Symmetric Graph Properties Have Independent Edges

Transcription

1 Symmetric Graph Properties Have Independent Edges Paris Siminelakis PhD Student, Stanford EE Joint work with Dimitris Achlioptas, UCSC CS

2 Random Graphs p 1-p G(n,m) G(n,p)

3 Random Graphs p 1-p G(n,m) G(n,p)

4 Random Graphs p 1-p G(n,m) G(n,p)

5 Random Graphs p 1-p G(n,m) G(n,p)

6 Coupling as Simulation p 1-p G(n,m) G(n,p)

7 Coupling as Simulation p m(g) edges G(n,m) 1-p G(n,p)

8 Coupling as Simulation m(g) edges G(n,m) p G m(g) 1-p G(n,p)

9 Coupling as Simulation m(g) edges G(n,m) p G m(g) 1-p G(n,p)

10 Coupling as Simulation p p 1-p G(n,m) G(n,p)

11 Coupling as Simulation p p 1-p Chernoff Bound G(n,m) G(n,p)

12 Concentration m(g) 1-p 2 (1±ε) n p/2 G(n,m) p p G(n,p)

13 Concentration p G m(g) m(g) 1-p 2 (1±ε) n p/2 G(n,m) G(n,p)

14 Concentration + Conditioning p G m(g) m(g) 1-p 2 (1±ε) n p/2 G(n,m) G(n,p)

15 Concentration + Conditioning Coupling p m(g) 1-p G(n,m) G(n,p)

16 Coupling G(n,m) by G(n,p) with p(m) = 2m /n2 G(n,(1-ε) p) G(n,m) G(n,(1+ε) p)

17 Coupling G(n,m) by G(n,p) with p(m) = 2m /n2 G G(n,(1-ε) p) G(n,m) G(n,(1+ε) p)

18 Coupling G(n,m) by G(n,p) with p(m) = 2m /n2 G- G G(n,(1-ε) p) G(n,m) G+ G(n,(1+ε) p)

19 Sanwiching G(n,m) by G(n,p) G- G G(n,(1-ε) p) G(n,m) G+ G(n,(1+ε) p)

20 Sandwichability of G(n,m) G- G G(n,(1-ε) p) G(n,m) G+ G(n,(1+ε) p)

21 Sandwichability of G(n,m) Kn 0

22 Sandwichability of G(n,m) Kn G(n,m) 0

23 Sandwichability of G(n,m) Kn Uniform Measure S={ G: E(G) = m} 0

24 Sandwichability of G(n,m) Kn Uniform Measure S={ G: E(G) = m} approximated by G(n,p) 0

25 Sandwichability of Graph Properties Kn Uniform Measure S subset of graphs S 0

26 Sandwichability of Graph Properties Kn Uniform Measure S subset of graphs S approximated by G(n,Q) 0

27 Motivating Example [D.Achlioptas] n arbitrary points Finite spool of wire B

28 Motivating Example [D.Achlioptas] Random Graph with given wire? n arbitrary points Finite spool of wire B

29 Motivating Example [D.Achlioptas] Random Graph with given wire? n arbitrary points Finite spool of wire B P( r ) = exp( λ r)

30 Our Approach Random Graphs subject to Constraints Geometric Analogues of Erdos-Renyi Graphs No independence assumptions Maximally Agnostic aka Uniform Measure

31 Our Approach Random Graphs subject to Constraints Geometric Analogues of Erdos-Renyi Graphs No independence assumptions Maximally Agnostic aka Uniform Measure Tools to Analyze such objects as if Edges were Independent!!

32 Partition Symmetry P ={P1,..., Pk} of all edges... Edge Profile of x G in {0,1} E m(x) = (m1(x),..., mk(x)) mmi(x) == x E(G) IPi P (G) i i

33 Partition Symmetry P ={P1,..., Pk} of all edges... Edge Profile of x G in {0,1} E m(g) = (m1(g),..., mk(g)) mi(g) = E(G) Pi

34 Partition Symmetry P ={P1,..., Pk} of all edges... Edge S={ Profile of x G in {0,1} E = (m G:m(G) m(g) in1(g), C }..., mk(g)) mi(g) = E(G) Pi Definition: P-symmetric set

35 Partition Symmetry S={ G: m(g) in C }

36 Partition Symmetry S={ G: m(g) in C } Full symmetry k=1 G(n,m)

37 Partition Symmetry S={ G: m(g) in C } Full symmetry k=1 C = {m}

38 Partition Symmetry S={ G: m(g) in C } Full symmetry k=1 C=[a, b]

39 Partition Symmetry S={ G: m(g) in C } Full symmetry k=1 No symmetry k= n2 / 2 G(n,m) arbitrary S

40 Stochastic Block Model p11 p12 p13 Vertex partition V={V1,..., Vq} p21 p22 p23 k = q2 / 2 p31 p32 p33 Szemeredi Regularity Lemma Approximation in Cut norm Graph sequence G0, G1,, GT, SBM sequence SBM0, SBM1,, SBMT,

41 Stochastic Block Model p11 p12 p13 Vertex partition V={V1,..., Vq} p21 p22 p23 k = q2 / 2 p31 p32 p33 Szemeredi Regularity Lemma Approximation ApproximationininCut Cutnorm norm Graph sequence G0, G1,, GT, SBM sequence SBM0, SBM1,, SBMT,

42 Stochastic Block Model p11 p12 p13 Vertex partition V={V1,..., Vq} p21 p22 p23 k = q2 / 2 p31 p32 p33 Szemeredi Regularity Lemma Approximation in Cut norm Graph sequence SBM sequence G0, G1,, GT, Graph Limits [LS 06] SBM, SBM,, SBM, 0 1 T

43 Stochastic Block Model p11 p12 p13 Vertex partition V={V1,..., Vq} p21 p22 p23 k = q2 / 2 p31 p32 p33 Szemeredi Regularity Lemma Approximation in Cut norm Graph sequence SBM sequence G0, G1,, GT, Graph Limits [LS 06] SBM, SBM,, SBM, 0 1 T

44 Geometric Partitions Distance function d(u,v)

45 Geometric Partitions Distance function d(u,v) Ri = [ (1+δ)i-1, (1+δ)i )

46 Geometric Partitions Distance function d(u,v) Ri = [ (1+δ)i-1, (1+δ)i )

47 Geometric Partitions Distance function d(u,v) Ri = [ (1+δ)i-1, (1+δ)i ) R0 R1 Ri

48 Geometric Partitions Distance function d(u,v) Ri = [ (1+δ)i-1, (1+δ)i ) R0 R1 P ={P1,..., Pk} Pi = {(u,v): d(u,v) in Ri } Ri

49 Geometric Partitions Distance function d(u,v) Ri = [ (1+δ)i-1, (1+δ)i ) R0 R1 Ri P ={P1,..., Pk} Pi = {(u,v): d(u,v) in Ri } k = log( Dmax / Dmin )

50 Language of Partitions Block models p11 p12 p13 p21 p22 p23 p31 p32 p33 Geometry Abstract Geometry R0 R1 Ri

51 Language of Partitions Block models Geometry p11 p12 p13 p21 p22 p S={ G: m(g) in C } R p31 p32 p33 23 Abstract Geometry 0 R1 Ri

52 Uniform Measure and P-symmetry S={ G: m(g) in C } P = { P1, P2, P3 } n = 6, k = 3

53 Uniform Measure and P-symmetry S={ G: m(g) in {m*} } P = { P1, P2, P3 } n = 6, k = 3

54 Uniform Measure and P-symmetry S={ G: m(g) in {m*} } m1* = 4 m2* = 2 m3* = 2

55 Uniform Measure and P-symmetry S={ G: m(g) in {m*} } m1* = 4 m2* = 2 m3* = 2

56 Uniform Measure and P-symmetry S={ G: m(g) in {m*} } m1* = 4 m2* = 2 m3* = 2

57 Uniform Measure and P-symmetry S={ G: m(g) in {m*} } m1* = 4 G(P1,m1*) m2* = 2 G(P2,m2*) m3* = 2 G(P3,m3*)

58 Uniform Measure and P-symmetry S={ G: m(g) in {m*} } Coupling m1* = 4 G(P1,m1*) q1 m2* = 2 G(P2,m2*) q2 m3* = 2 G(P3,m3*) q3

59 Uniform Measure and P-symmetry S={ G: m(g) in {m*} } Coupling G(P1,m1*) q1 m2* = 2 G(P2,m2*) q2 m3* = 2 G(P3,m3*) q3 Union Bound m1* = 4

60 Uniform Measure and P-symmetry S={ G: m(g) in {m*} } m* = (m1*, m2*, m3*}

61 Uniform Measure and P-symmetry S={ G: m(g) in {m*} } m* = (m1*, m2*, m3*} Coupling q* = (q1*, q2*, q3*}

62 Uniform Measure and P-symmetry S={ G: m(g) in C} m in C Coupling? q* = (q1*, q2*, q3*}

63 Uniform Measure and P-symmetry S={ G: m(g) in C} Co m n in C ce of ntr mcoupling at i? (G on q* = (q *, )q! *, q *} 1 2 3

64 Concentration of Edge-profiles m3 m1 m2 C

65 Concentration of Edge-profiles m3 m1 m2 max s. t. Ent( m ) m in C C

66 Concentration of Edge-profiles m3 m1 m2 max s. t. Ent( m ) m in Conv(C) Conv(C)

67 Concentration of Edge-profiles m3 m* m1 m2 max s. t. Ent( m ) m in Conv(C) Conv(C)

68 Concentration of Edge-profiles m3 m* m1 m2 Conv(C) max s. t. Ent( m ) m in Conv(C) μ(s) = min {mi*, Pi - mi* }

69 Concentration of Edge-profiles m3 m* m1 m2 Conv(C) max s. t. Ent( m ) m in Conv(C) μ(s) = min {mi*, Pi - mi* } Theorem [Achlioptas, S, 15] For a convex P-symmetric set S and for every ε > r(s) P( mi(g) - mi* > ε mi* ) < exp(-μ(s)[ε2 λ(s)])

70 Proof of Theorem Theorem [Achlioptas, S, 15] For a convex P-symmetric set S and for every ε > r(s) P( mi(g) - mi* > ε mi* ) < exp(-μ(s)[ε2 λ(s)]) 1. Count # graphs for given m 2. Solve convex optimization problem get m* 3. Bound rate of decay away of m* 4. Round m* to get a vector in C + quantify loss 5. Integrate outside of ``good set of edge-profiles 6. Union Bound

71 Proof of Theorem Theorem [Achlioptas, S, 15] For a convex P-symmetric set S and for every ε > r(s) P( mi(g) - mi* > ε mi* ) < exp(-μ(s)[ε2 λ(s)]) 1. Count # graphs for given m 2. Solve convex optimization problem get m* 3. Bound rate of decay away of m* 4. Round m* to get a vector in C + quantify loss 5. Integrate outside of ``good set of edge-profiles 6. Union Bound

72 Proof of Theorem Theorem [Achlioptas, S, 15] For a convex P-symmetric set S and for every ε > r(s) P( mi(g) - mi* > ε mi* ) < exp(-μ(s)[ε2 λ(s)]) 1. Count # graphs for given m 2. Solve convex optimization problem get m* 3. Bound rate of decay away of m* 4. Round m* to get a vector in C + quantify loss 5. Integrate outside of ``good set of edge-profiles 6. Union Bound

73 Proof of Theorem Theorem [Achlioptas, S, 15] For a convex P-symmetric set S and for every ε > r(s) P( mi(g) - mi* > ε mi* ) < exp(-μ(s)[ε2 λ(s)]) 1. Count # graphs for given m 2. Solve convex optimization problem get m* 3. Bound rate of decay away of m* 4. Round m* to get a vector in C + quantify loss 5. Integrate outside of ``good set of edge-profiles 6. Union Bound

74 Proof of Theorem Theorem [Achlioptas, S, 15] For a convex P-symmetric set S and for every ε > r(s) P( mi(g) - mi* > ε mi* ) < exp(-μ(s)[ε2 λ(s)]) 1. Count # graphs for given m 2. Solve convex optimization problem get m* 3. Bound rate of decay away of m* 4. Round m* to get a vector in C + quantify loss 5. Integrate outside of ``good set of edge-profiles 6. Union Bound

75 Proof of Theorem Theorem [Achlioptas, S, 15] For a convex P-symmetric set S and for every ε > r(s) P( mi(g) - mi* > ε mi* ) < exp(-μ(s)[ε2 λ(s)]) 1. Count # graphs for given m 2. Solve convex optimization problem get m* 3. Bound rate of decay away of m* 4. Round m* to get a vector in C + quantify loss 5. Integrate outside of ``good set of edge-profiles 6. Union Bound

76 Proof of Theorem Theorem [Achlioptas, S, 15] For a convex P-symmetric set S and for every ε > r(s) P( mi(g) - mi* > ε mi* ) < exp(-μ(s)[ε2 λ(s)]) 1. Count # graphs for given m 2. Solve convex optimization problem get m* 3. Bound rate of decay away of m* 4. Round m* to get a vector in C + quantify loss 5. Integrate outside of ``good set of edge-profiles 6. Union Bound

77 Applications m3 m * 1. Graphs defined by Linear Programs S = { G: X m(g) < b } X in Rm x k m1 m2 P( ei ) = exp( XiTλ) 2. Kleinberg s Navigability for Set Systems [Achlioptas, S 2015].

78 Applications m3 m * 1. Graphs defined by Linear Programs S = { G: X m(g) < b } X in Rm x k m1 m2 P( ei ) = exp( XiTλ) 2. Kleinberg s Navigability for Set Systems [Achlioptas, S 2015].

79 Take Home Message Random Graphs subject to Constraints Geometric Analogues of Erdos-Renyi Graphs Independence = Symmetry + Concentration P-symmetry concept to be further exploited

80 Future Work Generalize notion of symmetry and extend beyond uniform measure Geometric coding of Edge Partitions for Inference and Sampling. Applications in Learning Theory, Approximation Algorithms, Average Case Analysis.

81 Maximum Entropy in Computer Science Max-Min Fair Allocation of Indivisible Goods Asadpour, Saberi STOC 2007 An O(log n/ loglog n)- approx Algorithm for the Asymmetric TSP Asadpour, Goemans, Madry, Oveis Gharan, Saberi. SODA 2010 Randomized Rounding Approach to the Traveling Salesman Problem Oveis Gharan, Saberi, Singh FOCS 2011 The Entropy Rounding Method in Approximation Algorithms Rothvoß SODA 2012 Entropy, Optimization and Counting Singh, Vishnoi STOC 2014

82 Boltzmann s Heuristic Fixed Energy Uniform Measure Maximum Entropy Realistic? Rigorous?