Graphs, Vectors, and Matrices Daniel A. Spielman Yale University. AMS Josiah Willard Gibbs Lecture January 6, 2016

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1 Graphs, Vectors, and Matrices Daniel A. Spielman Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016

2 From Applied to Pure Mathematics Algebraic and Spectral Graph Theory Sparsification: approximating graphs by graphs with fewer edges The Kadison-Singer problem

3 A Social Network Graph

4 A Social Network Graph

5 A Social Network Graph vertex or node edge = pair of nodes

6 A Social Network Graph vertex or node edge = pair of nodes

7 A Big Social Network Graph

8 A Graph G =(V,E) V = vertices, E = edges, pairs of vertices

9 The Graph of a Mesh

10 Examples of Graphs

11 Examples of Graphs

12 How to understand large-scale structure Draw the graph Identify communities and hierarchical structure Use physical metaphors Edges as resistors or rubber bands Examine processes Diffusion of gas / Random Walks

13 The Laplacian quadratic form of G =(V,E) x : V! R (a,b)2e (x(a) x(b)) 2

14 The Laplacian quadratic form of G =(V,E) x : V! R (a,b)2e (x(a) x(b))

15 The Laplacian quadratic form of G =(V,E) x : V! R (a,b)2e (x(a) x(b)) (0.5) 2 1 (0.5) (0) 2 (0.5) 2 (0.5) 2 (0.5) 2 0 (0.5)

16 The Laplacian matrix of G =(V,E) x : V! R (a,b)2e (x(a) x(b)) 2 = x T Lx

17 Graphs as Resistor Networks View edges as resistors connecting vertices Apply voltages at some vertices. Measure induced voltages and current flow. 1V 0V

18 Graphs as Resistor Networks Induced voltages minimize, subject to constraints. (a,b)2e 1V (x(a) x(b)) 2 0V

19 Graphs as Resistor Networks Induced voltages minimize, subject to constraints. (a,b)2e 1V 0.5V 1V (x(a) x(b)) 2 0V 0.5V 0V 0.375V 0.625V

20 Graphs as Resistor Networks Induced voltages minimize, subject to constraints. (a,b)2e 1V 0V 0V 0.5V (0.5) 2 (0.5) 2 (0.375) 2 0.5V (0.125) V (0.5) 2 (0.5) 2 (0.125) 2 (0.25) 2 (x(a) x(b)) 2 1V (0.375) V

21 Graphs as Resistor Networks Induced voltages minimize, subject to constraints. (a,b)2e 0V 0.5V 0.5V (x(a) x(b)) 2 Effective conductance = current flow with one volt 0V (0.5) 2 (0.5) 2 (0.375) 2 (0.125) V (0.5) 2 (0.5) 2 (0.125) 2 (0.25) 2 1V (0.375) 2 1V 0.625V

22 Weighted Graphs Edge (a, b) assigned a non-negative real weight w a,b 2 R measuring strength of connection 1/resistance x T Lx = (a,b)2e w a,b (x(a) x(b)) 2

23 Spectral Graph Drawing (Hall 70) Want to map V! R with most edges short Edges are drawn as curves for visibility.

24 Spectral Graph Drawing (Hall 70) Want to map Minimize V! R x T Lx = with most edges short (x(a) x(b)) 2 (a,b)2e to fix scale, require x(a) 2 =1 a

25 Spectral Graph Drawing (Hall 70) Want to map Minimize V! R x T Lx = with most edges short (x(a) x(b)) 2 (a,b)2e to fix scale, require a x(a) 2 =1 kxk =1

26 Courant-Fischer Theorem 1 = min x6=0 kxk=1 x T Lx v 1 = arg min x6=0 kxk=1 x T Lx Where and v 1 1 is the smallest eigenvalue of L is the corresponding eigenvector.

27 Courant-Fischer Theorem 1 = min x6=0 kxk=1 x T Lx v 1 = arg min x6=0 kxk=1 x T Lx Where and For v 1 1 is the smallest eigenvalue of L is the corresponding eigenvector. x T Lx = (a,b)2e (x(a) x(b)) 2 1 =0 and v 1 is a constant vector

28 Spectral Graph Drawing (Hall 70) Want to map V! R with most edges short Minimize x T Lx = (x(a) x(b)) 2 (a,b)2e Such that kxk =1 and x(a) =0 a

29 Spectral Graph Drawing (Hall 70) Want to map V! R with most edges short Minimize x T Lx = (x(a) x(b)) 2 Such that kxk =1 (a,b)2e and Courant-Fischer Theorem: solution is v 2, the eigenvector of 2, the second-smallest eigenvalue a x(a) =0

30 Spectral Graph Drawing (Hall 70) (x(a) x(b)) 2 = area under blue curves (a,b)2e

31 Spectral Graph Drawing (Hall 70) (x(a) x(b)) 2 = area under blue curves (a,b)2e kxk =1 0= a x(a)

32 Space the points evenly

33 And, move them to the circle

34 Finish by putting me back in the center

35 Spectral Graph Drawing (Hall 70) Want to map V! R 2 with most edges short 2 x(a) x(b) Minimize y(a) y(b) Such that (a,b)2e kxk =1 kyk =1 and and a a x(a) =0 y(a) =0

36 Spectral Graph Drawing (Hall 70) Want to map V! R 2 with most edges short 2 x(a) x(b) Minimize y(a) y(b) (a,b)2e Such that kxk =1 and 1 T x =0 and kyk =1 1 T y =0

37 Spectral Graph Drawing (Hall 70) Want to map V! R 2 with most edges short 2 x(a) x(b) Minimize y(a) y(b) (a,b)2e Such that kxk =1 and 1 T x =0 kyk =1 and 1 T y =0 and x T y =0, to prevent x = y

38 Spectral Graph Drawing (Hall 70) Minimize (a,b)2e x(a) y(a) x(b) y(b) 2 Such that kxk =1 kyk =1 1 T x =0 1 T y =0 and x T y =0 Courant-Fischer Theorem: solution is x = v 2,y = v 3, up to rotation

39 Spectral Graph Drawing (Hall 70) Arbitrary Drawing 9 7 Spectral Drawing 8

40 Spectral Graph Drawing (Hall 70) Original Drawing Spectral Drawing

41 Spectral Graph Drawing (Hall 70) Original Drawing Spectral Drawing

42 Dodecahedron Best embedded by first three eigenvectors

43 Spectral drawing of Erdos graph: edge between co-authors of papers

44 When there is a nice drawing: Most edges are short Vertices are spread out and don t clump too much 2 is close to 0 When is big, say there is no nice picture of the graph 2 > 10/ V 1/2

45 Expanders: when 2 is big Formally: infinite families of graphs of constant degree d and large Examples: random d-regular graphs Ramanujan graphs Have no communities or clusters. Incredibly useful in Computer Science: Act like random graphs (pseudo-random) Used in many important theorems and algorithms 2

46 Good Expander Graphs d -regular graphs with 2,..., n d Courant-Fischer: for all 1 T x =0 kxk =1 x T L G x d

47 Good Expander Graphs d -regular graphs with Courant-Fischer: for all 2,..., n d 1 T x =0 kxk =1 x T L G x d For K n, the complete graph on n vertices 2,..., n = n, so for 1 T x =0 kxk =1 x T L Kn x = n L Kn n d L G

48 Good Expander Graphs L Kn n d L G

49 Sparse Approximations of Graphs (S-Teng 04) A graph H is a sparse approximation of if H has few edges and L H L G G few: the number of edges in H is or, where O(n) O(n log n) n = V 1 1+ apple xt L H x x T L G x L H L G if apple 1+ for all x

50 Sparse Approximations of Graphs (S-Teng 04) A graph H is a sparse approximation of if H has few edges and L H L G G few: the number of edges in H is or, where O(n) O(n log n) n = V 1 1+ apple xt L H x x T L G x L H L G if apple 1+ for all x 1 1+ L G 4 L H 4 (1 + )L G Where M 4 f M if x T Mx apple x T f Mx for all x

51 Sparse Approximations of Graphs (S-Teng 04) A graph H is a sparse approximation of if H has few edges and L H L G G few: the number of edges in H is or, where O(n) O(n log n) n = V 1 1+ apple xt L H x x T L G x L H L G if apple 1+ for all x 1 1+ L G 4 L H 4 (1 + )L G Where M 4 f M if x T Mx apple x T f Mx for all x

52 Sparse Approximations of Graphs The number of edges in or H is, where O(n) O(n log n) n = V 1 1+ L G 4 L H 4 (1 + )L G (S-Teng 04) Where M 4 f M if x T Mx apple x T f Mx for all x

53 Why we sparsify graphs To save memory when storing graphs. To speed up algorithms: flow problems in graphs (Benczur-Karger 96) linear equations in Laplacians (S-Teng 04)

54 Graph Sparsification Theorems For every G =(V,E,w), there is a H =(V,F,z) s.t. L G L H and F apple (2 + ) 2 n/ 2 (Batson-S-Srivastava 09)

55 Graph Sparsification Theorems For every G =(V,E,w), there is a H =(V,F,z) s.t. L G L H and F apple (2 + ) 2 n/ 2 (Batson-S-Srivastava 09) By careful random sampling, can quickly get F apple O(n log n/ 2 ) (S-Srivastava 08)

56 Laplacian Matrices x T L G x = L G = L 1,2 = = (x(a) x(b)) 2 (a,b)2e (a,b)2e L a,b

57 Laplacian Matrices x T L G x = (x(a) x(b)) 2 L G = (a,b)2e (a,b)2e L a,b = u a,b u T a,b u a,b = a b (a,b)2e

58 Laplacian Matrices x T L G x = (x(a) x(b)) 2 L G = = (a,b)2e (a,b)2e = (a,b)2e u a,b U L a,b u a,b u T a,b U T u a,b = a b

59 Matrix Sparsification M = U U T fm = 1 (1 + ) M 4 f M 4 (1 + )M

60 Matrix Sparsification M = U U T fm = subset of vectors, scaled up 1 (1 + ) M 4 f M 4 (1 + )M

61 Matrix Sparsification M = U U T fm = subset of vectors, scaled up 1 (1 + ) M 4 f M 4 (1 + )M

62 Matrix Sparsification M = U U T = i u i u T i fm = = i s i u i u T i 1 (1 + ) M 4 f M 4 (1 + )M most s i =0

63 Simplification of Matrix Sparsification 1 (1 + ) M 4 M f 4 (1 + )M is equivalent to 1 (1 + ) I 4 M 1/2 MM f 1/2 4 (1 + )I

64 Simplification of Matrix Sparsification 1 (1 + ) I 4 M 1/2 MM f 1/2 4 (1 + )I Set v i = M 1/2 u i We need s i v i vi T I i v i vi T = I i

65 Simplification of Matrix Sparsification 1 (1 + ) I 4 M 1/2 MM f 1/2 4 (1 + )I Set v i = M 1/2 u i v i vi T = I i Decomposition of the identity Parseval frame Isotropic Position (vi T t) 2 = ktk 2 i

66 Matrix Sparsification by Sampling For v 1,...,v m 2 R n v i Choose If choose s i = (Rudelson 99, Ahlswede-Winter 02, Tropp 11) with with probability p i kv i k 2 v i, set s i =1/p i ( 1/p i with probability p i i v i v T i 0 with probability 1 p i = I " # E s i v i v T i = i v i v T i i

67 Matrix Sparsification by Sampling For v 1,...,v m 2 R n v i Choose If choose s i = (Rudelson 99, Ahlswede-Winter 02, Tropp 11) with ( 1/p i with probability p i i v i v T i 0 with probability 1 p i = I with probability p i kv i k 2 v i, set s i =1/p i (effective conductance) " # E s i v i v T i = i v i v T i i

68 Matrix Sparsification by Sampling For v 1,...,v m 2 R n v i Choose If choose s i = (Rudelson 99, Ahlswede-Winter 02, Tropp 11) with ( 1/p i with probability p i i v i v T i 0 with probability 1 p i = I with probability p i = C(log n) kv i k 2 / 2 v i, set s i =1/p i " # E s i v i v T i = i v i v T i i

69 Matrix Sparsification by Sampling For v 1,...,v m 2 R n v i Choose If choose (Rudelson 99, Ahlswede-Winter 02, Tropp 11) with i v i vi T = I with probability p i = C(log n) kv i k 2 / 2 v i, set s i =1/p i With high probability, choose and s i v i vi T i O(n log n/ 2 ) I vectors

70 Optimal (?) Matrix Sparsification For v 1,...,v m 2 R n Can choose (2 + ) 2 n/ 2 and nonzero values for the with v i vi T = I s i (Batson-S-Srivastava 09) vectors so that i i s i v i v T i I

71 Optimal (?) Matrix Sparsification For v 1,...,v m 2 R n Can choose (2 + ) 2 n/ 2 and nonzero values for the with v i vi T = I s i (Batson-S-Srivastava 09) vectors so that i i s i v i v T i I s i

72 Optimal (?) Matrix Sparsification For v 1,...,v m 2 R n Can choose (2 + ) 2 n/ 2 and nonzero values for the with v i vi T = I s i (Batson-S-Srivastava 09) vectors so that i i s i v i v T i I s i 1/ kv i k 2

73 The Kadison-Singer Problem 59 Equivalent to: Anderson s Paving Conjectures ( 79, 81) Bourgain-Tzafriri Conjecture ( 91) Feichtinger Conjecture ( 05) Many others Implied by: Weaver s KS 2 conjecture ( 04)

74 Weaver s Conjecture: Isotropic vectors v 4 v 1 v 2 t i v i v T i = I v 3 v 2 v 1 v 4 v 3 for every unit vector (vi T t) 2 =1 t i

75 Partition into approximately ½-Isotropic Sets S 1 S 2

76 Partition into approximately ½-Isotropic Sets S 1 S 2 1/4 apple P i2s j (v T i t)2 apple 3/4

77 Partition into approximately ½-Isotropic Sets S 1 S 2 1/4 apple P i2s j (v T i t)2 apple 3/4 1/4 apple eigs( P i2s j v i v T i ) apple 3/4

78 Partition into approximately ½-Isotropic Sets S 1 S 2 1/4 apple P i2s j (v T i t)2 apple 3/4 1/4 apple eigs( P i2s j v i v T i ) apple 3/4 () eigs( P i2s j v i v T i ) apple 3/4 because i2s 1 v i v T i = I i2s 2 v i v T i

79 Big vectors make this difficult S 1 S 2

80 Big vectors make this difficult S 1 S 2

81 Weaver s Conjecture KS 2 There exist positive constants and so that if all kv i k 2 apple and P vi v T i = I then exists a partition into S 1 and S 2 with eigs( P i2s j v i v T i ) apple 1

82 Theorem (Marcus-S-Srivastava 15) For all > 0 if all kv i k 2 apple and P vi v T i = I then exists a partition into S 1 and S 2 with eigs( P i2s j v i v T i ) apple

83 We want eigs 0 v i vi T 0 Bi2S 0 v i vi T i2s 2 1 C A apple

84 We want roots 0 poly 0 v i vi T 0 Bi2S 0 v i vi T i2s 2 11 CC AA apple

85 We want roots 0 poly 0 v i vi T 0 Bi2S 0 v i vi T i2s 2 11 CC AA apple Consider expected polynomial of a random partition.

86 Proof Outline 1. Prove expected characteristic polynomial has real roots 2. Prove its largest root is at most 1/ Prove is an interlacing family, so exists a partition whose polynomial has largest root at most 1/2+3

87 Interlacing Polynomial interlaces p(x) = Q d i=1 (x i) q(x) = Q d 1 i=1 (x i) if 1 apple 1 apple 2 apple d 1 apple d 1 apple d Example: q(x) = d dx p(x)

88 Common Interlacing and p 2 (x) have a common interlacing if can partition the line into intervals so that each contains one root from each polynomial p 1 (x) 1 2 ) ( ) ( ) ( ) ( 3 d 1

89 Common Interlacing If p 1 and p 2 have a common interlacing, 1 max-root (p i ) apple max-root (E i [ p i ]) for some i. 2 ) ( ) ( ) ( ) ( 3 d 1 Largest root of average

90 Common Interlacing If p 1 and p 2 have a common interlacing, 1 max-root (p i ) apple max-root (E i [ p i ]) for some i. 2 ) ( ) ( ) ( ) ( 3 d 1 Largest root of average

91 Without a common interlacing (x + 1)(x + 2) (x 1)(x 2)

92 Without a common interlacing x 2 +4 (x + 1)(x + 2) (x 1)(x 2)

93 Without a common interlacing (x + 4)(x 1)(x 8) (x + 3)(x 9)(x 10.3)

94 Without a common interlacing (x + 4)(x 1)(x 8) (x +3.2)(x 6.8)(x 7) (x + 3)(x 9)(x 10.3)

95 Common Interlacing If p 1 and p 2 have a common interlacing, 1 max-root (p i ) apple max-root (E i [ p i ]) for some i. 2 ) ( ) ( ) ( ) ( 3 d 1 Largest root of average

96 Common Interlacing have a common interlacing iff p 1 (x)+(1 )p 2 (x) is real rooted for all 0 apple apple 1 p 1 (x) and p 2 (x) d 1 ) ) ( ) ( ) ( (

97 Interlacing Family of Polynomials {p } 2{1,2} n is an interlacing family if its members can be placed on the leaves of a tree so that when every node is labeled with the average of leaves below, siblings have common interlacings E i,j [ p i,j ] E i [ p 1,i ] E i [ p 2,i ] p 1,1 p 1,2 p 2,1 p 2,2

98 Interlacing Family of Polynomials {p } 2{1,2} n is an interlacing family if its members can be placed on the leaves of a tree so that when every node is labeled with the average of leaves below, siblings have common interlacings E i,j [ p i,j ] E i [ p 1,i ] E i [ p 2,i ] p 1,1 p 1,2 p 2,1 p 2,2 have a common interlacing

99 Interlacing Family of Polynomials {p } 2{1,2} n is an interlacing family if its members can be placed on the leaves of a tree so that when every node is labeled with the average of leaves below, siblings have common interlacings E i,j [ p i,j ] E i [ p 1,i ] E i [ p 2,i ] p 1,1 p 1,2 p 2,1 p 2,2 have a common interlacing

100 Interlacing Family of Polynomials Theorem: There is a so that max-root(p ) apple max-root(e p ) E i,j [ p i,j ] E i [ p 1,i ] E i [ p 2,i ] p 1,1 p 1,2 p 2,1 p 2,2

101 Interlacing Family of Polynomials Theorem: There is a so that max-root(p ) apple max-root(e p ) E i,j [ p i,j ] E i [ p 1,i ] E i [ p 2,i ] p 1,1 p 1,2 p 2,1 p 2,2 have a common interlacing

102 Interlacing Family of Polynomials Theorem: There is a so that max-root(p ) apple max-root(e p ) E i,j [ p i,j ] E i [ p 1,i ] E i [ p 2,i ] p 1,1 p 1,2 p 2,1 p 2,2 have a common interlacing

103 Our family is interlacing E S1,S poly 0 v i vi T 0 Bi2S 0 v i vi T i2s 2 1 C A Form other polynomials in the tree by fixing the choices of where some vectors go

104 Summary 1. Prove expected characteristic polynomial has real roots 2. Prove its largest root is at most 1/ Prove is an interlacing family, so exists a partition whose polynomial has largest root at most 1/2+3

105 To learn more about Laplacians, see My class notes from Spectral Graph Theory and Graphs and Networks My web page on Laplacian linear equations, sparsification, etc. To learn more about Kadison-Singer Papers in Annals of Mathematics and survey from ICM. Available on ariv and my web page

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