Calculate Thresholds of Diffusion:

Exploratory Network Analysis with PajekWANG Chengjun

wangchj04@gmail.com 2010.12.24

Chapter 8 diffusion

The characteristics of DiffusionDiffusion is a special case of

brokerageTime dimensionRelationships as channelThe combination of structural

positions & adoption time

Example

Empirical dataInnovations of new mathematics

method in 1950, Allegheny County, Pennsylvania, U.S.A.

School superintendents as gatekeepers

Nomination method: ask the respondents to indicate their three best friends

The social network is named modern math network

Codes

Read data ------------------------------------------------------------------------------ Reading Network --- E:\lingfei wu\pajek125\ESNAdata\

Chapter8\ModMath.net ------------------------------------------------------------------------------ Reading Partition --- E:\lingfei wu\pajek125\ESNAdata\

Chapter8\ModMath_adoption.clu ------------------------------------------------------------------------------

Data display

ModMath_adoption.cluthe adoption time of the network

MODMATH.NET

*Vertices 38 1 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4

MODMATH_ADOPTION.CLU

4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6

*Vertices 38 1 "v1" 0.0500

0.5346 0.5000 2 "v2" 0.2300

0.7423 0.5000 3 "v3" 0.2300

0.6038 0.5000 ........... *Arcs *Edges 2 32 1 2 23 1 2 3 1 ……………….

Draw> Draw-Partition or Ctrl+P

Layers> in y directionLayers> Optimize layers in x directionMove > Fix >yOptions> Transform> Rotate 2D

Contagion:passing on innovations via social tiesTwo-step flow modelFirst phase: Mass media inform and

influence opinion leadersSecond phase: opinion leaders

influence potential adoptersDiffusion of innovationsOpinion leaders use social relations

to influence their contactsAdvice and friendship relations

S-shape diffusion curve

Hypotheses

Personal characteristics The type of innovations Perceived risk of innovations Network structure: In a dense network an innovation spreads more easily and

faster than in a sparse network, In an unconnected network diffusion will be slower and less

comprehensive than in a connected network, In a bi-component diffusion will be faster than in

components with cut-points or bridges, The larger the neighborhood of a person within the network,

the earlier s/he will adopt an innovation, A central position is likely to lead to early adoption, Diffusion from a central vertex is faster than from a vertex in

the margins of the network.

Draw the diffusion curve:Info> Partition

Dimension: 38 The lowest value: 1 The highest value: 6

The highest clusters values:

Rank Vertex Cluster Id -------------------------------- 1 38 6 v38

Frequency distribution of cluster numbers:

Cluster Freq Freq% CumFreq CumFreq% Representative

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

1 1 2.6316 1 2.6316 v1 2 4 10.5263 5 13.1579 v2 3 10 26.3158 15 39.4737

v6 4 12 31.5789 27 71.0526

v16 5 8 21.0526 35 92.1053

v28 6 3 7.8947 38 100.0000

v36

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

Sum 38 100.0000

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Diffusion curve

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Exercise 1

Create a random networkNet> Random Network> Vertices Output

DegreeOut-degree 1 or 2No multiple lines Pick a vertex as the source of diffusion

process Assume a vertex will adopt at the first

time point after it has established direct contact with an adopter

Net> k-Neighbours> AllInfo>Partition

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diffusion curve of random network

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Adoption threshold

Everyone is unequally susceptible to contagion Two approaches to evaluate innovativeness: Adoption categories Classify people by their adoption time: Innovators, early adopters,

early majority, late adopters, laggards. It’s useful to identify the social and demographic characteristics Threshold categories: The threshold is his or her exposure at the time of adoption The exposure of a vertex in a network at a particular moment is

the proportion of its neighbors who have adopted before that time

Some people are easily persuaded (more susceptible) than others However, individual thresholds are computed after the fact, which

is a hindsight and not informative. They should be validated by other indicators of innovativeness.

Calculate the exposureWe first choose time 2 (1959), and calculate the exposure at the time 2.And then, calculate time 3, time4, time 5, time6

1. Symmetrize the directed network into undirected one

Net> Transform> Arcs->Edges>ALL

2. Identify the adopters at the selected time Partition> Binarize (fill in 1-2) Adoption time 1 & 2 are assigned a

score of 1, and others are assigned a score of 0.

Partition> Make vector or Ctrl + VDraw> Draw-vector

Dra

w

vect

or

FILL IN 1-2

3. Compute numbers of adopters in each actor’s neighborhood

Operations> Vector> Summing up neighbors

4. Divide the number of adopters in the neighborhood by its total number of neighbors

Because we defined exposure as the percentage of neighbors who have adopted.

Vectors> First vector Net> Partitions> Degree Partition> Make vector (do not normalize) Vectors> Second vector Vectors> Divide First by Second Options> Read/Write>0/0

There aren’t the submenus of first vector and second

vector in PAJEK125 !!!!!!!

Using macro: execute prior procedures of computing the exposure

Macro> Play Options> Read/Write>0/0Making new macro:Macro> Record----- Macro> Record

Time 1-2Draw > Draw VectorSet size= vector value

Time 1-3

Time 1-4

Time 1-5

Time 1-6

Calculate the thresholdsResults supplied by the author

Change the original undirected network into directed network Read Project Operations> Transform> Direction>

Lower-Higher

Change the original undirected network into directed network

The right method

Threshold=in-degree/ all-degree in-degree is the in-degree of network which is directed

and having no multiple lines and no lines within classes

all-degree is the all-degree of network which is undirected and having n0 multiple lines

Because the original network is undirected and having no multiple lines, so we can calculate all-degree directly.

To obtain the in-degree, we should re-read original network and change it into directed one which has no lines within classes first, and then we can calculate in-degree directly.

Using the submenu “divide first by second” in the menu of “Vectors”, we can get the threshold.

Draw the vectors, and “mark vertices using” “vector values”.

Procedures of making threshold Record macro Read project Draw partition Net> partitions > Degree> ALL Vectors> Second vectors Read project Operations> Transform> Direction Net> partitions > Degree> Input Vectors> First vectors Vectors> Divide First by Second Draw> Draw-vector Record macro

Macro report

NETBEGIN 1 CLUBEGIN 1 PERBEGIN 1 CLSBEGIN 1 HIEBEGIN 1 VECBEGIN 1

Msg Reading Pajek Project File --- E:\lingfei wu\pajek125\ESNAdata\Chapter8\ModMath.paj Msg Reading Network --- ModMath_directed.net Msg Reading Network --- ModMath.net Msg Reading Partition --- ModMath_adoption.clu N 9999 RDPAJ ? N 2 LAYERSNX 2 1 Msg Optimizing total length of lines ... Msg All degree centrality of 2. ModMath.net (38) C 2 DEGC 2 [2] (38) N 3 ETOAINC 2 1 1 DEL (38) Msg Input degree centrality of 3. Directed Network [INC DEL] of N2 according to C1 (38) C 3 DEGC 3 [0] (38) V 3 DIVV 2 1 (38)

Save the macro as “threshold new”

Critical massNet> Transform> Arcs->Edges> AllNet> Vector> Centrality> BetweennessInfo > Vector

Threshold lag

A threshold lag is a period in which an actor does not adopt although he or she is exposed at the level at which he or she will adopt later.

The critical mass of a diffusion process is the minimum number of adopters needed to sustain a diffusion process.

V28 and V29 undergoes a threshold lag, respectively (we can tell that from the pic of thresholds).

Critical mass was reached in 1960 (which can be get from the table of acceleration).

Using the macro of threshold lag : find the actors who encounter threshold lags.

Export the threshold dataTools> SPSS> Locate SPSSTools> SPSS> Send to SPSS

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Thank you and welcome for your comments