A simulation model is presented that represents belief change, based on Thagard’s theory of explanatory coherence, within a population of agents who are connected by a social network. In this model there are a fixed number of represented beliefs, each of which are either held or not by each agent. These beliefs are to different extents coherent with each other – this is modelled using a coherence function from possible sets of core beliefs to [-1,1]. The social influence is achieved through gaining of a belief from another agent across a social link. Beliefs can be lost by being dropped from an agent’s store. Both of these processes happen with a probability related to the change in coherence that would result in an agent’s belief store. A resulting measured “opinion” can be retrieved in a number of ways, here as a weighted sum of a pattern of the core beliefs – opinion is thus an outcome and not directly processed by agents. Results suggest that a reasonable rate of copy and drop processes and a well connected network are required to achieve consensus, but given that, the approach is effective at producing consensuses for many compatibility functions. However, there are some belief structures where this is difficult.

1. Modelling Belief Change in a Population Using Explanatory Coherence Bruce EdmondsCentre for Policy ModellingManchester Metropolitan University

2. 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 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 2

3. 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 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 3

4. 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”functionfrom 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 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 4

5. Processes Each iteration the following occurs: Copying: each arc is selected; a belief at the source randomly selected; then copied to destination with a probabilitylinearly proportional to the change in coherency it would cause Dropping: each node is selected; a random belief is selected and then dropped with a probabilitylinearly proportional to the change in coherency it would cause -11 change has probability of 1 1-1 change has probability of 0 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 5

6. Coherency Function Not just binary consistency/inconsistency but a range of values in between too (hence name) Could be mapped onto individuals’ reports of (in)coherence between beliefs Can allow a mapping from a formal logic to a coherency function so that model dynamics roughly matches reasonable belief revision Thus if we know AB and B↔C then Cn might be constrained by Cn({A, B})≥Cn({A}) and Cn({B, C})<0... ...so if there are any B’s around then a node with {A} in its belief set will likely to become {A, B} and a node with {B,C} will probably drop one of B or C Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 6

7. 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 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 7

8. Example – the randomly assigned coherency function just specified ABC -0.54 AB BC AC -0.37 0.87 0.75 A -0.81 B C 0.19 -0.56  -0.65 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 8

9. 5 different coherency functions Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 9

10. “Density” of A for different sized networks – Fixed Random Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 10

11. “Density” of C for different sized networks – Fixed Random Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 11

12. Number of Beliefs Disappeared over time, different sized networks – Fixed Random Fn Number of Beliefs Disappeared by time 500 Network Size Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 12

13. Av. Av. Resultant Opinion Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 13

14. Av. Consensus, Each Function Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 14

15. Zero Function ABC 0 AB BC AC 0 0 0 A B C 0 0 0  0 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 15

16. Consensus – Zero Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 16

17. Av. Resultant Opinion – Fixed Random Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 17

18. The Fixed Random Fn ABC -0.54 AB BC AC -0.37 0.87 0.75 A -0.81 B C 0.19 -0.56  -0.65 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 18

19. Consensus – Fixed Random Function Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 19

20. Single Function ABC -1 AB BC AC -0.5 -0.5 -0.5 A B C 1 1 1  0 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 20

21. Consensus – Single Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 21

22. Av. Resultant Opinion – Single Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 22

23. Prevalence of Belief Sets Example – Single Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 23

24. Double Function ABC -1 AB BC AC 1 1 1 A B C 0 0 0  -1 Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 24

25. Consensus – Double Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 25

26. Prevalence of Belief Sets Example – Double Fn Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 26

27. 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) Ability to capture polarisation? To do! Modelling Belief Change in a Population Using Explanatory Coherence, Bruce Edmonds, CODYN@ECCS, Vienna, September 2011, slide 27

28. The End Bruce Edmonds http://bruce.edmonds.name Centre for Policy Modelling http://cfpm.org These slides have been uploaded to http://slideshare.com