The cutoff phenomenon for random walk on random directed graphs

Transcription

1 The cutoff phenomenon for random walk on random directed graphs Justin Salez Joint work with C. Bordenave and P. Caputo

2 Outline of the talk

3 Outline of the talk 1. The cutoff phenomenon for Markov chains

4 Outline of the talk 1. The cutoff phenomenon for Markov chains 2. Random walk on directed graphs

5 Outline of the talk 1. The cutoff phenomenon for Markov chains 2. Random walk on directed graphs 3. Main results

6 Outline of the talk 1. The cutoff phenomenon for Markov chains 2. Random walk on directed graphs 3. Main results

7 Markov chain mixing

8 Markov chain mixing Any Markov chain with irreducible, aperiodic transition matrix P on a finite state space X converges to its stationary law π = πp:

9 Markov chain mixing Any Markov chain with irreducible, aperiodic transition matrix P on a finite state space X converges to its stationary law π = πp: (x, y) X 2, P t (x, y) t π(y).

10 Markov chain mixing Any Markov chain with irreducible, aperiodic transition matrix P on a finite state space X converges to its stationary law π = πp: (x, y) X 2, P t (x, y) t π(y). Distance to equilibrium at time t: D tv (t) := max x X Pt (x, ) π( ) tv

11 Markov chain mixing Any Markov chain with irreducible, aperiodic transition matrix P on a finite state space X converges to its stationary law π = πp: (x, y) X 2, P t (x, y) t π(y). Distance to equilibrium at time t: D tv (t) := max x X Pt (x, ) π( ) tv Mixing times (0 < ε < 1): t mix (ε) := min{t 0: D tv (t) ε}

12 Example: card shuffling

13 Example: card shuffling X = S n

14 Example: card shuffling X = S n ; P = top-to-random shuffle

15 Example: card shuffling X = S n ; P = top-to-random shuffle ; π = uniform

16 Example: card shuffling X = S n ; P = top-to-random shuffle ; π = uniform Theorem (Aldous-Diaconis 86). For any fixed 0 < ε < 1, t mix(ε) n log n n 1.

17 Example: card shuffling X = S n ; P = top-to-random shuffle ; π = uniform Theorem (Aldous-Diaconis 86). For any fixed 0 < ε < 1, t mix(ε) n log n n 1. at t = 0.99 n log n, the deck is not mixed at all (D tv (t) 1)

18 Example: card shuffling X = S n ; P = top-to-random shuffle ; π = uniform Theorem (Aldous-Diaconis 86). For any fixed 0 < ε < 1, t mix(ε) n log n n 1. at t = 0.99 n log n, the deck is not mixed at all (D tv (t) 1) at t = 1.01 n log n, the deck is completely mixed (D tv (t) 0)

19 Example: card shuffling X = S n ; P = top-to-random shuffle ; π = uniform Theorem (Aldous-Diaconis 86). For any fixed 0 < ε < 1, t mix(ε) n log n n 1. at t = 0.99 n log n, the deck is not mixed at all (D tv (t) 1) at t = 1.01 n log n, the deck is completely mixed (D tv (t) 0) Theorem (Diaconis-Fill-Pitman 90). For any fixed λ R, D tv (n log n + λn + o(n))

20 Example: card shuffling X = S n ; P = top-to-random shuffle ; π = uniform Theorem (Aldous-Diaconis 86). For any fixed 0 < ε < 1, t mix(ε) n log n n 1. at t = 0.99 n log n, the deck is not mixed at all (D tv (t) 1) at t = 1.01 n log n, the deck is completely mixed (D tv (t) 0) Theorem (Diaconis-Fill-Pitman 90). For any fixed λ R, D tv (n log n + λn + o(n)) n Φ(λ) with Φ: R (0, 1) decreasing from Φ( ) = 1 to Φ(+ ) = 0.

21 Ubiquity of the cutoff phenomenon

22 Ubiquity of the cutoff phenomenon Cutoff has been shown to arise in various contexts, including

23 Ubiquity of the cutoff phenomenon Cutoff has been shown to arise in various contexts, including Card shuffling (Aldous, Diaconis, Shahshahani...)

24 Ubiquity of the cutoff phenomenon Cutoff has been shown to arise in various contexts, including Card shuffling (Aldous, Diaconis, Shahshahani...) Birth-and-death chains (Diaconis, Saloff-Coste...)

25 Ubiquity of the cutoff phenomenon Cutoff has been shown to arise in various contexts, including Card shuffling (Aldous, Diaconis, Shahshahani...) Birth-and-death chains (Diaconis, Saloff-Coste...) Random walks on finite groups (Chen, Saloff-Coste...)

26 Ubiquity of the cutoff phenomenon Cutoff has been shown to arise in various contexts, including Card shuffling (Aldous, Diaconis, Shahshahani...) Birth-and-death chains (Diaconis, Saloff-Coste...) Random walks on finite groups (Chen, Saloff-Coste...) Glauber dynamics (Levin, Lubetzky, Luczak, Peres, Sly...)

27 Ubiquity of the cutoff phenomenon Cutoff has been shown to arise in various contexts, including Card shuffling (Aldous, Diaconis, Shahshahani...) Birth-and-death chains (Diaconis, Saloff-Coste...) Random walks on finite groups (Chen, Saloff-Coste...) Glauber dynamics (Levin, Lubetzky, Luczak, Peres, Sly...) Random walks on sparse graphs

28 Ubiquity of the cutoff phenomenon Cutoff has been shown to arise in various contexts, including Card shuffling (Aldous, Diaconis, Shahshahani...) Birth-and-death chains (Diaconis, Saloff-Coste...) Random walks on finite groups (Chen, Saloff-Coste...) Glauber dynamics (Levin, Lubetzky, Luczak, Peres, Sly...) Random walks on sparse graphs Random regular graphs (Lubetzky, Sly 10)

29 Ubiquity of the cutoff phenomenon Cutoff has been shown to arise in various contexts, including Card shuffling (Aldous, Diaconis, Shahshahani...) Birth-and-death chains (Diaconis, Saloff-Coste...) Random walks on finite groups (Chen, Saloff-Coste...) Glauber dynamics (Levin, Lubetzky, Luczak, Peres, Sly...) Random walks on sparse graphs Random regular graphs (Lubetzky, Sly 10) Ramanujan graphs (Lubetzky, Peres 15)

30 Ubiquity of the cutoff phenomenon Cutoff has been shown to arise in various contexts, including Card shuffling (Aldous, Diaconis, Shahshahani...) Birth-and-death chains (Diaconis, Saloff-Coste...) Random walks on finite groups (Chen, Saloff-Coste...) Glauber dynamics (Levin, Lubetzky, Luczak, Peres, Sly...) Random walks on sparse graphs Random regular graphs (Lubetzky, Sly 10) Ramanujan graphs (Lubetzky, Peres 15) Trees (Basu, Hermon, Peres 15)

31 Ubiquity of the cutoff phenomenon Cutoff has been shown to arise in various contexts, including Card shuffling (Aldous, Diaconis, Shahshahani...) Birth-and-death chains (Diaconis, Saloff-Coste...) Random walks on finite groups (Chen, Saloff-Coste...) Glauber dynamics (Levin, Lubetzky, Luczak, Peres, Sly...) Random walks on sparse graphs Random regular graphs (Lubetzky, Sly 10) Ramanujan graphs (Lubetzky, Peres 15) Trees (Basu, Hermon, Peres 15) Random graphs with given degrees (Berestycki, Lubetzky, Peres, Sly 15 and Ben-Hamou, S. 15)

32 Ubiquity of the cutoff phenomenon Cutoff has been shown to arise in various contexts, including Card shuffling (Aldous, Diaconis, Shahshahani...) Birth-and-death chains (Diaconis, Saloff-Coste...) Random walks on finite groups (Chen, Saloff-Coste...) Glauber dynamics (Levin, Lubetzky, Luczak, Peres, Sly...) Random walks on sparse graphs Random regular graphs (Lubetzky, Sly 10) Ramanujan graphs (Lubetzky, Peres 15) Trees (Basu, Hermon, Peres 15) Random graphs with given degrees (Berestycki, Lubetzky, Peres, Sly 15 and Ben-Hamou, S. 15) Still, this phenomenon is far from being completely understood.

33 Ubiquity of the cutoff phenomenon Cutoff has been shown to arise in various contexts, including Card shuffling (Aldous, Diaconis, Shahshahani...) Birth-and-death chains (Diaconis, Saloff-Coste...) Random walks on finite groups (Chen, Saloff-Coste...) Glauber dynamics (Levin, Lubetzky, Luczak, Peres, Sly...) Random walks on sparse graphs Random regular graphs (Lubetzky, Sly 10) Ramanujan graphs (Lubetzky, Peres 15) Trees (Basu, Hermon, Peres 15) Random graphs with given degrees (Berestycki, Lubetzky, Peres, Sly 15 and Ben-Hamou, S. 15) Still, this phenomenon is far from being completely understood. In particular, very few results outside the reversible world...

34 1. The cutoff phenomenon for Markov chains 2. Random walk on directed graphs 3. Main results

35 Random walk on a digraph 3 5 X = {1,..., 6}

36 Random walk on a digraph 3 5 X = {1,..., 6} How long does it take for the walk to mix?

37 Random walk on a digraph 3 5 X = {1,..., 6} How long does it take for the walk to mix? What does the stationary distribution π look like?

38 Motivation: ranking algorithms (credit: the opte project)

39 Random digraph with given degrees (Cooper-Frieze 04)

40 Random digraph with given degrees (Cooper-Frieze 04) Goal: generate a random digraph G on X = {1,..., n} with given in-degrees {d x } x X and out-degrees {d + x } x X (equal sum m)

41 Random digraph with given degrees (Cooper-Frieze 04) Goal: generate a random digraph G on X = {1,..., n} with given in-degrees {d x } x X and out-degrees {d + x } x X (equal sum m)

42 Random digraph with given degrees (Cooper-Frieze 04) Goal: generate a random digraph G on X = {1,..., n} with given in-degrees {d x } x X and out-degrees {d + x } x X (equal sum m)

43 Simulation: n = , (d +, d ) = (3, 2), (3, 4), (4, 4)

44 Simulation: n = , (d +, d ) = (3, 2), (3, 4), (4, 4)

45 Distribution of the stationary masses {nπ(x): x X }

46 Distribution of the stationary masses {nπ(x): x X }

47 A glimpse at the eigenvalues

48 1. The cutoff phenomenon for Markov chains 2. Random walk on directed graphs 3. Main results

49 Cutoff and profile

50 Cutoff and profile Sparse regime: 2 d ± x with fixed as n

51 Cutoff and profile Sparse regime: 2 d x ± with fixed as n µ := 1 m x X d x log d + x

52 Cutoff and profile Sparse regime: 2 d ± x µ := 1 m x X d x log d + x Theorem 1 (cutoff): set t n = log n µ. with fixed as n

53 Cutoff and profile Sparse regime: 2 d ± x µ := 1 m x X d x log d + x Theorem 1 (cutoff): set t n = log n µ. with fixed as n D tv (λt n + o(t n )) P n { 1 if λ < 1 0 if λ > 1

54 Cutoff and profile Sparse regime: 2 d ± x with fixed as n µ := 1 dx log d x +, σ 2 := 1 m m x X Theorem 1 (cutoff): set t n = log n µ. x X d x ( log d + x µ ) 2 D tv (λt n + o(t n )) P n { 1 if λ < 1 0 if λ > 1

55 Cutoff and profile Sparse regime: 2 d ± x with fixed as n µ := 1 dx log d x +, σ 2 := 1 m m x X Theorem 1 (cutoff): set t n = log n µ. x X d x ( log d + x µ ) 2 D tv (λt n + o(t n )) P n { 1 if λ < 1 0 if λ > 1 Theorem 2 (profile): set w n = σ 2 log n µ 3

56 Cutoff and profile Sparse regime: 2 d ± x with fixed as n µ := 1 dx log d x +, σ 2 := 1 m m x X Theorem 1 (cutoff): set t n = log n µ. x X d x ( log d + x µ ) 2 D tv (λt n + o(t n )) P n { 1 if λ < 1 0 if λ > 1 Theorem 2 (profile): set w n = σ 2 log n µ 3 ( log log n).

57 Cutoff and profile Sparse regime: 2 d ± x with fixed as n µ := 1 dx log d x +, σ 2 := 1 m m x X Theorem 1 (cutoff): set t n = log n µ. x X d x ( log d + x µ ) 2 D tv (λt n + o(t n )) P n { 1 if λ < 1 0 if λ > 1 Theorem 2 (profile): set w n = σ 2 log n µ 3 ( log log n). D tv (t n + λw n + o(w n )) P n Φ(λ)

58 Cutoff and profile Sparse regime: 2 d ± x with fixed as n µ := 1 dx log d x +, σ 2 := 1 m m x X Theorem 1 (cutoff): set t n = log n µ. x X d x ( log d + x µ ) 2 D tv (λt n + o(t n )) P n { 1 if λ < 1 0 if λ > 1 Theorem 2 (profile): set w n = σ 2 log n µ 3 ( log log n). D tv (t n + λw n + o(w n )) P Φ(λ) = 1 n 2π λ e u2 2 du

59 Sensitivity to initial condition

60 Sensitivity to initial condition how does all this depend on the choice of the initial vertex?

61 Sensitivity to initial condition how does all this depend on the choice of the initial vertex? Theorem 3 (vertex irrelevance): previous results unchanged if D tv (t) := min x X Pt (x, ) π tv

62 Sensitivity to initial condition how does all this depend on the choice of the initial vertex? Theorem 3 (vertex irrelevance): previous results unchanged if D tv (t) := min x X Pt (x, ) π tv What about a more spread-out initial law, e.g. ν(x) := d x m?

63 Sensitivity to initial condition how does all this depend on the choice of the initial vertex? Theorem 3 (vertex irrelevance): previous results unchanged if D tv (t) := min x X Pt (x, ) π tv What about a more spread-out initial law, e.g. ν(x) := d x m? Theorem 4 (constant-time relaxation): for fixed t 0, νp t π tv 2 ϱt + o P (1) with ϱ 2 := 1 dx m d + x X x 1 2

64 Sensitivity to initial condition how does all this depend on the choice of the initial vertex? Theorem 3 (vertex irrelevance): previous results unchanged if D tv (t) := min x X Pt (x, ) π tv What about a more spread-out initial law, e.g. ν(x) := d x m? Theorem 4 (constant-time relaxation): for fixed t 0, νp t π tv 2 ϱt + o P (1) with ϱ 2 := 1 dx m d + x X x 1 2 Corollary: π(x) is determined by the local geometry around x only!

65 Distribution of the stationary masses {nπ(x): x X }

66 Distribution of the stationary masses {nπ(x): x X } d W (L, L ) = sup f Lip 1 (R) R f dl f dl R

67 Distribution of the stationary masses {nπ(x): x X } d W (L, L ) = sup f Lip 1 (R) R f dl f dl R Theorem 5 (asymptotics for the equilibrium masses): ( ) 1 P d W δ n nπ(x), L 0. n x

68 Distribution of the stationary masses {nπ(x): x X } d W (L, L ) = sup f Lip 1 (R) R f dl f dl R Theorem 5 (asymptotics for the equilibrium masses): ( ) 1 P d W δ n nπ(x), L 0. n x L P 1 (R) determined by the recursive distributional equation 1 d + I d I k=1 X k law = X1, in which (X k ) k 1 are i.i.d and independent of I, P(I = x) = d+ x m.

69 Thank you!