Belief Change via Social Influence and Explanatory Coherence

1. Belief Change via Social Influence and Explanatory Coherence Bruce EdmondsCentre for Policy ModellingManchester Metropolitan University

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. Illustration A B C B Copying A A C C Dropping Opinion dynamics models, Nania, Edinburgh, August 2007, http://cfpm.org/nania slide-8

9. 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

10. Consensus with different connectivity (bi-directional arcs)

11. Consensus of different coherence functions (20 uni-directional arcs) 20 arcs - 10 nodes - 3 tags - cr .5 dr .5 - init prob 0.5 - diff uni-nets - selection con fns- PD Nania Final Meeting, Edinburgh, August 2008, http://cfpm.org/nania slide-11

12. Consensus of different coherence functions (10 bi-directional arcs) 10 arcs - 10 nodes - 3 tags - cr .5 dr .5 - init prob 0.5 - diff Bi-nets - selection con fns- PD Nania Final Meeting, Edinburgh, August 2008, http://cfpm.org/nania slide-12

13. Consensus of different coherence functions (20 uni-directional arcs, only drop incoherent) 20 arcs - 10 nodes - 3 tags - cr .5 dr .5 - init prob 0.5 - diff uni-nets - selection con fns- ODI Nania Final Meeting, Edinburgh, August 2008, http://cfpm.org/nania slide-13

14. Example – fixed random coherency function – Fixed Random Fn -0.54 ABC -0.37 AB AC 0.75 0.87 BC -0.81 A B 0.19 -0.56 C -0.65 

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

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

17. Number of Beliefs Disappeared over time, different sized networks – Fixed Random Fn Number of Beliefs Disappeared by time 500 Network Size

18. Av. Resultant Opinion – Fixed Random Fn

19. Consensus – Fixed Random Function

20. Zero Function ABC 0 AB AC BC 0 0 0 A B C 0 0 0  0

21. Consensus – Zero Fn

22. Single Function ABC -1 AB AC BC -0.5 -0.5 -0.5 A B C 1 1 1  0

23. Consensus – Single Fn

24. Av. Resultant Opinion – Single Fn

25. Prevalence of Belief Sets Example – Single

26. Double Function ABC -1 AB AC BC 1 1 1 A B C 0 0 0  -1

27. Consensus – Double Fn

28. Prevalence of Belief Sets Example – Double Fn

29. Av. Av. Resultant Opinion

30. Av. Consensus, Each Function

31. Effect of Number of Beliefs and Hardness of Coherency Function

32. Effect of Number of Agents, Drop Rat and Coherency Hardness

33. Effect of Network Structure with Contrasting Coherency Functions

34. 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

35. 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

36. 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 “ACS model -v2.2.nlogo”