Transitivity

In-Degree

Information Centrality

Out-Degree

Two-step Reach

Reciprocity

Betweenness

Positional Analysis

Block models

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Role set characteristics: Core

🡪 Secondary Core Branch

Federated

Friends

Segmented

Peers

Power Centrality

Closeness Centrality

Total Degree

Ego Density

Ego Transitivity

In-Degree

Information Centrality

Out-Degree

Two-step Reach

Reciprocity

Betweenness

Power Centrality

Closeness Centrality

Total Degree

Ego Density

Ego Transitivity

In-Degree

Information Centrality

Out-Degree

Two-step Reach

Reciprocity

Betweenness

Lieutenants

Asymmetric

Bridges

Power Centrality

Closeness Centrality

Total Degree

Ego Density

Ego Transitivity

In-Degree

Information Centrality

Out-Degree

Two-step Reach

Reciprocity

Betweenness

Positional Analysis

Block models

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Role set characteristics: Core 🡪 Leading Crowd

Power Centrality

Closeness Centrality

Total Degree

Ego Density

Ego Transitivity

In-Degree

Information Centrality

Out-Degree

Two-step Reach

Reciprocity

Betweenness

Leading

Crowd

Aloof

Elites

Power Centrality

Closeness Centrality

Total Degree

Ego Density

Ego Transitivity

In-Degree

Information Centrality

Two-step Reach

Reciprocity

Betweenness

Out-Degree

Positional Analysis

Block models

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(8)

(4)

(6)

(7)

Uninvolved

Outsiders

Hangers-on

Aloofs

Leading Crowd

Segmented

Peers

Lieutenants

Federated Friends

Popular Loner

Positional Analysis

Block models

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Role positions largely cross-cut demographics and behaviors:

Positional Analysis

Block models

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Wave 1 to Wave 2 Mobility

(row normalized mobility table; cells shaded by ratio of

observed to expected if >2.5)

Positional Analysis

Block models

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Wave 2 to Wave 3 Mobility

(row normalized mobility table; cells shaded by ratio of

observed to expected if >2.5)

Positional Analysis

Block models

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Wave 3 to Wave 4 Mobility

(row normalized mobility table; cells shaded by ratio of

observed to expected if >2.5)

Positional Analysis

Block models

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Wave 4 to Wave 5 Mobility

(row normalized mobility table; cells shaded by ratio of

observed to expected if >2.5)

Positional Analysis

Block models

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Individual mobility models*

Log-odds:

Female

Statistically significant cell values as bold OR

Girls tend to migrate to the “Federated Friends” position (blue column), and are unlikely to leave (red row)

*Setting level random effects multinomial logistic regression models conditional on origin, including wave, sex, race, slun, deviance, church, school attachment, grades, school-level reciprocity, transitivity, degree centralization, structural cohesion & hierarchy score.

Positional Analysis

Block models

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Individual mobility models*

Log-odds:

Female

Statistically significant cell values as bold OR

Girls are not generally drawn to the Leading crowd (no column effect), but once there, likely to stay

*Setting level random effects multinomial logistic regression models conditional on origin, including wave, sex, race, slun, deviance, church, school attachment, grades, school-level reciprocity, transitivity, degree centralization, structural cohesion & hierarchy score.

Positional Analysis

Block models

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Individual mobility models*

Log-odds:

School Lunch

Statistically significant cell values as bold OR

High-status poor is unstable, likely to become popular loners or outsiders, and outside status is unlikely to move to anything else.

*Setting level random effects multinomial logistic regression models conditional on origin, including wave, sex, race, slun, deviance, church, school attachment, grades, school-level reciprocity, transitivity, degree centralization, structural cohesion & hierarchy score.

Positional Analysis

Block models

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Individual mobility models*

*Setting level random effects multinomial logistic regression models conditional on origin, including wave, sex, race, slun, deviance, church, school attachment, grades, school-level reciprocity, transitivity, degree centralization, structural cohesion & hierarchy score.

Log-odds:

Deviant

Statistically significant cell values as bold OR

Deviant kids are more likely to move from periphery positions to core positions: deviance confers status…

Positional Analysis

Block models

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Individual mobility models*

Log-odds:

Grades (GPA)

Statistically significant cell values as bold OR

*Setting level random effects multinomial logistic regression models conditional on origin, including wave, sex, race, slun, deviance, church, school attachment, grades, school-level reciprocity, transitivity, degree centralization, structural cohesion & hierarchy score.

…but good grades also predict being in the most central position, and are protective against moving to peripheral position.

Positional Analysis

Block models

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Positional Analysis

Block models

Alternative approach: Cluster node metrics.

Automorphic equivalence: Same pattern of ties to all others in the network. Nodes are indistinguishable on any summary metric. Example: Sports team positions

This implies that if you can characterize a node’s position within a (set of) network(s) as a vector of summary scores, then you can simply cluster that vector to find nodes that are similar across a wide set. This can be very effective for large networks.

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It's also possible to use simple-to-calculate scores in unique ways.

So while its time and space/bandwidth consuming to run a full triad-based structural equivalence model over a giant network; you can calculate a host of local and bridging sorts of scores, then cluster those to get positions quickly.

55.6%

13.6%

12%

12.5%

3.4%

1%

1.4%

0.5%

0.06%

Low activity

retweet bridging

Low activity,

Pendants

Low activity,

Mixed bridge

high activity

retweet bridging

Quote Bridges

Reply

Bridges

(wgt)

Active group

members

Local Authorities,

fighters

Superstar hubs

(each x value is a within & between community involvement score)

Role profile plots

Positional Analysis

Block models

Alternative approach: Cluster node metrics.

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It's also possible to use simple-to-calculate scores in unique ways.

So while its time and space/bandwidth consuming to run a full triad-based structural equivalence model over a giant network; you can calculate a host of local and bridging sorts of scores, then cluster those to get positions quickly.

Roles overlaid on a single community

Role 9

Role 8

Role 7

Homo SocioNeticus: Scaling the cognitive foundations of online social behavior” Defense Agency Research Projects Agency (DARPA), Mark Orr, PI

Positional Analysis

Block models

Alternative approach: Cluster node metrics.

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Positional Analysis

Block models

Implementation notes

Structural Equivalence:

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Positional Analysis

Block models

Implementation notes

Structural Equivalence:

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Positional Analysis

Block models

Implementation notes

Structural Equivalence:

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Positional Analysis

Block models

Implementation notes

Structural Equivalence:

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Positional Analysis

Block models

Implementation notes

Structural Equivalence:

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Positional Analysis

Block models

Conclusions

Positional approaches are under-used but promising ways to think about networks & health.

Simplest way in is via node metrics or combinations of metrics that capture a unique pattern of ties within the path/flow system of the (perhaps multiple) networks.

Class based approaches are a little more involved, but not terribly so and modern approaches to classification and clustering make it much simpler than in the past.

Still requires a fair amount of investigator judgement.

Models are difficult due to non-independence, but do provide a way to capture some of the otherwise unobserved structure in the network.

Tests for unobserved autocorrelation might be helpful (stay tuned for more on that!).