Complex Graphs and Networks Lecture 3: Duplication models for biological networks

Transcription

1 Complex Graphs and Networks Lecture 3: Duplication models for biological networks Linyuan Lu University of South Carolina BASICS2008 SUMMER SCHOOL July 27 August 2, 2008

2 Overview of talks Lecture 1: Overview and outlines Lecture 2: Generative models - preferential attachment schemes Lecture 3: Duplication models for biological networks Lecture 4: The rise of the giant component Lecture 5: The small world phenomenon: average distance and diameter Lecture 6: Spectrum of random graphs with given degrees Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 2 / 47

3 The power law The number of vertices of degree k is approximately proportional to k β for some positive β. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 3 / 47

4 The power law The number of vertices of degree k is approximately proportional to k β for some positive β. A power law graph is a graph whose degree sequence satisfies the power law. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 3 / 47

5 Power law distribution Left: The collaboration graph follows the power law degree distribution with exponent β 3.0 Right: An IP graph follows the power law degree distribution with exponent β 2.4 Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 4 / 47

6 Power law in ecological networks P (k) k β Lecture 3: Duplication models for biological networks Jordan and Scheuring, Oikos, 2002 Linyuan Lu (University of South Carolina) 5 / 47

7 Ecological networks Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 6 / 47

8 Functional associations of proteins P(k) k 1.6 Snel, Bork & Huynen, PNAS 99, (2002) Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 7 / 47

9 A map of protein-protein interactions in saccharomyces cerevisiae Jeong, Mason, Barabasi, Oltwai, Nature, 2001 Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 8 / 47

10 Biological networks versus non-biological networks Biological Networks β Yeast Gene Expression 1.5 Yeast Protein-Protein Maps 1.5, 1.7, 2.1 E. Coli Metabolic Map 1.7, 2.1 Ecology 1.7, 2.1 Other Networks β WWW Graphs 2.1 (in), 2.5 (out) Collaboration Graphs 3 Call Graphs 2.2 Costars Graph of Actors 2.3 Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 9 / 47

11 A critical threshold β = 2 Range 1 < β < 2 2 < β Average degree Unbounded Bounded Examples Biological networks Non-biological networks Known models evolution None Many Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 10 / 47

12 The reference [1.] Fan Chung and Linyuan Lu, T. Gregory Dewey, and David J. Galas. Duplication models for biological networks, Journal of Computational Biology, 10, No. 5, (2003), Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 11 / 47

13 Genome evolution Susumu Ohno s insight The best source of new genes is old genes, and that s where they come from! Gene duplication can include the duplication of regulatory regions - both nodes and edges are duplicated. This may not be the only way to use old information for new purposes, but it s a major one. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 12 / 47

14 Genomic duplications in saccharomyces cerevisiae Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 13 / 47

15 Gene regulatory graphs Genes Nodes, cis regulatory sites Edges. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 14 / 47

16 Partial duplication Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 15 / 47

17 Continue Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 16 / 47

18 Partial-duplication model Evolution of graphs Construct G t+1 from G t, G t 1 G t G t+1 - Select a random vertex u of G t uniformly. - Add a new vertex v and the edge uv. - For each neighbor w of u, with probability p, add an edge wv independently. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 17 / 47

19 Full duplication verse partial duplication Full duplication 1 Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 18 / 47

20 Full duplication verse partial duplication Full duplication 1 1 Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 18 / 47

21 Full duplication verse partial duplication Full duplication 1 1 Partial duplication 1 Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 18 / 47

22 Full duplication verse partial duplication Full duplication 1 1 Partial duplication 1 p 1 1 p p Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 18 / 47

23 Recurrence formula f(i, t) the number of vertices with degree i at time t. Type Probability Value d t u = i d t+1 1+pi u = i t d t u = i 1 d t+1 1+(i 1)p u = i ( 1 t d t u = j d t+1 v = i j i 1) p i 1 (1 p) j i+1 1 Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 19 / 47

24 Recurrence formula f(i, t) the number of vertices with degree i at time t. Type Probability Value d t u = i d t+1 1+pi u = i t d t u = i 1 d t+1 1+(i 1)p u = i ( 1 t d t u = j d t+1 v = i j i 1) p i 1 (1 p) j i+1 1 E(f(i,t + 1)) = (1 1 + pi )E(f(i,t)) t 1 + (i 1)p + E(f(i 1,t)) t + ( ) j p i 1 (1 p) j i+11 i 1 t E(f(j,t)). j i 1 Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 19 / 47

25 Heuristic analysis Substitute E(f(i, t)) by a i t. a i (t + 1) = (1 1 + pi )a i t t 1 + (i 1)p + a i 1 t t + ( ) j p i 1 (1 p) j i+11 i 1 t a jt. j i 1 Let t, we have (2+ip)a i = (1+p(i 1))a i 1 + j i 1 ( ) j a j p i 1 (1 p) j i+1. i 1 Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 20 / 47

26 Recurrence formula for a i Replace i by i + 1. We get (2 + (i + 1)p)a i+1 = (1 + pi)a i + j i a j ( j i ) p i (1 p) j i. Here a 0 = 0. (No isolated vertex.) Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 21 / 47

27 Lemma 1 Lemma:For fixed k and large x, we have ( x c ) ( k 1 x = (1 + O( k) x k ))(1 k x )c. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 22 / 47

28 Lemma 1 Lemma:For fixed k and large x, we have ( x c ) ( k 1 x = (1 + O( k) x k ))(1 k x )c. Proof: ( x c ( k x k ) ) = Γ(x c + 1)/Γ(x c + 1 k) Γ(x + 1)/Γ(x + 1 k) 1 = (1 + O( x k )) (x + 1) c (x + 1 k) c 1 = (1 + O( x k ))(1 k x )c. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 22 / 47

29 Lemma 2 Lemma: For large i, we have ( j ( ) j i ) β p i (1 p) j i = p β 1 + O( 1 i i ). j i Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 23 / 47

30 Lemma 2 Lemma: For large i, we have ( j ( ) j i ) β p i (1 p) j i = p β 1 + O( 1 i i ). j i Proof: ( j ( ) j i ) β p i (1 p) j i = i j i k=0 (1 + k i ) β ( i + k k ) p i (1 p) k Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 23 / 47

31 Lemma 2 Lemma: For large i, we have ( j ( ) j i ) β p i (1 p) j i = p β 1 + O( 1 i i ). j i j i Proof: ( j ( ) j i ) β p i (1 p) j i = i (1 + k ( i + k i ) β k k=0 = (1 + O( 1 i )) ( i + k β k k=0 ) p i (1 p) k ) p i (1 p) k Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 23 / 47

32 Lemma 2 = (1 + O( 1 i ))pi k=0 ( i + k β k ) (1 p) k Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 24 / 47

33 Lemma 2 = (1 + O( 1 i ))pi k=0 = (1 + O( 1 i ))pi k=0 ( ) i + k β (1 p) k k ( ) β i 1 ( 1) k (1 p) k k Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 24 / 47

34 Lemma 2 = (1 + O( 1 i ))pi k=0 = (1 + O( 1 i ))pi k=0 ( ) i + k β (1 p) k k ( ) β i 1 ( 1) k (1 p) k k = (1 + O( 1 i ))pi (1 (1 p)) β i 1 = (1 + O( 1 i ))pβ 1. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 24 / 47

35 Heuristic analysis II Heuristically, we let a i Ci β for large i. we have a i+1 a i (1 β i ) a j a i ( j i ) β. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 25 / 47

36 Heuristic analysis II Heuristically, we let a i Ci β for large i. we have a i+1 a i (1 β i ) a j a i ( j i ) β. (2+(i+1)p)(1 β i ) = (1+ip)+ j i Apply Lemma 2 and simplify it. We get 1 + p = pβ + p β 1. ( j i ) β ( j i ) p i (1 p) j i. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 25 / 47

37 Results Theorem (Chung, Dewey, Galas, Lu, 2002) Almost surely, the partial duplication model with selection probability p generates power law graphs with the exponent β satisfying p(β 1) = 1 p β 1. In particular, if 1 2 < p < 1 then β < 2. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 26 / 47

38 What we need to do Show the limit lim t f(t,i) t = a i exits. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 27 / 47

39 What we need to do f(t,i) Show the limit lim t t = a i exits. Show a i satisfies i=1 a i = 1 and (2 + (i + 1)p)a i+1 = (1 + pi)a i + j i a j ( j i ) p i (1 p) j i. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 27 / 47

40 What we need to do f(t,i) Show the limit lim t t = a i exits. Show a i satisfies i=1 a i = 1 and (2 + (i + 1)p)a i+1 = (1 + pi)a i + j i a j ( j i ) p i (1 p) j i. Show a i ci β and β satisfies p(β 1) = 1 p β 1. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 27 / 47

41 A tricky way Because the ratio f(t,i) a i t oscillates a lot, it is very hard to f(t,i) prove lim i a i t = 1 directly. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 28 / 47

42 A tricky way Because the ratio f(t,i) a i t oscillates a lot, it is very hard to f(t,i) prove lim i a i t = 1 directly. Let g(t, i) = 1 i t k=1 E(f(t, i)) be the expected number of vertices with degree at most i at time t. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 28 / 47

43 A tricky way Because the ratio f(t,i) a i t oscillates a lot, it is very hard to f(t,i) prove lim i a i t = 1 directly. Let g(t, i) = 1 i t k=1 E(f(t, i)) be the expected number of vertices with degree at most i at time t. By definition, for fixed t, g(t, i) is an increasing function of i. In particular, g(t, i) = 1, for all i t 0. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 28 / 47

44 A tricky way Because the ratio f(t,i) a i t oscillates a lot, it is very hard to f(t,i) prove lim i a i t = 1 directly. Let g(t, i) = 1 i t k=1 E(f(t, i)) be the expected number of vertices with degree at most i at time t. By definition, for fixed t, g(t, i) is an increasing function of i. In particular, g(t, i) = 1, for all i t 0. We will show lim t g(t, i) = i k=0 a i. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 28 / 47

45 Recurrence formula for g(t, i) Lemma For i 1 and t n 0, g(t, i) satisfies the following recurrence formula. g(t + 1, i) = (1 2 + pi 1 + pi )g(t, i) + g(t, i 1) t + 1 t g(t, j)f(j, i 1, p). t + 1 j i Here g(t, 0) = 0 and F(j, i, p) = i ( j ) k=0 k p k (1 p) j k i k=0 ( j+1 ) k p k (1 p) j+1 k. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 29 / 47

46 Remark The lemma can be proved by induction on i and the following tool. Abel summation identity (c j c j 1 )d j = j=1 c j (d j d j+1 ) c 0 d 1. j=1 Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 30 / 47

47 The result Lemma:For all i, the limit lim t g(t, i) exists. We have i lim g(t, i) = t k=1 a k. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 31 / 47

48 The result Lemma:For all i, the limit lim t g(t, i) exists. We have i lim g(t, i) = t k=1 a k. Proof: Let s i = i k=1 a k and h(t) = sup{ g(t,i) s i } i 1. We claim that h(t) 1, for all t 1. h(t) is a decreasing function of t. lim t h(t) = 1. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 31 / 47

49 Proof The first item is from the following observation: h(t) g(t, t) s t = 1 s t 1. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 32 / 47

50 Proof The first item is from the following observation: h(t) g(t, t) s t = 1 s t 1. To show h(t + 1) h(t), it is sufficient to show For i 1, we have g(t + 1, i) h(t)s i, for all i. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 32 / 47

51 Proof g(t + 1, i) = (1 2 + pi 1 + ip )g(t, i) + g(t, i 1) t + 1 t g(t, j)f(j, i 1, p) t + 1 j i (1 2 + pi t + 1 )h(t)s i ip t + 1 h(t)s i h(t)s j F(j, i 1, p) t + 1 = h(t)s i. j i Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 33 / 47

52 Proof g(t + 1, i) = (1 2 + pi 1 + ip )g(t, i) + g(t, i 1) t + 1 t g(t, j)f(j, i 1, p) t + 1 j i (1 2 + pi t + 1 )h(t)s i ip t + 1 h(t)s i h(t)s j F(j, i 1, p) t + 1 = h(t)s i. j i g(t + 1, i) Thus, h(t + 1) = sup{ } i 1 h(t). Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 33 / 47 s i

53 Proof The function h(t) monotone decreases and lower-bounded by 1. Therefore, the limit lim t h(t) exists. We denote the limit by c. We have c = lim t h(t) 1. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 34 / 47

54 Proof The function h(t) monotone decreases and lower-bounded by 1. Therefore, the limit lim t h(t) exists. We denote the limit by c. We have c = lim t h(t) 1. If c = 1, then lim t g(i, t) = s i. We are done. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 34 / 47

55 Proof by contradiction Suppose, to the contrary, that c > 1. We note lim s i = i a i = 1. i=0 Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 35 / 47

56 Proof by contradiction Suppose, to the contrary, that c > 1. We note lim s i = i i=0 a i = 1. There exists an index i 0 so that s i c for i i 0. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 35 / 47

57 Proof by contradiction Suppose, to the contrary, that c > 1. We note lim s i = i i=0 a i = 1. There exists an index i 0 so that s i c for i i 0. For any t and i i 0, we have g(t, i) s i 1 s i 1 + c 2. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 35 / 47

58 Proof by contradiction Thus, the supreme h(t) is always achieved at some index i(t) < i 0. Let c i = a j j i 0 a i F(j, i 1, p). We define δ to the minimum value of c 0, c 1,..., c i0. It is clear that δ > 0. We can prove g(t, i) s i h(t 1) (c 1)δ 2t. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 36 / 47

59 Proof by contradiction Thus, the supreme h(t) is always achieved at some index i(t) < i 0. Let c i = a j j i 0 a i F(j, i 1, p). We define δ to the minimum value of c 0, c 1,..., c i0. It is clear that δ > 0. We can prove g(t, i) s i h(t 1) (c 1)δ 2t. Since the superium h(t) is always achieved at some index i(t) < i 0, we have Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 36 / 47

60 Proof by contradiction g(t, i) h(t) = min{, 0 i i 0 } s i h(t 1) (c 1)δ 2t. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 37 / 47

61 Proof by contradiction In particular, we have g(t, i) h(t) = min{, 0 i i 0 } s i h(t 1) (c 1)δ 2t. h(t) h(0) (c 1)δ 2 t k=1 1 k. Since the harmonic sum diverges, for t large enough, h(t) < 0. Contradiction to h(t) 1, for all t. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 37 / 47

62 Generating function Lemma: Let F(z) = i=1 a iz i. Then (2/z 1)F(z) F(pz + 1 p) + p(1 z)f (z) = 0. Proof: We have F(pz + 1 p) = = = j=0 j=0 i=0 a j (pz + 1 p) j a j Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 38 / 47 j i=0 z i j=i ( ) j p i (1 p) j i z i i ( ) j a j p i (1 p) j i. i

63 Generating function F(pz + 1 p) = = = 2 i=0 i=0 i=0 z i j=i a j ( j i ) p i (1 p) j i z i[ (2 + p(i + 1))a i+1 (1 + pi)a i ] i=0 a i+1 z i + p a i z i p (i + 1)a i+1 z i i=0 i=0 ia i z i = 2F(z)/z + pf (z) F(z) pzf (z). Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 39 / 47

64 The average degree Rewritten as F(pz + 1 p) (2/z 1)F(z) 1 z = pf (z). Take the limit as z 1. we have pf (1) + F (1) 2 = pf (1). F (1) = 2 1 2p. If p < 1 2, then the expected average degree is 2 1 2p. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 40 / 47

65 Maximal degree Theorem For the duplication model, with probability at least 1 te c, the maximum degree t is at most Θ(ct p ). Moreover, suppose the maximum degree of the initial graph G t0 is t0. Then with probability at least 1 te c, we have t ( t0 + 1 p + c c( t p ) + c ( ) p t 3 ). t 0 Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 41 / 47

66 Concentration result For the duplication model G(p), for any vertex v with degree born before time t 1, the degree d v (t) at time t satisfies: 1. With probability at least 1 e c, for some c > 0, we have d v (t) (d v (t 1 )+ 1 p + 16c c(d v(t 1 ) + 1 p )+4c 3 )( t t 1 ) p 1 p 2. With probability at least 1 e c, for some c > 0, we have d v (t) (d v (t 1 )+ 1 p c c(d v(t 1 ) + 1 p ) c 3 )( t t 1 ) p 1 p Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 42 / 47

67 The number of edges If p < 1 2, almost surely the number of edges is t 1 2p + O(t2p log t). Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 43 / 47

68 The number of edges If p < 1 2, almost surely the number of edges is t 1 2p + O(t2p log t). If p > 1 2, almost surely the number of edges is O(t 2p log t). Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 43 / 47

69 The mixed model - With probability q, take a partial duplication step with the selection probability p. - With probability 1 q, take a full duplication step. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 44 / 47

70 Result on Mixed model Theorem 2 (Chung, Dewey, Galas, Lu, 2002) Almost surely, the mixed model generates power law graphs with the exponent β satisfying β(1 q) + pq(β 1) = 1 qp β 1. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 45 / 47

71 Summary - The biological networks follow the power law degree distributions with exponent β in the range (1, 2), which is different from non-biological networks. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 46 / 47

72 Summary - The biological networks follow the power law degree distributions with exponent β in the range (1, 2), which is different from non-biological networks. - The (partial) duplications are probably the main force shaping the biological networks. Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 46 / 47

73 Summary - The biological networks follow the power law degree distributions with exponent β in the range (1, 2), which is different from non-biological networks. - The (partial) duplications are probably the main force shaping the biological networks. - The partial duplication model is able to generate the power law graphs with exponent in the full range (1, ). Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 46 / 47

74 Overview of talks Lecture 1: Overview and outlines Lecture 2: Generative models - preferential attachment schemes Lecture 3: Duplication models for biological networks Lecture 4: The rise of the giant component Lecture 5: The small world phenomenon: average distance and diameter Lecture 6: Spectrum of random graphs with given degrees Lecture 3: Duplication models for biological networks Linyuan Lu (University of South Carolina) 47 / 47