Networks & Health
Intro & overview
Outline
Social Network Data
Introduction
We live in a connected world:
“To speak of social life is to speak of the association between people – their associating in work and in play, in love and in war, to trade or to worship, to help or to hinder. It is in the social relations men establish that their interests find expression and their desires become realized.”
Peter M. Blau
Exchange and Power in Social Life, 1964
*1934, NYTime. Moreno claims this work was covered in “all the major papers” but I can’t find any other clips…
*
Introduction
We live in a connected world:
"If we ever get to the point of charting a whole city or a whole nation, we would have … a picture of a vast solar system of intangible structures, powerfully influencing conduct, as gravitation does in space. Such an invisible structure underlies society and has its influence in determining the conduct of society as a whole."
J.L. Moreno, New York Times, April 13, 1933
High Schools as Networks
Introduction
Introduction
Countryside High School, by grade
Introduction
Countryside High School, by race
And yet, standard social science analysis methods do not take this space into account.
“For the last thirty years, empirical social research has been dominated by the sample survey. But as usually practiced, …, the survey is a sociological meat grinder, tearing the individual from his social context and guaranteeing that nobody in the study interacts with anyone else in it.”
Allen Barton, 1968 (Quoted in Freeman 2004)
Moreover, the complexity of the relational world makes it impossible to identify social connectivity using only our intuition.
Social Network Analysis (SNA) provides a set of tools to empirically extend our theoretical intuition of the patterns that compose social structure.
Introduction
Social network analysis is:
Introduction
But scientists are starting to take network seriously:
“Networks”
Introduction
“Networks”
“Obesity”
Introduction
But scientists are starting to take network seriously: why?
Introduction
…and NSF is investing heavily in it.
Social Determinants of Health
“…social determinants of health refers to the complex, integrated, and overlapping social structures and economic systems that include social and physical environments and health services.” (CDC, 2010)
WHO Commission on Social Determinants of Health Conceptual Framework
Introduction
Social Determinants of Health
Social effects hold promising multiplier effects:
Introduction
A general embeddedness rubric for network models…
Outcome
+
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Balanced
Opposition
Segregation, political polarization, feuds, wars, etc.
Diffusion of health behavior
Introduction
Key Questions
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:
Connections & Positions: Network Problems
Classic characterization of epidemic spread is with Ro.
Ro depends on:
The spread of any epidemic depends on the number of secondary cases per infected case, known as the reproductive rate (R0). R0 depends on the probability that a contact will be infected over the duration of contact (β), the likelihood of contact (c), and the duration of infectiousness (D).
Why do networks matter?
Two fundamental mechanisms: Connections example
Isolated vision
Why do networks matter?
Two fundamental mechanisms: Connections example
Connected vision
Why do networks matter?
Why do networks matter?
Two fundamental mechanisms: Connections example
The structure of a network captures who is connected to who:
Ro is knowable if your disease relevant contact network (nd) is:
In all cases we can get (near) exact solutions for diffusion in these situations.
…but these situations almost never happen in the real world.
Why do networks matter?
Two fundamental mechanisms: Connections example
Standard models fail because:
There is fundamental science to be done explicating how nd affects epidemic risk.
The structure of a network captures who is connected to who:
Why do networks matter?
Two fundamental mechanisms: Connections example
Why do networks matter?
Two fundamental mechanisms: Connections example
Perhaps more important nd is conditioned by the social embeddedness of actors in other social systems:
nd = f(beliefs, practices, family, work, politics, …,etc)
nd = f(Human Social Systems)
Core systems:
We need to study more than nd to understand how disease networks function. nd=f(Human Social Systems)
Why do networks matter?
Two fundamental mechanisms: Connections example
Provides food for
Romantic Love
Bickers with
Why do networks matter?
Two fundamental mechanisms: Positions
Positional network mechanisms : Networks matter because of the way they capture role behavior and social exchange. Networks as Roles.
C
P
X
Y
Parent
Parent
Child
Child
Child
Provides food for
Romantic Love
Bickers with
Why do networks matter?
Two fundamental mechanisms: Positions
Positional network mechanisms : Networks matter because of the way they capture role behavior and social exchange. Networks as Roles.
C
P
X
Y
Why do networks matter?
Scope of Social Networks & Health
For those who want a deeper more systematic review:
English language Articles indexed in Web of Science Social Science Citation Index on: ("health" or "well being" or "medicine") and "network*").
18572 papers 2000 - 2018.
Why do networks matter?
Scope of Social Networks & Health
Bibliographic Similarity Networks: 1-step neighborhood of a single paper
Why do networks matter?
Scope of Social Networks & Health
Bibliographic Similarity Networks: 2-step neighborhood of a single paper
Why do networks matter?
Scope of Social Networks & Health
Since the net is large…
Use a force-directed layout to display the full space & overlay clusters….
Why do networks matter?
Scope of Social Networks & Health
The example paper…
Modularity:
Top-Level: 0.798 @ 32 Clusters
2nd Level: 0.785 @ 150 Clusters
Rogers:
Valente: Various
Christakis & Fowler
Add Health
Pescosolido
Social Networks & Health Grant Landscape
1940 currently active NIH grants that include network elements
Social Networks & Health Grant Landscape
1940 currently active NIH grants that include network elements
Social Networks & Health Grant Landscape
1940 currently active NIH grants that include network elements
Color is funding mechanism; brown=R01 which is about 42%)
Social Networks & Health Grant Landscape
1940 currently active NIH grants that include network elements
Topic Map
This workshop!
Introduction
Network Research Lifecycle
Overview
SN&H Program
Overview
SN&H Program
Overview
SN&H Program
Overview
SN&H Program
Overview
SN&H Program
Overview
SN&H Program
Strategy?
SN&H Program
Does it feel like a bit much!?
Strategy?
SN&H Program
Code is (reasonably) easy to generate
Strategy?
SN&H Program
Code is (reasonably) easy to generate
Strategy?
SN&H Program
Code is (reasonably) easy to generate.
It’s knowing what question to ask that’s difficult.
…Breathe…
The unit of interest in a network are the combined sets of actors and their relations.
We represent actors with points and relations with lines.
Actors are referred to variously as:
Nodes, vertices or points
Relations are referred to variously as:
Edges, Arcs, Lines, Ties
Example:
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Social Network Data
In general, a relation can be:
Binary or Valued
Directed or Undirected
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Undirected, binary
Directed, binary
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Undirected, Valued
Directed, Valued
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Social Network Data
In general, a relation can be: (1) Binary or Valued (2) Directed or Undirected
Social Network Data
Basic Data Elements
The social process of interest will often determine what form your data take. Conceptually, almost all of the techniques and measures we describe can be generalized across data format, but you may have to do some of the coding work yourself….
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b
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Directed,
Multiplex categorical edges
We can examine networks across multiple levels:
1) Ego-network
- May include estimates of connections among alters
2) Partial network
- Ego networks plus some amount of tracing to reach contacts of contacts
Social Network Data
Basic Data Elements: Levels of analysis
3) Complete or “Global” data
- Data on all actors within a particular (relevant) boundary
Ego-Net
Global-Net
Best Friend
Dyad
Primary
Group
Social Network Data
Basic Data Elements: Levels of analysis
2-step
Partial network
Social Network Data
Social network data are substantively divided by the number of modes in the data.
1-mode data represents edges based on direct contact between actors in the network. All the nodes are of the same type (people, organization, ideas, etc). Examples:Communication, friendship, giving orders, sending email.
This is commonly what people think about when thinking about networks: nodes having direct relations with each other.
Social Network Data
Social network data are substantively divided by the number of modes in the data.
2-mode (bipartite) data represents nodes from two separate classes, where all ties are across classes. Examples:
People as members of groups
People as authors on papers
Words used often by people
Events in the life history of people
The two modes of the data represent a duality: you can project the data as people connected to people through joint membership in a group, or groups to groups through common membership.
There may be multiple relations of multiple types connecting your nodes.
Bipartite networks imply a constraint on the mixing, such that ties only cross classes.
Here we see a tie connecting each woman with the party she attended (Davis data)
Social Network Data
Basic Data Elements: Modes
Social Network Data
Basic Data Elements: Modes
Bipartite networks imply a constraint on the mixing, such that ties only cross classes.
Here we see a tie connecting each woman with the party she attended (Davis data)
Bipartite
Person Projection
Event Projection
Social Network Data
Basic Data Elements: Modes
Number of modes are general:
1-mode (direct) : Person to Person
2-mode (bipartite): Person to event
3-mode (tripartite): person to event to event-type
i.e. students, programs, staff
authors, papers, topics
persons, animals, parasites
…
n-mode
Logically these ideas are clearly extensible. In practice, it’s a question of whether the overhead of analyzing as a mode is better than treating it as metadata on the primary relations of interest.
Social Network Data
Multi-layer networks
A recent generalization of multiplex networks and multi-mode networks is the multi-layer network. A multi-layer network is a network that has qualitatively different classes of nodes (like multi-mode networks) and qualitatively different types of relations (like multiplex). Multiplex and multi-mode networks can be subsumed under the multi-layer formalism.
See Kivela, Mikko, Alex Arenas Marc Bartheelemy, James P. Gleeson, Yamir Moreno, and Mason Porter. “Multilayer Networks” Journal of complex Networks 2:203-271. https://doi.org/10.1093/comnet/cnu016
Social Network Data
Basic Data Elements: summary
Social Network Data
(not so basic) Data Elements
From pictures to matrices
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Undirected, binary
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An undirected graph and the �corresponding matrix is symmetric.
The traditional way to store & represent network data is with an adjacency matrix.�
The matrix (X) at right represents an undirected binary network. Each node (a-e) is listed on both the row and the column.
The ith row and the jth column (Xij) records the value of a tie from node i to node j. For example, the line between nodes a and b is represented as an entry in the first row and second column (red at right).
Because the graph is undirected the ties sent are the same as the ties receive, so every entry above the diagonal equals the entries below the diagonal.
Basic Data Structures
Social Network Data
Directed, binary
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A directed graph and the �corresponding matrix is asymmetrical.
Directed graphs, on the other hand,�are asymmetrical.
We can see that Xab =1 and Xba =1, �therefore a “sends” to b and b “sends” to a. ��However, Xbc=0 while Xcb=1; therefore,�c “sends” to b, but b does not reciprocate.
Basic Data Structures
Social Network Data
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1
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2
1
A directed graph and the �corresponding matrix is asymmetrical.
Directed graphs, on the other hand,�are asymmetrical.
We can see that Xab =1 and Xba =1, �therefore a “sends” to b and b “sends” to a. ��However, Xbc=0 while Xcb=1; therefore,�c “sends” to b, but b does not reciprocate.
Basic Data Structures
Social Network Data
Directed, Valued
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From matrices to lists (binary)
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a b
b a c
c b d e
d c e
e c d
a b
b a
b c
c b
c d
c e
d c
d e
e c
e d
Adjacency List
Arc List
Social network analysts also use adjacency lists and arc lists�to more efficiently store network data.
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Basic Data Structures
Social Network Data
From matrices to lists (valued)
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a b
b a c
c b d e
d c e
e c d
a b 1
b a 1
b c 2
c b 2
c d 3
c e 5
d c 3
d e 1
e c 5
e d 1
Adjacency List
Arc List
Social network analysts also use adjacency lists and arc lists�to more efficiently store network data.
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Basic Data Structures
Social Network Data
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1
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a 1
b 1 2
c 2 3 1
d 3 1
e 5 1
contact
value
Working with two-mode data
A person-to-group adjacency matrix is rectangular, with one mode (persons, say) down rows and the other (groups, say) across columns
1 2 3 4 5
A 0 0 0 0 1
B 1 0 0 0 0
C 1 1 0 0 0
D 0 1 1 1 1
E 0 0 1 0 0
F 0 0 1 1 0
A =
Each column is a group, each row a person, and the cell = 1 if the person in that row belongs to that group.
You can tell how many groups two people both belong to by comparing the rows: Identify every place that both rows = 1, sum them, and you have the overlap.
Basic Data Structures
Social Network Data
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--
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--
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--
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--
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--
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0
Social Network Metrics:
Volume
*in valued graphs, sometimes called average tie strength, but I think that’s potentially misleading, as it reads like average given that its nonzero
Degree: Number of links adjacent to a node
Social Network Metrics: Basic Volume
Strength: Sum of links adjacent to a node
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1 2 1 2 1 7
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1 3 5 4 1 14
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6
Note that in igraph there are different choices:
Mode=
For the central node here:
d=igraph::degree(g, mode =“in”) 🡪 3
d=igraph::degree(g, mode =“out”) 🡪 3
d=igraph::degree(g, mode =“all”) 🡪 6
If you want the unique number of nodes they are adjacent to for “all” then do:
d=igraph::degree(as.undirected(g), mode =“all”) 🡪 4
If you have multiple relations, each relation counts for degree. So if that’s not the behavior you want…use degree(simplify(g)).
Degree: Number of links adjacent to a node
Social Network Metrics: Basic Volume
While conceptually simple, note some subtle bits in igraph.
Social Network Metrics: Basic Volume
Degree distribution underlies most everything else in the network. Be sure you know your degree distribution and that it’s sensible
Social Network Metrics: Basic Volume
Degree distribution underlies most everything else in the network. Be sure you know your degree distribution and that it’s sensible
Social Network Metrics: Basic Volume
Degree distribution underlies most everything else in the network. Be sure you know your degree distribution and that it’s sensible
Network Building Blocks
Dyad Census & Reciprocity
Where g is the number of nodes, M is the number of mutual dyads, L is the number of lines and L2 is the sum of the squared degree distribution.
Rho: 0.38
Network Building Blocks
Dyad Census & reciprocity
We can combine directionality and degree:
A quick-and-simple measure of prominence would be in-degree net of mutual ties
A similar measure of gregariousness would be out-degree net of mutual ties.
A similar measure of tendencies toward intimacy would be the node-level proportion of ties that are mutual.
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.
There are 16 possible triads on a binary directed graph:
Network Building Blocks
Triad Census and Hierarchy
003
(0)
012
(1)
102
021D
021U
021C
(2)
111D
111U
030T
030C
(3)
201
120D
120U
120C
(4)
210
(5)
300
(6)
Triad Census: The periodic table of social elements
Network Building Blocks
Triad Census & reciprocity
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
Triad Census: The periodic table of social elements
Network Building Blocks
Triad Census & reciprocity
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
Network Building Blocks
Triad Distribution
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
Network Building Blocks
Triad Distribution
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
Network Building Blocks
Triad Distribution
An Example of the triad census
Network Building Blocks
Triad Distribution
Network Building Blocks
Transitivity Scores
Intuitively, “transitivity” and “clustering” are metrics for the tendency to observe closed triads. Transitivity refers specifically to a consistent ordering.
030T
120U
300
Lots of these:
021D
021C
111U
Few of these:
030C
120C
210
These make my head hurt
Network Building Blocks
Transitivity Scores
Watts clustering coefficient:
Classic Transitivity ratio:
transitivity(g, type=“ratio”)
transitivity(g, type=“global”)
Original (1998) clustering coefficient:
Where k is degree, and e is number of edges between neighbors. Ci = ego-network density.
Mean(transitivity(g, type=“local”))
Measuring Networks: Connectivity
Redundancy (Local)
Local redundancy is known as “clustering” or “transitivity” - that one’s friends are friends with each other.
Density is the proportion of pairs tied, excluding ego.
Transitivity is the proportion of two-step ties that are closed (Friend of a Friend is a friend)
Density
Transitivity
Transitivity
No ego
0 0 0
0.4 0.71 1.0
1 1 1
0.7 0.78 .64
Structural Indices based on the distribution of triads
The observed distribution of triads can be fit to 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
μT= the expected value of T
ΣT = the variance-covariance matrix for T
Network Building Blocks
Triad Distribution
Triad:
003
012
102
021D
021U
021C
111D
111U
030T
030C
201
120D
120U
120C
210
300
BA
Triad Micro-Models:
BA: Ballance (Cartwright and Harary, ‘56) CL: Clustering Model (Davis. ‘67)
RC: Ranked Cluster (Davis & Leinhardt, ‘72) R2C: 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 p1-p4: Johnsen, 1986. Process Agreement
Models.
CL
RC
R2C
TR
HC
39+
p1
p2
p3
p4
Network Building Blocks
Weighting vectors
PRC{300,102, 003, 120D, 120U, 030T, 021D, 021U} Ranked Cluster:
M
M
N*
M
M
N*
M
A*
A*
A*
A*
A*
A*
A*
A*
1
1
1
1
1
1
1
1
1
0
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
And many more...
Network Building Blocks
Triad Distribution
d
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c
Indirect connections are what make networks systems. One actor can reach another if there is a path in the graph connecting them.
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c
e
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f
b
f
a
In a directed graph, paths are directed, leading to a distinction between strong and weak reachability.
Social Network Metrics: Connectivity
Reachability
If you can trace a sequence of relations from one actor to another, then the two are reachable. If there is at least one path connecting every pair of actors in the graph, the graph is connected and is called a component.
Intuitively, a component is the set of people who are all connected by a chain of relations.
Social Network Metrics: Connectivity
This example contains many components.
Social Network Metrics: Connectivity
Because relations can be directed or undirected, components come in two flavors:
For a graph with any directed edges, there are two types of components:
Strong components consist of the set(s) of all nodes that are mutually reachable
Weak components consist of the set(s) of all nodes where at least one node can reach the other.
Social Network Metrics: Connectivity
There are only 2 strong components with more than 1 person in this network.
Components are the minimum requirement for social groups. As we will see later, they are necessary but not sufficient
All of the major network analysis software identifies strong and weak components
Social Network Metrics: Connectivity
Partner
Distribution
Component
Size/Shape
Emergent Connectivity in “low-degree” networks
Example: Small local changes can create cohesion cascades
Based on work supported by R21-HD072810 (NICHD, Moody PI), R01 DA012831-05 (NIDA Morris, Martina PI)
Connections: Diffusion
Average degree, degree distribution & connectivity
a
Geodesic distance is measured by the smallest (weighted) number of relations separating a pair:
Actor “a” is:
1 step from 4
2 steps from 5
3 steps from 4
4 steps from 3
5 steps from 1
a
Social Network Metrics: Connectivity
Distance
a b c d e f g h i j k l m
------------------------------------------
a. . 1 2 . . . . . . . . 2 1
b. 3 . 1 . . . . . . . . 1 2
c. . . . . . . . . . . . . .
d. 4 3 1 . 1 2 1 . 2 . . 2 3
e. 3 2 2 1 . 1 2 . 1 . . 1 2
f. 4 3 3 2 1 . 3 . 2 . . 2 3
g. 5 4 4 3 2 1 . . 3 . . 3 4
h. . . . . . . . . 1 . . . .
i. . . . . . . . . . . . . .
j. . . . . . . . . 1 . . . .
k. . . . . . . . . 1 . . . .
l. 2 1 2 . . . . . . . . . 1
m. 1 2 3 . . . . . . . . 1 .
b
c
d
g
f
e
k
i
j
h
l
m
a
When the graph is directed, distance is also directed (distance to vs distance from), following the direction of the tie.
Social Network Metrics: Connectivity
Distance
As a graph statistic, the distribution of distance can tell you a good deal about how close people are to each other (we’ll see this more fully when we get to closeness centrality).
The diameter of a graph is the longest geodesic, giving the maximum distance. We often use the l, or the mean distance between every pair to characterize the entire graph.
For example, all else equal, we would expect rumors to travel faster through settings where the average distance is small.
Geodesic distance is just the simplest kind of distance – there are others, and they are interesting!
Measuring Networks: Connectivity
Distance
Node Connectivity
As size of cut-set
0
1
2
3
Structural Cohesion:
A network’s structural cohesion is equal to the minimum number of actors who, if removed from the network, would disconnect it.
Measuring Networks: Connectivity
Redundancy Global: Structural Cohesion
0
1
2
3
Node Connectivity
As number of node-independent paths
Measuring Networks: Connectivity
Redundancy Global: Structural Cohesion
Structural Cohesion:
A network’s structural cohesion is equal to the minimum number of actors who, if removed from the network, would disconnect it.
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4
Nestedness Structure
Cohesive Blocks
Depth
Sociogram
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9
Cohesive Blocking
The arrangement of subsequently more connected sets by branches and depth uniquely characterize the connectivity structure of a network
Measuring Networks: Connectivity
Redundancy Global: Structural Cohesion
Conceptually centrality identifies nodes in the ‘center’ of the network.
In practice, ‘center’ is complicated, as there are multiple dimensions to be central on.
The standard centrality measures capture a wide range of “importance” in a network:
Measuring Networks
Centrality
Measuring Networks
Centrality
..and just a teaser of all the elements we are leaving out:
Schoch lists over 100 different measures…
Measuring Networks
Centrality
Degree Centrality: Number of lines adjacent to each node. Captures local importance.
Strength: Sum of the edges adjacent to each node
In-degree: number of lines pointing at node
Out-degree: number of lines pointing from node
mutual-degree: number of reciprocated ties
Closeness Centrality: inverse of the average number of steps from one node to another
Usually geodesic distance, so sensitive to random ties
Betweenness Centrality: Number of times a node sits on the shortest paths between all other nodes.
Captures ability to bridge different parts of the network
Eigenvector/Power Centrality: Degree weighted by the degree (weighted by the degree…) of nodes each node is connected to.
Being central amongst the central.
Be careful using igraph for centrality on weighted networks. It probably doesn’t do what you think it does.
Measuring Networks
Centrality
In many cases, igraph treats weights as “costs” so walks through the graph cumulate – like miles in a trip.
a
b
c
d
5
8
1
2
The a-b-c path will be length 13, but the a d c path will be length 3. IF you want to represent who is closer (i.e. weights are strengths), then “13” is “closer” than “3.”
So INVERT the distances first for betweenness and closeness (path based)
Homophily
Homophily is the tendency for social contacts to be similar.
Homophily is the tendency for social contacts to be similar.
Homophily is the tendency for social contacts to be similar.
Grade Mixing Matrix
Internal ties
External ties
Measuring Homophily
A value of -1 indicates perfect homophily (all ties are internal), a value of 1 indicates perfect heterophily (all ties are external), and a value near 0 indicates a balance between homophilous and heterophilous ties.
Within-group fraction
E-I Index
Assortativity index
Edge-wise correlation of attributes
Segregation Index
(Freeman, L. C. 1972. "Segregation in Social Networks." Sociological Methods and Research 6411-30.)
Freeman asked how we could identify segregation in a social network. Theoretically, he argues, if a given attribute (group label) does not matter for social relations, then relations should be distributed randomly with respect to the attribute. Thus, the difference between the number of cross-group ties expected by chance and the number observed measures segregation.
Measuring Homophily
(Analogous to a chi-square test on a contingency table)
Calculation notes in the hidden slides.
One problem with the segregation index is that it is not ‘margin free.’ That is, if you were to change the distribution of the category of interest (say race) by a constant but not the core association between race and friendship choice, you can get a different segregation level.
One antidote to this problem is to use odds ratios. In this case, and odds ratio tells us the relative likelihood that two people in the same category will choose each other as friends. Familiar from logistic regression.
Odds Ratios
Measuring Homphily
We can also measure the extent that ties fall within groups with the modularity score. This is the most common current scoring, it’s a linear transform of the segregation index.
Where:
m is the number of edges
k is the degree
Aij is the edge weight between ij
δ(cicj) is 1 if in the same group
γ is the resolution parameter
Modularity of 0 means random mixing. This has become standard for community detection.
Modularity
Questions?
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Gabriel Varela
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Tom Wolff
Liann Tucker
Rhodes iiD
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DUPRI
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NICHD: 5R25HD079352
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