Belief Change via Social Influence and Explanatory Coherence

Bruce EdmondsCentre for Policy Modelling

Manchester Metropolitan University

Context

• Dissatisfied with representing beliefs at “opinions” as a point on a continuous scale, since this confuses a measured effect with an underlying mechanism

• Wish to combine something of a cognitive model with social process of influence, since scientific belief is a combination of social and cognitive processes

• This is a “thought experiment” only, I hope that someone will point me to data to enable its assessment and development

Explanatory Coherence

• Thagard (1989)• A network in which beliefs are nodes, with

different relationships (the arcs) of consonance and dissonance between them

• Leading to a selection of a belief set with more internal coherency (according to the dissonance and consonance relations)

• Can be seen as an internal fitness function on the belief set (but its very possible that individuals have different functions)

• The idea of the presented model is to add a social contagion process to this

Adding Social Influence

• The idea is that a belief may be adopted by an actor from another with whom they are connected, if by doing so it increases the coherency of their set of beliefs

• Thus the adoption process depends on the current belief set of the receiving agent

• Belief revision here is done in a similar basis, beliefs are dropped depending on whether this increases internal coherence

• Opinions can be recovered in a number of ways, e.g. a weighted sum of belief presence or the change in coherence OR the change in coherence in the presence of a probe belief

Model Basics

• Fixed network of nodes and arcs• There are, n, different beliefs {A, B, ....}

circulating• Each node, i, has a (possibly empty) set of

these “beliefs” that it holds• There is a fixed “coherency” function from

possible sets of beliefs to [-1, 1]• Beliefs are randomly initialised at the start• Beliefs are copied along links or dropped by

nodes according to the change in coherency that these result in

Coherency Function

• Gives a measure of the extent to which different sets of beliefs are coherent

• Assumes a background of shared beliefs• Thus {A}0.5 and {B}{0.7} but {A, B}-0.4 if

beliefs A and B are mutually inconsistent• Different coherency functions will be applicable

to different sets of ‘foreground’ candidate beliefs and backgrounds of shared beliefs

• The probability of gaining a new belief from another or dropping an existing belief in this model is dependent on whether it increases or decreases the coherency of the belief set

Processes

Each iteration the following occurs:• Copying: each arc is selected; a belief at the

source randomly selected; then copied to destination with a probability linearly proportional to the change in coherency it would cause

• Dropping: each node is selected; a random belief is selected and then dropped with a probability linearly proportional to the change in coherency it would cause

• -11 change has probability of 1• 1-1 change has probability of 0

Illustration

Opinion dynamics models, Nania, Edinburgh, August 2007, http://cfpm.org/nania slide-8

A

B

C

AB

Copying

C

C

Dropping

A

Example of the use of the coherency function• coherency({}) = -0.65• coherency({A}) = -0.81• coherency({A, B}) = -0.37• coherency({A, B, C}) = -0.54• coherency({A, C}) = 0.75• coherency({B}) = 0.19• coherency({B, C}) = 0.87• coherency({C}) = -0.56• A copy of a “C” making {A, B} change to {A, B, C} would

cause a change in coherence of (-0.37--0.54 = 0.17)• Dropping the “A” from {A, C} causes a change of -1.31

Consensus with different connectivity (bi-directional arcs)

0

5

10

5 200

5

10

15

20

25

Consensus of different coherence functions (20 uni-directional arcs)

Nania Final Meeting, Edinburgh, August 2008, http://cfpm.org/nania slide-11

0

5

10

5 200

decreasingFunction

doubleFunction

increasingFunction

randomConsistency

singleFunction

zeroConsistency

20 arcs - 10 nodes - 3 tags - cr .5 dr .5 - init prob 0.5 - diff uni-nets - selection con fns- PD

Consensus of different coherence functions (10 bi-directional arcs)

Nania Final Meeting, Edinburgh, August 2008, http://cfpm.org/nania slide-12

0

5

10

5 200

decreasingFunction

doubleFunction

increasingFunction

randomConsistency

singleFunction

zeroConsistency

10 arcs - 10 nodes - 3 tags - cr .5 dr .5 - init prob 0.5 - diff Bi-nets - selection con fns- PD

Consensus of different coherence functions (20 uni-directional arcs, only drop incoherent)

Nania Final Meeting, Edinburgh, August 2008, http://cfpm.org/nania slide-13

0

5

10

5 200

decreasingFunction

doubleFunction

increasingFunction

randomConsistency

singleFunction

zeroConsistency

20 arcs - 10 nodes - 3 tags - cr .5 dr .5 - init prob 0.5 - diff uni-nets - selection con fns- ODI

Example – fixed random coherency function – Fixed Random Fn

A B C

ABC

AB BCAC

-0.65

-0.81 0.19 -0.56

-0.54

-0.37 0.870.75

“Density” of A for different sized networks – Fixed Random Fn

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5 80 155 230 305 380 455

5

10

15

20

25

“Density” of C for different sized networks – Fixed Random Fn

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5 80 155 230 305 380 455

5

10

15

20

25

Number of Beliefs Disappeared over time, different sized networks – Fixed Random Fn

0

0.5

1

1.5

2

2.5

3

5 10 15 20 25 30 35 40 45 50Nu

mb

er

of B

elie

fs D

isa

ppe

are

d b

y tim

e 5

00

Network Size

Av. Resultant Opinion – Fixed Random Fn

Consensus – Fixed Random Function

Zero Function

A B C

ABC

AB BCAC

0

0 0 0

0

0 00

Consensus – Zero Fn

Single Function

A B C

ABC

AB BCAC

0

1 1 1

-1

-0.5 -0.5-0.5

Consensus – Single Fn

Av. Resultant Opinion – Single Fn

Prevalence of Belief Sets Example – Single

Double Function

A B C

ABC

AB BCAC

-1

0 0 0

-1

1 11

Consensus – Double Fn

Prevalence of Belief Sets Example – Double Fn

Av. Av. Resultant Opinion

Av. Consensus, Each Function

Effect of Number of Beliefs and Hardness of Coherency Function

Effect of Number of Agents, Drop Rat and Coherency Hardness

Effect of Network Structure with Contrasting Coherency Functions

Comparing with Evidence

• Lack of available cross-sectional AND longitudinal opinion studies in groups

• But it can be compared with broad hypotheses– Consensus only appears in small groups (balance of

beliefs in bigger ones)– Big steps towards agreement appears due to the

disappearance of beliefs– (Mostly) network structure does not matter– Relative coherency of beliefs matters– Different outcomes can result depending on what gets

dropped (in small groups)• How the model responds to different agents with

different consistency functions not yet examined

Future Work

• Validation! Finding suitable data sets where the coherency function can be estimated and time series of outcomes can be obtained

Possible extensions of model:• Making the model less noisy with a threshold for

coherency change (a minimum change of coherency for a change to occur)

• Agents with different coherency functions interacting in the same network

• Changing social network, maybe with belief homophily so that one is more likely to influence those with more similar beliefs

The End

Bruce Edmonds

http://bruce.edmonds.name

Centre for Policy Modelling

http://cfpm.org

A version of these slides is at: http://slideshare.com/BruceEdmonds

The simulation is available at: http://cfpm.org/models

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