Network Level Indicators

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Network Level Indicators

• Birds eye view of network
• Image matrix example of network level
• Many network level measures
• Some would argue this is the most appropriate
  level of analysis

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Size

• Number of nodes (people) in the network
• Matters because as size increases
• Density decreases
• Clustering increases
• Reflects network boundary
• Should always be included as a covariate

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Density

• Structural property
• Given by

• Should always be included as covariate as well

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Density Size Negatively Correlated

• In STEP study we have data from 24 coalitions at
  baseline
• We correlated size and density and discovered a
  negative association as predicted
• R-0.69

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Reciprocity (Mutuality, Symmetry)

• Mutual ties A ? B then B?A
• Some relations are inherently symmetric or
  asymmetric
• Who did you have lunch with?
• Who did you go to for advice?
• Reciprocity is calculated as the percent of ties
  that are reciprocated

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Triads Transitivity

• Holland Leinhardt introduced the concept of
  triads and a triad census
• In a directed graph there are 16 possible triads
• A?B B?C A?C
• A?B B?C C?A
• .
• One can do a triad census of a network
  calculating the percent of triads of each type in
  the network

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MAN (Mutual, Asymmetric, Null) Census
003
012
102
021D
021U
021C
111D
111U
030T
030C
201
120D
120U
120C
210
300
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Triads Transitivity (cont.)

• Most often concerned with transitivity
• A transitive triad occurs if
• A?B B?C
• Implies
• A?C
• Transitivity implies balance, and balance theory
  is one of the foundations of many behavioral
  theories
• It is believed that people seek balance both
  toward others and objects (Heider)
• If a person is imbalanced, this creates cognitive
  dissonance and people will try to reduce
  cognitive dissonance (Festinger)

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Transitive Triad
C
B
A
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Transitivity

• The percent of transitive triads provides a
  measure of cohesion
• In the STEP study we found an average of 17 of
  triads were transitive.

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4 Nodes?

• One might expect the next level of analysis to
  increase to 4 nodes, as reciprocity was 2 nodes,
  and triads 3 nodes, but
• 4 nodes takes us to groups (this is where cycles
  come in)
• And back to the lecture on groups

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Diameter/Ave. Path Length

• Diameter Length of the longest path in the
  network
• Ave path length/characteristic path length
• Average of all the distances between nodes
• A measure of network size

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Average and Maximum Change in Cohesion for each
Link Removed
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Cohesion Measure of how close everyone is, on
average, in the network
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Unconnected Nodes

• Distances are important to calculate in networks
• What about unconnected nodes
• Distance equals infinity
• Creates intractable math calculations
• Substitute some finite number
• Defensible on the grounds that if a node is
  included in a network it is reachable because it
  is in the same set
• Might not be reachable because of measurement
  error
• Might not be reachable because of instrumentation
  (e.g., 5 closest friends)

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What to substitute for unconnected nodes?

• Choices
• N-1
• Advantages is the maximum theoretical distance
  between nodes in any network
• N
• Advantages is linearly related to max distance
  and would be the distance if a node were deleted
• Max. path length plus 1
• Advantages is intuitively more meaningful
• Most Use N-1

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Clustering

• Watts re-introduced the clustering coefficient
• Average of the individual personal network
  densities

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Personal Network Density
x
x
A
y
y
B
z
z
PN Density 1/6 16.7
PN Density 3/6 50.0
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Centralization

• The degree ties are focused on one or a few
  people
• Index ranges from 0 to 1 with 1 being perfectly
  centralized.
• Recall Centralized network are scale free
  networks

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Examples of Dense Networks (Density36.4)
Decentralized (9.1)
Centralized (50.9)
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Examples of Sparse Networks (Density18.2)
Decentralized (0.0)
Centralized (87.3)
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Centralization Can Be Calculated On All
Centrality Measures

• Centralization Degree

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Centralization (cont.)

• Similar formulas exist for Centralization
  Closeness, Betweenness, Integration
• Can also be calculated by taking the standard
  deviation of the centrality scores.

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Core Periphery Structures

• CP Networks have cores of densely connected
  people and a
• Periphery of those loosely connected to the core
  and to each other
• Can test whether networks have a C-P structure

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Core-Periphery Analysis

• A network with a perfect CP structure will have
  all core nodes connected and peripheral ones
  connected only to the core
• Construct this idealized matrix and correlate the
  ideal with the empirical.
• Correlation coefficient is a measure of the CP

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Childrens Health Insurance of Greater LA
(CP0.29)
? Missing Periphery ? Core
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Network Structure Behavior

• Size clearly matters, large networks
• difficult to coordinate organize
• Norms unclear or diffuse
• Diffusion takes longer
• Small networks
• Easy to coordinate
• Information and behaviors of others are known
• Information can travel quickly, but
• Small networks are not powerful

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Density

• We discussed earlier the possible curvilinear
  relationship
• Reciprocity At the individual level,
  reciprocated relationship should be more likely
  associated with behavioral transmission People
  more likely influenced by reciprocated
  relationships
• On the other hand, advice seeking is asymmetric
  and one more likely to model those they seek
  advice from
• Thus, at individual level, reciprocity affects on
  behavior depend on relationship and behavior

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Data from STEP
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Reciprocity Transitivity

• Networks with high levels of reciprocity
• Diffusion within faster but
• Diffusion between groups slower
• Transitive triads also more likely to
• Increase homogeneity of opinions
• Facilitate diffusion within groups, but inhibit
  diffusion of outside ideas

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Clustering

• High rates of clustering are even more indicative
  of closed subgroups
• Clustering will inhibit spread between groups but
  accelerate it within groups
• Higher clustering will increase the importance of
  bridges that connect clusters

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Centralization

• Centralized networks should/could have fastest
  diffusion
• Central nodes are key players in the process
• Central nodes are gatekeepers
• Other properties may interact with centralization

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Core Periphery

• Diffusion more likely to occur in the core
• Take a while for behaviors to filter to the
  periphery
• Many innovation may come from the periphery then
  percolate to the core
• Core groups can keep infectious diseases endemic
  to communities STDs, HIV, etc.

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2 Mode Data

• Recall that data on events, organizations, etc.
  can be used to construct 2 mode networks
• E.g., in this class students come from different
  departments
• Can construct a network based on shared dept.
  affiliations

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Transposing a Matrix
Matrix A
Matrix A (transpose)
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Excel File
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Steps

• Read into UCINET as excel file
• Input this file Data\affiliations\dept06
• Creates 1 mode data person by person
• And creates 1 mode dept by dept

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Dept 06 PxP
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Do They Correlate?

• Dept affiliations may lead to who knows whom
• We can correlate the 2 matrices
• Procedure to do so is know as QAP Quadratic
  Assignment Procedure
• This procedures accounts for the dependencies in
  the rows and columns
• QAP Reg. coefficient between knowing and
  department affiliation is 0.30

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