11

Network Level Indicators

• Bird’s 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

22

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

33

Density

• Structural property

• Given by

)1(

nn

lD

• Should always be included as covariate as well

44

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

55

Reciprocity (Mutuality, Symmetry)

• Mutual ties: A B then BA• 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:

)1()1(

)1(&)1(

jiij

jiij

AorA

AAR

66

Triads & Transitivity

• Holland & Leinhardt introduced the concept of triads and a triad census

• In a directed graph there are 16 possible triads:– AB BC AC

– AB BC CA

– ….

• One can do a triad census of a network calculating the percent of triads of each type in the network

7

MAN (Mutual, Asymmetric, Null) Census

003 012 102 021D

021U 021C 111D 111U

030T 030C 201 120D

120U 120C 210 300

88

Triads & Transitivity (cont.)

• Most often concerned with transitivity• A transitive triad occurs if:

– AB BC – Implies– AC

• 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)

9

Transitive Triad

A B

C

1010

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.

1111

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

1212

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

13

Average and Maximum Change in Cohesion for each Link Removed

-4-2

02

4pb

dmax

/pbd

el/p

bam

ax/p

badd

0 .2 .4 .6 .8 1density

pbdmax pbdel

pbamax pbadd

14

Cohesion: Measure of how close everyone is, on average, in the network

14

)1(

1

nn

dCohesion

ij

1515

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)

1616

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

1717

Clustering

• Watts re-introduced the clustering coefficient:

• Average of the individual personal network densities:

1818

Personal Network Density

PN Density = 1/6 = 16.7% PN Density = 3/6 = 50.0%

A

z

x

y

z

x

yB

1919

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

2020

Examples of Dense Networks (Density=36.4%)

Decentralized (9.1%) Centralized (50.9%)

2121

Examples of Sparse Networks (Density=18.2%)

Decentralized (0.0%) Centralized (87.3%)

2222

Centralization Can Be Calculated On All Centrality Measures:

• Centralization Degree:

23

))((2

nn

CCMaxCD DiiD

2323

Centralization (cont.)

• Similar formulas exist for Centralization Closeness, Betweenness, Integration

• Can also be calculated by taking the standard deviation of the centrality scores.

2424

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

25

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

26

Children’s Health Insurance of Greater LA (CP=0.29)

Provider

CBO

Provider

CBO

Other Provider

Phil

CBOCBO

Policy

Health_Plan

Health_PlanGovt

GovtGovt

Govt

School

Govt

Phil

CBO

Acad

Phil

Policy

Phil

Acad

Phil

Acad

CBO

CBOCBO

School

▲ Missing■ Periphery● Core

2727

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

2828

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

2929

Data from STEP

3030

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

3131

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

3232

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

3333

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.

3434

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

35

Transposing a Matrix

35

Event A Event B Event C

Person 1 1 0 1

Person 2 1 1 0

Person 3 0 1 0

Person N 0 0 1

Matrix A

Person 1 Person 2 Person 3 Person N

Event A 1 1 0 0

Event B 0 1 1 0

Event C 1 0 0 1

Matrix A’ (transpose)

3636

Excel File

ID SPPD ASC IPR Other

1 1 0 0 0

2 0 0 1 0

3 0 0 1 0

4 1 0 0 0

5 0 0 1 0

6 1 0 0 0

7 0 1 0 0

8 0 1 0 0

9 0 0 1 0

10 0 1 0 0

11 0 0 1 0

12 0 0 0 1

13 1 0 0 0

14 0 1 0 0

15 0 0 0 1

16 1 0 0 0

17 1 0 0 0

18 0 0 1 0

19 0 0 1 0

3737

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

3838

Dept 06 PxP

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

1617

18

19

3939

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