A Minimal Dynamic Logic for Threshold Influence

Transcription

1 A for Threshold Influence Zoé Christoff Joint work with Alexandru Baltag, Rasmus K. Rendsvig and Sonja Smets M&M 2014 Lund - May /29

2 Social Network agents k j b c a i h d g f e 2/29

3 Social Network agents + links k j b c a i h d g f e 2/29

4 Diffusion Phenomena in Social Networks Diffusion of a behaviour : product fashion opinion infection facebook-like information... 3/29

5 Threshold Influence Adopt whenever enough network-neighbours have adopted it already. 4/29

6 Example: threshold = 1/4 new behaviour old behaviour j b c i a m h d k g f e 5/29

7 Example: threshold = 1/4 new behaviour old behaviour j b c i a m h d k g f e 5/29

8 Example: threshold = 1/4 new behaviour old behaviour j b c i a m h d k g f e 5/29

9 Example: threshold = 1/4 new behaviour old behaviour j b c i a m h d k g f e 5/29

10 Example: threshold = 1/4 new behaviour old behaviour j b c i a m h d k g f e Complete cascade! 5/29

11 Example: threshold = 1/2 new behaviour old behaviour j b c i a m h d k g f e 6/29

12 Example: threshold = 1/2 new behaviour old behaviour j b c i a m h d k g f e 6/29

13 Example: threshold = 1/2 new behaviour old behaviour j b c i a m h d k g f e 6/29

14 Example: threshold = 1/2 new behaviour old behaviour j b c i a m h d k g f e 6/29

15 Example: threshold = 1/2 new behaviour old behaviour j b c i a m h d k g f e 6/29

16 Questions Does the process stabilize? 7/29

17 Questions Does the process stabilize? How fast is the diffusion? 7/29

18 Questions Does the process stabilize? How fast is the diffusion? What properties of networks facilitate/slow down diffusion? 7/29

19 Questions Does the process stabilize? How fast is the diffusion? What properties of networks facilitate/slow down diffusion? When is a cascade complete? 7/29

20 Goal Model threshold influence using logic: 8/29

21 Goal Model threshold influence using logic: 1. Minimal dynamic logic for standard threshold models 8/29

22 Goal Model threshold influence using logic: 1. Minimal dynamic logic for standard threshold models 2. Epistemic extension for epistemic threshold models 8/29

23 Graph Definition (Network) A network (or graph) is a pair (A,N), where: A is a non-empty finite set of agents N : A P(A) assigns a set N(a) to each a A, such that: a / N(a) (Irreflexivity), b N(a) if and only if a N(b) (Symmetry). 9/29

24 Model Definition (Threshold model) A threshold model is a tuple M = (A,N,B,θ) where: (A,N) is a network, B A is a behaviour θ [0,1] is a uniform adoption threshold. 10/29

25 Update Definition (Threshold model update) Let M = (A,N,B,θ) be a threshold model. The updated model M = (A,N,B,θ), where: B = B {a A : N(a) B N(a) θ}. 11/29

26 Minimal Definition (Threshold influence language L TIA +) ϕ := N ab B a ϕ ϕ ϕ [adopt]ϕ where a,b A + for some finite A + 12/29

27 Threshold Models Definition (threshold model) A threshold model is a tuple M = (A,N,B,θ) where: (A,N) is a network, B A is a behaviour θ [0,1] is a uniform adoption threshold 13/29

28 Threshold Models of bounded size Models which the language L TIA + can fully describe: Definition (A + -threshold model) An A + - threshold model is a tuple M = (A,N,B,θ) where: (A,N) is a network, such that A A + B A is a behaviour θ [0,1] is a uniform adoption threshold 13/29

29 Threshold Models of bounded size Models which the language L TIA + can fully describe: Definition (A + -threshold model) An A + - threshold model is a tuple M = (A,N,B,θ) where: (A,N) is a network, such that A A + B A is a behaviour θ [0,1] is a uniform adoption threshold Definition (C A +) We denote by C A + the class of all A + -threshold models. 13/29

30 Threshold Models of bounded size Models which the language L TIA + can fully describe: Definition (A + -threshold model) An A + - threshold model is a tuple M = (A,N,B,θ) where: (A,N) is a network, such that A A + B A is a behaviour θ [0,1] is a uniform adoption threshold Definition (C A +) We denote by C A + the class of all A + -threshold models. Definition (C A + θ) We denote by C A + θ the class of all A + -threshold models with fixed threshold θ. 13/29

31 Semantics Definition (Truth clauses for L TIA +) Let M = (A,N,B,θ) be an A + -threshold-model and N ab,b a,ϕ,ψ L TIA +. 14/29

32 Semantics Definition (Truth clauses for L TIA +) Let M = (A,N,B,θ) be an A + -threshold-model and N ab,b a,ϕ,ψ L TIA +. M N ab iff b N(a) 14/29

33 Semantics Definition (Truth clauses for L TIA +) Let M = (A,N,B,θ) be an A + -threshold-model and N ab,b a,ϕ,ψ L TIA +. M N ab iff b N(a) M B a iff a B 14/29

34 Semantics Definition (Truth clauses for L TIA +) Let M = (A,N,B,θ) be an A + -threshold-model and N ab,b a,ϕ,ψ L TIA +. M N ab iff b N(a) M B a iff a B M ϕ iff M ϕ 14/29

35 Semantics Definition (Truth clauses for L TIA +) Let M = (A,N,B,θ) be an A + -threshold-model and N ab,b a,ϕ,ψ L TIA +. M N ab iff b N(a) M B a iff a B M ϕ iff M ϕ M ϕ ψ iff M ϕ and M ψ 14/29

36 Semantics Definition (Truth clauses for L TIA +) Let M = (A,N,B,θ) be an A + -threshold-model and N ab,b a,ϕ,ψ L TIA +. M N ab iff b N(a) M B a iff a B M ϕ iff M ϕ M ϕ ψ iff M ϕ and M ψ M [adopt]ϕ iff M ϕ, where M = (A,N,B,θ), with: B = B {a A : N(a) B N(a) θ}. 14/29

37 Axiomatization Network axioms 15/29

38 Axiomatization Network axioms N aa Irreflexivity N ab N b a Symmetry 15/29

39 Axiomatization Network axioms N aa Irreflexivity N ab N b a Symmetry Reduction axioms [adopt]n ab N ab 15/29

40 Axiomatization Network axioms N aa Irreflexivity N ab N b a Symmetry Reduction axioms [adopt]n ab N ab [adopt] ϕ [adopt]ϕ [adopt]ϕ ψ [adopt]ϕ [adopt]ψ 15/29

41 Axiomatization Network axioms N aa Irreflexivity N ab N b a Symmetry Reduction axioms [adopt]n ab N ab [adopt] ϕ [adopt]ϕ [adopt]ϕ ψ [adopt]ϕ [adopt]ψ [adopt]b a B a N ab N ab B b ) ( {G N A + : G N θ} b N b/ N b G 15/29

42 Axiomatization Network axioms N aa Irreflexivity N ab N b a Symmetry Reduction axioms [adopt]n ab N ab [adopt] ϕ [adopt]ϕ [adopt]ϕ ψ [adopt]ϕ [adopt]ψ [adopt]b a B a N ab N ab B b ) ( {G N A + : G N θ} b N b/ N b G Theorem (Completeness) For every ϕ L TIA +, = CA + θ ϕ iff LTIA + θ ϕ 15/29

43 Expressing stability? Stable model M ϕ [adopt]ϕ for all ϕ L TIA +. 16/29

44 Expressing stability? Stable model M ϕ [adopt]ϕ for all ϕ L TIA +. M B a [adopt]b a for all a A +. 16/29

45 Expressing stabilization? Stabilizing model For some n N: M [adopt] n ϕ [adopt] n+1 ϕ for all ϕ L TIA +. 17/29

46 Expressing stabilization? Stabilizing model For some n N: M [adopt] n ϕ [adopt] n+1 ϕ for all ϕ L TIA +. M [adopt] n B a [adopt] n+1 B a for all a A +. 17/29

47 Expressing stabilization? Stabilizing model For some n N: M [adopt] n ϕ [adopt] n+1 ϕ for all ϕ L TIA +. M [adopt] n B a [adopt] n+1 B a for all a A +. All models stabilize in at most A 1 steps M [adopt] A 1 B a [adopt] A B a for all a A +. 17/29

48 Clusters Definition (Cluster of density d) Given a network (A,N) a cluster of density d is any group C A such that for all i C, N(i) C d. N(i) Any network contains at least one cluster of density 1, namely A and that each singleton {a} A is a cluster of density 0 (by irreflexivity). 18/29

49 Example: clusters o,n,i,l : cluster of density 3/4 o j b c n i h a f l k g e d 19/29

50 Example: clusters o,n,i,l : cluster of density 3/4 c,f,d,a,e: cluster of density 2/3 o j b c n i h a f l k g e d 19/29

51 When is a cascade complete (for θ 0)? When (and only when) there is no cluster of density greater than 1 θ in A\B 20/29

52 Back to our initial example: threshold = 1/4 There is no red cluster of density > 3/4 new behaviour old behaviour j b c i a m h d k g f e 21/29

53 Back to our initial example: threshold = 1/4 There is no red cluster of density > 3/4 new behaviour old behaviour j b c i a m h d k g f e 21/29

54 Back to our initial example: threshold = 1/4 There is no red cluster of density > 3/4 new behaviour old behaviour j b c i a m h d k g f e 21/29

55 Back to our initial example: threshold = 1/4 There is no red cluster of density > 3/4 new behaviour old behaviour j b c i a m h d k g f e 21/29

56 Back to our initial example: threshold = 1/4 There is no red cluster of density > 3/4 new behaviour old behaviour j b c i a m h d k g f e Complete cascade! 21/29

57 Back to our initial example: threshold = 1/2 a,b,c,d,m,e,f form a cluster of density d = 5/7 > 1/2 new behaviour old behaviour j b c i a m h d k g f e 22/29

58 Back to our initial example: threshold = 1/2 a,b,c,d,m,e,f form a cluster of density d = 5/7 > 1/2 new behaviour old behaviour j b c i a m h d k g f e 22/29

59 Back to our initial example: threshold = 1/2 a,b,c,d,m,e,f form a cluster of density d = 5/7 > 1/2 new behaviour old behaviour j b c i a m h d k g f e 22/29

60 Back to our initial example: threshold = 1/2 a,b,c,d,m,e,f form a cluster of density d = 5/7 > 1/2 new behaviour old behaviour j b c i a m h d k g f e 22/29

61 Back to our initial example: threshold = 1/2 a,b,c,d,m,e,f form a cluster of density d = 5/7 > 1/2 new behaviour old behaviour j b c i a m h d k g f e 22/29

62 Talking about clusters C A is a cluster of density d in (A,N) iff M = (A,N,B,θ) satisfies: i C {N A: N C ( d} j N Nij j N Nij) N 23/29

63 Existence of a -cluster There exists a cluster of density d: C d := C A i C {N A: N C ( d} j N Nij j N Nij) N There exists a cluster of density greater than d: C >d := C A i C {N A: N C ( >d} j N Nij j N Nij) N 24/29

64 Existence of a B-cluster There exists a cluster of density d: C d := C A i C {N A: N C ( d} j N Nij j N Nij) N There exists a cluster of density greater than d: C >d := C A i C {N A: N C ( >d} j N Nij j N Nij) N There exists a cluster of density greater than d in which nobody has adopted : C >d B := C A i C {N A: N C ( >d} j N Nij j N N Nij Bi) 24/29

65 A cascade is complete (for θ 0) iff there is no cluster of density greater than 1 θ in A\B [adopt] A 1 B i C >(1 θ) B. i A 25/29

66 Future research Use the minimal logic to prove (new?) things about the threshold influence dynamics. Consider less minimal cases: drop constraints on the network structure unadoption (stabilization becomes more interesting), several behaviours, non-uniform thresholds, etc. 26/29

67 Conclusion: Blind influence vs Informed influence? SO FAR: Agents react automatically to their environnment. BUT WHAT IF... the behaviour of agents depends on what they KNOW/SEE about the behaviour of others around them? 27/29

68 Conclusion: Maybe blind influence is not so blind after all Under some natural assumptions about the knowledge of agents in a network... 28/29

69 Conclusion: Maybe blind influence is not so blind after all Under some natural assumptions about the knowledge of agents in a network... the dynamics of diffusion among informed agents reduces to the diffusion among our blind agents! 28/29

70 Conclusion: Maybe blind influence is not so blind after all Under some natural assumptions about the knowledge of agents in a network... the dynamics of diffusion among informed agents reduces to the diffusion among our blind agents!...stay tuned for Rasmus! 28/29

71 THANK YOU! 29/29