11 Contagion

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How Networks Shape Attitudes and Attitudes Shape Networks Diffusion and Contagion

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Transcript of 11 Contagion

How Networks Shape Attitudes and Attitudes Shape

NetworksDiffusion and Contagion

GridWorlds

Net Worlds

Contagion and Diffusion

• We choose our friends, and our friends choose us

• We learn from them, they learn from us

• Thus…

• Formation of networks and formation of attitudes are inextricably linked

Cont…

• We shall observe two processes that shape dynamic networks:

• Social contagion = diffusion of attributes shaped by network structure

• Formation of homophily groups = network structure shaped by attributes

Attitudes are a function of two sources:a) Individual characteristics

•Gender, Age, Race, Education, Etc. Standard sociology

b) Interpersonal influences•Actors negotiate opinions with others

Basic Peer Influence Model

Freidkin claims in his Structural Theory of Social Influence that the theory has four benefits:

•Relaxes the simplifying assumption of actors who must either conform or deviate from a fixed consensus of others (public choice model)

•Does not necessarily result in consensus, but can have a stable pattern of disagreement

•Is a multi-level theory:•micro level: cognitive theory about how people weigh and combine other’s opinions•macro level: concerned with how social structural arrangements enter into and constrain the opinion-formation process

•Allows an analysis of the systemic consequences of social structures

Basic Peer Influence Model

Influenfce Model in English

• Every agents’ beliefs are affected by the beliefs of agents he is connected to

• At time t+1• Ego belief= weight *Ego belief at time t +

Sum(weight * belief of alter)

For all alters connected to ego

XBY =)1( (1)

)1()1()( )1( YWYY áá Tt −+= −(2)

Y(1) = an N x M matrix of initial opinions on M issues for N actors

X = an N x K matrix of K exogenous variable that affect YB = a K x M matrix of coefficients relating X to Y = a weight of the strength of endogenous interpersonal

influences (how much is ego influenced by alters)W = an N x N matrix of interpersonal influences

The same, in Matrix Form

Basic Peer Influence ModelFormal Model

XBY =)1( (1)

This is the standard sociology model for explaining anything: the General Linear Model.

It says that a dependent variable (Y) is some function (B) of a set of independent variables (X). At the individual level, the model says that:

∑=k

kiki BXY

Usually, one of the X variables is , the model error term.

Basic Peer Influence Model

)1()1()( )1( YWYY áá Tt −+= −(2)

This part of the model taps social influence. It says that each person’s final opinion is a weighted average of their own initial opinions

)1()1( Yá−

And the opinions of those they communicate with (which can include their own current opinions)

)1( −TáWY

Basic Peer Influence Model

The key to the peer influence part of the model is W, a matrix of interpersonal weights. W is a function of the communication structure of the network, and is usually a transformation of the adjacency matrix. In general:

∑ =

≤≤

jij

ij

w

w

1

10

Various specifications of the model change the value of wii, the extent to which one weighs their own current opinion and the relative weight of alters.

Basic Peer Influence Model

1 2

3

4

1 2 3 41 1 1 1 02 1 1 1 03 1 1 1 14 0 0 1 1

1 2 3 41 .33 .33 .33 02 .33 .33 .33 03 .25 .25 .25 .254 0 0 .50 .50

1 2 3 41 .50 .25 .25 02 .25 .50 .25 03 .20 .20 .40 .204 0 0 .33 .67

Even

2*self

1 2 3 41 .50 .25 .25 02 .25 .50 .25 03 .17 .17 .50 .174 0 0 .50 .50

degree

Self weight:

1 2 3 41 2 1 1 02 1 2 1 03 1 1 2 14 0 0 1 2

1 2 3 41 2 1 1 02 1 2 1 03 1 1 3 14 0 0 1 1

Basic Peer Influence Model

)1()1()( )1( YWYY áá Tt −+= −

Formal Properties of the model

When interpersonal influence is complete, model reduces to:

)1(

)1()1()(

01−

=

+=T

Tt

WY

YWYY

When interpersonal influence is absent, model reduces to:

)1(

)1()1()(

0

Y

YWYY

=

+= −Tt

(2)

Basic Peer Influence ModelSimple example

1 2

3

4

1 2 3 41 .33 .33 .33 02 .33 .33 .33 03 .25 .25 .25 .254 0 0 .50 .50

Y1357

= .8

T: 0 1 2 3 4 5 6 7 1.00 2.60 2.81 2.93 2.98 3.00 3.01 3.01 3.00 3.00 3.21 3.33 3.38 3.40 3.41 3.41 5.00 4.20 4.20 4.16 4.14 4.14 4.13 4.13 7.00 6.20 5.56 5.30 5.18 5.13 5.11 5.10

Basic Peer Influence ModelSimple example

1 2

3

4

1 2 3 41 .33 .33 .33 02 .33 .33 .33 03 .25 .25 .25 .254 0 0 .50 .50

Y1357

= 1.0

1.00 3.00 3.33 3.56 3.68 3.74 3.78 3.813.00 3.00 3.33 3.56 3.68 3.74 3.78 3.815.00 4.00 4.00 3.92 3.88 3.86 3.85 3.847.00 6.00 5.00 4.50 4.21 4.05 3.95 3.90

T: 0 1 2 3 4 5 6 7

Basic Peer Influence ModelExtended example: building intuition

Consider a network with three cohesive groups, and an initially random distribution of opinions:

(to run this model, use peerinfl1.sas)

Simulated Peer Influence:75 actors, 2 initially random opinions, Alpha = .8, 7 iterations

Simulated Peer Influence:75 actors, 2 initially random opinions, Alpha = .8, 7 iterations

Simulated Peer Influence:75 actors, 2 initially random opinions, Alpha = .8, 7 iterations

Simulated Peer Influence:75 actors, 2 initially random opinions, Alpha = .8, 7 iterations

Simulated Peer Influence:75 actors, 2 initially random opinions, Alpha = .8, 7 iterations

Simulated Peer Influence:75 actors, 2 initially random opinions, Alpha = .8, 7 iterations

Simulated Peer Influence:75 actors, 2 initially random opinions, Alpha = .8, 7 iterations

Simulated Peer Influence:75 actors, 2 initially random opinions, Alpha = .8, 7 iterations

Basic Peer Influence ModelNow weight in-group ties higher than between group ties

Simulated Peer Influence:75 actors, 2 initially random opinions, Alpha = .8, 7 iterations, in-group tie: 2

Consider the implications for populations of different structures. For example, we might have two groups, a large orthodox population and a small heterodox population. We can imagine the groups mixing in various levels:

Heterodox: 10 peopleOrthodox: 100 People

Light Heavy

Moderate

Light mixing

Light mixing

Light mixing

Light mixing

Light mixing

Light mixing

Moderate mixing

Moderate mixing

Moderate mixing

Moderate mixing

Moderate mixing

Moderate mixing

High mixing

High mixing

High mixing

High mixing

High mixing

High mixing

In an unbalanced situation (small group vs large group) the extent of contact can easily overwhelm the small group. Applications of this idea are evident in:

•Missionary work (Must be certain to send missionaries out into the world with strong in-group contacts)

•Overcoming deviant culture (I.e. youth gangs vs. adults)

Size Matters

In recent extensions (Friedkin, 1998), Friedkin generalizes the model so that alpha varies across people. We can extend the basic model by (1) simply changing to a vector (A), which then changes each person’s opinion directly, and (2) by linking the self weight (wii) to alpha.

)1()1()( )( YAIAWYY −+= −Tt

Were A is a diagonal matrix of endogenous weights, with 0 < aii < 1. A further restriction on the model sets wii = 1-aii

This leads to a great deal more flexibility in the theory, and some interesting insights. Consider the case of group opinion leaders with unchanging opinions (I.e. many people have high aii, while a few have low):

Factoring in Trust

Group 1 Leaders

Group 2 Leaders

Group 3 Leaders

Peer Opinion Leaders

Peer Opinion Leaders

Peer Opinion Leaders

Peer Opinion Leaders

Peer Opinion Leaders

Peer Opinion Leaders

Extensions of the Model

Time dependent : people likely value other’s opinions more early than later in a decision context Can be done in context of simulated annealing; Randomization in

Interact with XB: people’s self weights are a function of their behaviors & attributes

Make W dependent on structure of the network (weight transitive ties greater than intransitive ties, for example)

Time dependent W: The network of contacts does not remain constant, but is dynamic, meaning that influence likely moves unevenly through the network

Extensions of the Model

So we know…

• … Attitudes and knowledge are affected by the network structure

• But is the network affected by the attitudes?

Carley’s Construct Model

• An agent is…

• A: n x 1 - my neighbors in the network• B: m x 1 vector of beliefs that ego holds

• In matrix form:

Knowledge Network

• …or any 2-mode network

• “People x Attribute”

• “People x Resource”

• “People x Organization”

• …etc

How networks form

• Agents tend to communicated with either:• People similar to them (I.e. with similar

observed beliefs or attributes)

Or• People that can provide useful information

Need for Communicative Ease• Relative similarity RSij = how much I shares with J

divided by how much I shares with all others

• Bik is belief network (i knows information k)

• Expected interaction based on relative similarity• “A ratio of what ego shares with alter to what ego

shares with everybody else”

Need to Know

• Relative expertise REij = how much I thinks J knows that I does not know divided by how much I thinks all others know that I does not know

• Sik is knowledge network I knows information k• Expected interaction based on relative expertise

What do we get from this…

• RS and RE are equivalent to probability of interaction

• At each time, agents get a chance to interact based on the probabilities

• An interaction=creation of a network tie

• Ties decay with time if not reactivated

Structural Evolution

Knowledge Diffusion

• Percentage of shared beliefs in the network

Summary

• Friedkin’s Contagion Model derives belief structures from network structures

• Carley’s Construct Model derives network structures from belief structures

• Construct allows simultaneous manipulation of beliefs and network structure