Belief Revision in Social Networks

Transcription

1 Tsinghua University and University of Amsterdam 8-10 July 2015, IRIT, Université Paul Sabatier, Toulouse

2 Outline 1 2 Doxastic influence Finite state automaton Stability and flux Dynamics in the community 3 4

3 It is our friends that make our world Wherever we are, it is our friends that make our world. Henry Drummond ( )

4 Peer pressure I prefer p but my friends all prefer p Do I change my belief/preference?

5 General issues How does an agent change her belief with the influence from her friends? How to justify one s belief or belief change in social setting? How to model change of social relation and its epistemic impact? Can one find a way to inform all people in the network as fast as possile? (computational)

6 Main Reference Doxastic influence Finite state automaton Stability and flux Dynamics in the community, Jeremy Seligman, and Patrick Girard, "Logical Dynamics of Belief Change in the Community", Synthese, Volume 191, Issue 11, pp , 2014.

7 Doxastic influence Doxastic influence Finite state automaton Stability and flux Dynamics in the community Consider influence regarding a single proposition p. If I do not believe p and some significant number or proportion of my friends do believe it. How should I respond? (1) ignore their opinions and remain doxastically unperturbed. (2) If I am influenced to change my beliefs there are at least two ways: I may revise so that I too believe p (Rp) or (more cautiously) contract, removing p (Cp). Notations: [Rp]Bp; [Cp] B p. (Assuming success conditions)

8 Doxastic influence Doxastic influence Finite state automaton Stability and flux Dynamics in the community We draw a distinction between two kinds of influence: (a) influence that leads to revision (strong influence): Sp. (b) influence that leads to contraction (weak influence): Wp. We define a general operation of social influence regarding p (Ip) as the program if Sp then Rp else if Wp then C p; if S p then R p else if W p then Cp

9 Strong and weak influence Doxastic influence Finite state automaton Stability and flux Dynamics in the community I am strongly influenced to believe p iff ALL (at least one) of my friends believe p. I am weakly influenced to contract by belief in p iff NONE of my friends believe p. Strong and weak influence captured with the following axioms: Sϕ (FBϕ F Bϕ) W ϕ (F B ϕ F Bϕ)

10 Doxastic influence Finite state automaton Stability and flux Dynamics in the community Distribution of doxastic states: Example a : Bp b : B p c : Up d : Bp A network of four agents. Links between nodes indicate friendship (irreflexive and symmetric relation). Agent a believes p (written a : Bp) and has friends b and c ; agent b disbelieves p and has friends a and d; and so on.

11 Finite state automaton Doxastic influence Finite state automaton Stability and flux Dynamics in the community Wp Sp Up W p S p S p Sp B p S p Bp Sp

12 Threshhold influence: Example Doxastic influence Finite state automaton Stability and flux Dynamics in the community Example 1: a : Bp b : B p a:up b : Bp c : Up d : Bp Ip c : Bp d : Up Observation: In the configuration on the right, a and d are both strongly influenced to believe p, and so a further application of Ip would cause them to revise their beliefs, returning them to their previous doxastic states.

13 Stability and flux Doxastic influence Finite state automaton Stability and flux Dynamics in the community A community is stable if the operator Ip has no effect on the doxastic states of any agent in the community. Unanimity within the community is sufficient for stability but not necessary, as is shown below: Example 2: a : Bp b : Up c : Bp d : Up e : Bp f : B p g : B p h : Up

14 Doxastic influence Finite state automaton Stability and flux Dynamics in the community Stability and flux: one more example Configurations that never become stable will be said to be in flux: Example 3: a : Bp a : B p a : Bp b : B p Ip b : Bp Ip b : B p Ip...

15 Characterizing stability Doxastic influence Finite state automaton Stability and flux Dynamics in the community (B p Wp) (Up Sp) (Up S p) (Bp W p) Under the assumption of threshold influence, it is equivalent to (B p F B p F Bp) ( Bp B p FBp F Bp) (( Bp B p FB p F B p) (Bp F Bp F B p) A community is stable when every agent in the community satisfies this condition.

16 Doxastic influence Finite state automaton Stability and flux Dynamics in the community Dynamics I: Private belief change in a community Agents may change their minds for many reasons other than the influence of their friends opinions. This raises the question of if and how such changes are propagated to other members of the community. A very coherent community may resist all changes, ensuring that any agent who changes her mind unilaterally will soon be brought back into conformity. A less coherent community may be highly affected by the change, going into flux or even following the agent who changed her mind into a new stable configuration.

17 Isolated private belief change Doxastic influence Finite state automaton Stability and flux Dynamics in the community Example 4: a : Up b : Up a : Bp b : Up c : Up a Bp c : Up

18 More active resistence Doxastic influence Finite state automaton Stability and flux Dynamics in the community Unanimous belief within a community can be strong enough to resist private belief changes even further: Example 5: a : Bp b : Bp a : B p b : Bp a : Bp b : Bp c : Bp a B p c : Bp a : Sp c : Bp

19 location is critical Doxastic influence Finite state automaton Stability and flux Dynamics in the community For a community of undecided agents to be influenced by a private belief change, the location of the agent who comes to believe p is critical. Example 6: a : Up c : Up b : Up b Bp a : Up c : Up b : Bp a : Sp c : Sp a : Bp c : Bp b : Bp

20 Doxastic influence Finite state automaton Stability and flux Dynamics in the community Dynamics II: Gaining and losing friends Example 7: a : Up b : Bp c : Fa a : Up b : Bp a, c : Sp a : Bp b : Bp c : Up c : Up c : Bp We start with a stable distribution of opinions among three mutual friends, with only one believer. One friendship is broken, putting the mutual friend into a position of greater influence.

21 New friend joining in Doxastic influence Finite state automaton Stability and flux Dynamics in the community Example 8: a : B p b : Bp a : Sp b : S p a : Bp b : B p c : Fa c : Fb a : Bp b : B p c : Up c : Up c : Up The oscillating pair of friends at the top is calmed when an indifferent agent joins their circle. In one more step, they will all become undecided.

22 A different newcomer Doxastic influence Finite state automaton Stability and flux Dynamics in the community Example 9: a : Bp c : B p b : B p d : Fc a, c : Sp b : S p a : B p c : Bp b : Bp a : Sp a : Bp c : Bp b : Bp d : Bp d : Bp d : Bp

23 Adding justification We change our beliefs for some reasons. Two aspects to be considered: Instead of changing beliefs due to pure pressure, we want to connect agent s evidence to her beliefs. Friends are not always equal, we may trust some friends more than others.

24 Main reference Ongoing joint work with Alexandru Baltag and Sonja Smets.

25 Definition (Weighting justification model) A weighting justification model is a structure (S, E, w, V ) where a finite set S of possible worlds. a family E P(S) of non-empty subset e S ( / E), called evidence such that S is itself an evidence set (S E). A body of evidence is any consistent family of evidence sets, i.e. any G E such that G. We denote E P(E) the family of all bodies of evidence. w : E R. V a stardard valuation function. This is taken from Fiutek, other forms of the definition appeared in various works by Baltag, Smets, Christoff, and Hansen.

26 Adding epistemic and social components Definition (Epistemic weighting social network model) An epistemic weighting social network model is a structure (S, E, A, F, a, w, τ, V ) where S is a finite set of possible states; a is a binary epistemic indistingishable relation; V is a valuation function. E, a set of evidence. A is a finite set of agents. F : S (A P(A)). w : S A (E R). τ : S A (A R + ).

27 Some notations F s (a) for a s friends at s. wa(e) s for a s strength of her evidence e at s. τ s ab for a s trust towards b at s. s(a) = {t : s a t}.

28 Conditions (1) a is an equivalence relation (2) a F s (a) (3) s a s = F s (a) = F s (a) (3) e s(a) = = w s a(e) = 0 (4) s a s = w s a = w s a (5) s a s = τ s ab = τ s ab

29 Induced plausibility relation Definition (Induced plausibility relation) We define the notion of the largest body of evidence consistent with a given state s S and write it as E s := {e E s e} We can induce a plausbility relation on states directly from the weight comparison on E: for two states s, t S, we put s a t iff s a t and w a (s) w a (t) where w a (s) = {w s a(e) : e E s }.

30 Defining doxastic notions Given an epistemic weighting social network model, we can define the useful notions by using the plausibility order a : K a p = {s S s(a) P}. Best a P = {s P t a s for all t P}. B Q a P = {s S best a (Q s(a)) P}. a P = {s S t(s a t = t P)}.

31 Dynamic update Definition (Updated model) Given an epistemic weighting social network model M= (S, E, A, F, a, w, τ, V ), after one round of social communication, the updated model M is defined as follows: S, E, A, F, V and τ remain the same. { 0, if e s w a s (a) = (e) = b F s (a) τ ab s w b s(e), otherwise s a t s a t and w b s (e) = w b t (e) for all e, b F s (a) Note that friendship relation (F) and the trust weight τ can also change in a more complex setting.

32 Example Example 10: a : Up b : B p c : Bp w a (p) = 0, w a ( p) = 0 w b (p) = 0, w b ( p) = 1 w c (p) = 1, w c ( p) = 0

33 Example continued: communication Assuming τ aa = τ bb = τ cc = τ ab = τ ba = τ bc = τ cb = τ ac = τ ca = 1, after one-step communication, the updated model is (stable): Example 11: a : Up b : Up c : Up w a(p) = 1, w a( p) = 1 w b (p) = 1, w b ( p) = 1 w c(p) = 1, w c( p) = 1

34 With different trust weight Assuming τ aa = τ bb = τ cc =τ ab =τ ba = τ bc =τ cb = τ ca = 1, and τ ac = 2, after communication, Example 10 changes into: Example 12: a : Bp b : Up c : Up w a(p) = 2, w a( p) = 1 w b (p) = 1, w b ( p) = 1 w c(p) = 1, w c( p) = 1

35 In flux Assuming τ ab = τ ba = 2, and τ aa = τ bb = 1, consider Example 13: a : Bp a : B p a : Bp b : B p Ip b : Bp Ip b : B p Ip...

36 ?? s a t = τ s ba = τ t ba for all b F s (a) Does an agent know how much her friends trust her?

37 ?? s a t = τ s bc = τ t bc for all b, c F s (a) Does an agent a always know how much her friend b trusts her friend c?

38 Example: different strategies Example (Different strategies) Assume F s (a) = {b, c}, τab s = 10, τ ac s = 1. If a knows that and τbc s = 10, what would she do? keep updating as we proposed. optimistic: increase τ s ac pessimistic: lower τ s ab

39 Some remarks Though our models are based on the weighting changes (non-agm dynamics), but they can simulate AGM dynamics. We may need a set of update strategies, instead of one general rule. This may provide a way of interpreting "social" in epistemology.

40 Lots of interesting/open issues Incorporate dynamics of trust and friendship in the latter model Connecting to social epistemology: e.g. formation of public opinions in social network Representing the structure of social networks in logic: experts; skeptical vs. sheep rationality (CUNY, Stanford students)