Boolean Gossip Networks

Transcription

1 Boolean Gossip Networks Guodong Shi Research School of Engineering The Australian Na<onal University, Canberra, Australia ANU Workshop on Systems and Control,

2 Joint work with Bo Li and Hongsheng Qi, Academy of Mathema<cs and Systems Science, China Junfeng Wu, School of Control Automa<on, Zhejiang University, China and Alexandre Prou8ere, KTH Automa<c Control, Sweden 2

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5 Genes are DNA and RNA that are biological codes of molecules for their func<ons 4

6 Genes are DNA and RNA that are biological codes of molecules for their func<ons Genes, when turned on, are expressed into RNA and protein as func<onal gene products 4

7 Genes are DNA and RNA that are biological codes of molecules for their func<ons Genes, when turned on, are expressed into RNA and protein as func<onal gene products Some genes (or proteins) can control the expressions of other genes, which are known as regulator genes 4

8 Genes are DNA and RNA that are biological codes of molecules for their func<ons Genes, when turned on, are expressed into RNA and protein as func<onal gene products Some genes (or proteins) can control the expressions of other genes, which are known as regulator genes Gene regulators interact with each other forming coupled dynamical evolu<ons 4

9 Gene Regulatory Networks Genes are DNA and RNA that are biological codes of molecules for their func<ons Genes, when turned on, are expressed into RNA and protein as func<onal gene products Some genes (or proteins) can control the expressions of other genes, which are known as regulator genes Gene regulators interact with each other forming coupled dynamical evolu<ons 4

10 Gene Regulatory Networks [Ma et al. 2014] 5

11 Kauffman s Probabilis<c Boolean Network 6

12 Kauffman s Probabilis<c Boolean Network In a network of n nodes, each node holds a binary state at discre<zed <me instants 6

13 Kauffman s Probabilis<c Boolean Network In a network of n nodes, each node holds a binary state at discre<zed <me instants There are a finite number of mappings from n-dimensional binary space to itself 6

14 Kauffman s Probabilis<c Boolean Network In a network of n nodes, each node holds a binary state at discre<zed <me instants There are a finite number of mappings from n-dimensional binary space to itself Each node randomly selects one of the mappings describing how it interacts with the remainder of the network 6

15 Kauffman s Probabilis<c Boolean Network In a network of n nodes, each node holds a binary state at discre<zed <me instants There are a finite number of mappings from n-dimensional binary space to itself Each node randomly selects one of the mappings describing how it interacts with the remainder of the network x i (t) 2 {0, 1} 6

16 Kauffman s Probabilis<c Boolean Network In a network of n nodes, each node holds a binary state at discre<zed <me instants There are a finite number of mappings from n-dimensional binary space to itself Each node randomly selects one of the mappings describing how it interacts with the remainder of the network x i (t) 2 {0, 1} x i (t +1)=f i x 1 (t),...,x n (t) 6

17 Kauffman s Probabilis<c Boolean Network x i (t +1)=f i x 1 (t),...,x n (t) x i (t) 2 {0, 1} [Kauffman 1969] 7

18 Kauffman s Probabilis<c Boolean Network x i (t +1)=f i x 1 (t),...,x n (t) 8

19 Kauffman s Probabilis<c Boolean Network x i (t +1)=f i x 1 (t),...,x n (t) Finding a singleton abractor is NP hard! [Akutsu et al. 1998] 8

20 Kauffman s Probabilis<c Boolean Network x i (t +1)=f i x 1 (t),...,x n (t) Finding a singleton abractor is NP hard! [Akutsu et al. 1998] [Shmulevich et al. 2002; Brun et al. 2005; Cheng and Qi 2009; Chaves and Carta 2014; ] 8

21 Kauffman s Proposal Revisited x i (t +1)=f i x 1 (t),...,x n (t) 9

22 Kauffman s Proposal Revisited x i (t +1)=f i x 1 (t),...,x n (t) Locality of gene interac<ons: each regulator gene interacts only with a few (2 or 3) neighboring genes. element received just two inputs from other elements is biologically reasonable [Kauffman 1969] 9

23 A Boolean Gossip Network Model 10

24 Pairwise Boolean Interac<ons V={1,...,n} G= V, E x i (t) 2 {0, 1} 11

25 Pairwise Boolean Interac<ons i j 12

26 Pairwise Boolean Interac<ons x i (t) 2 {0, 1} i j 12

27 Pairwise Boolean Interac<ons x i (t) 2 {0, 1} x j (t) 2 {0, 1} i j 12

28 Pairwise Boolean Interac<ons x i (t) 2 {0, 1} x j (t) 2 {0, 1} i j [Karp et al. 2000, Boyd et al. 2006] 12

29 Pairwise Boolean Interac<ons 13

30 Pairwise Boolean Interac<ons x i (t) 2 {0, 1} x j (t) 2 {0, 1} i H = 1,..., 9, A,..., F j 14

31 Pairwise Boolean Interac<ons x i (t) 2 {0, 1} x j (t) 2 {0, 1} i H = 1,..., 9, A,..., F Admissible interac<on set C = C 1,..., C q j 14

32 Boolean Gossip i x i (t) 2 {0, 1} x j (t) 2 {0, 1} H = 1,..., 9, A,..., F C = C 1,..., C q j 15

33 Boolean Gossip i x i (t) 2 {0, 1} x j (t) 2 {0, 1} H = 1,..., 9, A,..., F C = C 1,..., C q j 8 >< x i (t +1)=x i (t) C x k j (t), x >: j (t +1)=x j (t) C x l i (t), x m (t +1)=x m (t), m /2 {i, j}. 15

34 Induced Markov Chain 16

35 Induced Markov Chain X t = x 1 (t)... x n (t) > 16

36 Induced Markov Chain X t = x 1 (t)... x n (t) > S n = [s 1...s n ]: s i 2 {0, 1},i2 V 16

37 Induced Markov Chain X t = x 1 (t)... x n (t) > S n = [s 1...s n ]: s i 2 {0, 1},i2 V ] P = P [s1...s n ][q 1...q 2 R 2 n 2 n : n P [s1...s n ][q 1...q n ] := P X t+1 =[q 1...q n ] X t =[s 1...s n ]. 16

38 Induced Markov Chain M G (C) =(S n,p) X t = x 1 (t)... x n (t) > S n = [s 1...s n ]: s i 2 {0, 1},i2 V ] P = P [s1...s n ][q 1...q 2 R 2 n 2 n : n P [s1...s n ][q 1...q n ] := P X t+1 =[q 1...q n ] X t =[s 1...s n ]. 16

39 Posi<ve Boolean Interac<ons 17

40 Posi<ve Boolean Opera<ons ^ _ 18

41 Posi<ve Boolean Opera<ons ^ _ C pst = {_, ^} 18

42 Conven<onal Machinery: Convergence Proposition. There exists a Bernoulli random variable x such that P lim x i (t) =x, for all i 2 V =1. t!1 The limit x satisfies E{x } = (I 2 n 2 Q) 1 R X 0 [1...1]. 19

43 Mean-field Approxima<on for Regular Graphs (t) = nx i=1 x i (t)/n 20

44 Mean-field Approxima<on for Regular Graphs (t) = nx i=1 x i (t)/n d ds (s) =p 2 2 (s)(1 (s)) (1 p ) 2 2 (s)(1 (s)) 20

45 Mean-field Approxima<on for Regular Graphs (t) = nx i=1 x i (t)/n d ds (s) =p 2 2 (s)(1 (s)) (1 p ) 2 2 (s)(1 (s)) (s) = (0) (1 (0))e 2(1 2p )s + (0). 20

46 Numerical Example Node Proportion with State Simulated Realization p * =0.49 ODE Approximation p =0.49 * Simulated Realization p =0.51 * ODE Approximation p =0.51 * Time t 21 x 10 4

47 Communica<on Classes 22

48 Communica<on Classes In a Markov chain, two states in the state space communicate with each other if they are accessible from each other. 22

49 Communica<on Classes In a Markov chain, two states in the state space communicate with each other if they are accessible from each other. Communica<on rela<onship forms an equivalence rela<on over the state space; the resul<ng equivalence classes are called communica<on classes. 22

50 Communica<on Classes In a Markov chain, two states in the state space communicate with each other if they are accessible from each other. Communica<on rela<onship forms an equivalence rela<on over the state space; the resul<ng equivalence classes are called communica<on classes. M G (C) =(S n,p) 22

51 Communica<on Classes In a Markov chain, two states in the state space communicate with each other if they are accessible from each other. Communica<on rela<onship forms an equivalence rela<on over the state space; the resul<ng equivalence classes are called communica<on classes. M G (C) =(S n,p) C (G) 22

52 Communica<on Classes Theorem There hold (i) C pst (G) = 2n if G is a line graph; (ii) C pst (G) = m +3ifGisacyclegraphwithn =2m; C pst (G) = m +2 if G is a cycle graph with n =2m +1; (iii) (iv) C pst (G) = 5 if G is neither a line nor a cycle, and contains no odd cycle; C pst (G) = 3 if G is not a cycle graph but contains an odd cycle. 23

53 Communica<on Classes C pst (G) = 2n 24

54 Communica<on Classes 25

55 Communica<on Classes C pst (G) = 3 25

56 General Boolean Interac<ons 26

57 General Boolean Opera<ons H = 1,..., 9, A,..., F C = C 1,..., C q 27

58 General Boolean Opera<ons H = 1,..., 9, A,..., F C = C 1,..., C q =

59 General Boolean Opera<ons H = 1,..., 9, A,..., F C = C 1,..., C q =65535 B := B 1 S B2 B 1 = C 6= { A } 2 2 H : { A } C { 2, 3, A, B} B 2 = C 2 2 H : { 2, B} C { 2, 3, A, B} 27

60 General Boolean Opera<ons H = 1,..., 9, A,..., F C = C 1,..., C q = elements! B := B 1 S B2 B 1 = C 6= { A } 2 2 H : { A } C { 2, 3, A, B} B 2 = C 2 2 H : { 2, B} C { 2, 3, A, B} 27

61 Communica<on Classes C (G) 28

62 Communica<on Classes C (G) C = C1,..., Cq 28

63 Communica<on Classes C (G) C = C1,..., Cq 28

64 Absorbing Chain Theorem Suppose C 2 B. Then M G (C) isanabsorbingmarkovchainifandonlyifg does not contain an odd cycle. 29

65 Absorbing Chain Theorem Suppose C 2 2 H \ B. Then M G (C) isanabsorbingmarkovchainifandonly if one of the following two conditions holds (i) C { 0, 1, 2, 3, 4, 5, 6, 7}; (ii) C { 1, 3, 5, 7, 9, B, D, F }. 30

66 Absorbing Chain Theorem Suppose C 2 2 H \ B. Then M G (C) isanabsorbingmarkovchainifandonly if one of the following two conditions holds (i) C { 0, 1, 2, 3, 4, 5, 6, 7}; (ii) C { 1, 3, 5, 7, 9, B, D, F }. Thank you! 30