From trees to seeds:

Transcription

1 From trees to seeds: on the inference of the seed from large random trees Joint work with Sébastien Bubeck, Ronen Eldan, and Elchanan Mossel Miklós Z. Rácz Microsoft Research Banff Retreat September 25, 2016.

2 Statistical inference in non-equilibrium networks Apple s inventor network over a 6-year period. Source: Kenedict. Given the current state of a network, what can we say about a previous state?

3 Randomly growing trees

4 Randomly growing trees Preferential attachment: P (u n = u) = d T n (u) 2n 2 Uniform attachment: P (u n = u) = 1 n

5 Randomly growing trees Preferential attachment: P (u n = u) = d T n (u) 2n 2 Uniform attachment: P (u n = u) = 1 n In general: P (u n = u) = (d T n (u)) α Z

6 Randomly growing trees Preferential attachment: P (u n = u) = d T n (u) 2n 2 Uniform attachment: P (u n = u) = 1 n In general: P (u n = u) = (d T n (u)) α Z Many other tree growth models...

7 The influence of the seed preferential attachment seed S 10 PA (n = 500, S 10 ) seed P 10 PA (n = 500, P 10 )

8 The influence of the seed uniform attachment seed S 10 UA (n = 500, S 10 ) seed P 10 UA (n = 500, P 10 )

9 Measuring the influence of the seed A crude measure: limit as a countably infinite tree

10 Measuring the influence of the seed A crude measure: limit as a countably infinite tree seed has no influence for PA or UA

11 Measuring the influence of the seed A crude measure: limit as a countably infinite tree seed has no influence for PA or UA But for superlinear attachment (α > 1), see Oliveira, Spencer (2005)

12 Measuring the influence of the seed A crude measure: limit as a countably infinite tree seed has no influence for PA or UA But for superlinear attachment (α > 1), see Oliveira, Spencer (2005) A finer measure: weak local limit (Benjamini-Schramm)

13 Measuring the influence of the seed A crude measure: limit as a countably infinite tree seed has no influence for PA or UA But for superlinear attachment (α > 1), see Oliveira, Spencer (2005) A finer measure: weak local limit (Benjamini-Schramm) seed has no influence for PA or UA

14 Measuring the influence of the seed A crude measure: limit as a countably infinite tree seed has no influence for PA or UA But for superlinear attachment (α > 1), see Oliveira, Spencer (2005) A finer measure: weak local limit (Benjamini-Schramm) seed has no influence for PA or UA See Rudas, Tóth, Valkó (2007) (PA trees) and Berger, Borgs, Chayes, Saberi (2014) (in general) for weak local limits.

15 Measuring the influence of the seed A much finer measure: total variation distance δ PA (S, T ) := lim n TV(PA(n, S), PA(n, T ))

16 Measuring the influence of the seed A much finer measure: total variation distance δ PA (S, T ) := lim n TV(PA(n, S), PA(n, T )) Hypothesis testing question: H 0 : R PA(n, S), H 1 : R PA(n, T ) Q: test with asymptotically (in n) non-negligible power?

17 Main results Preferential attachment: Theorem (Bubeck, Mossel, R.) If the degree profiles of S and T are different, and both have at least 3 vertices, then δ PA (S, T ) > 0. Theorem (Curien, Duquesne, Kortchemski, Manolescu) If S and T are non-isomorphic and both have at least 3 vertices, then δ PA (S, T ) > 0. Uniform attachment: Theorem (Bubeck, Eldan, Mossel, R.) If S and T are non-isomorphic and both have at least 3 vertices, then δ UA (S, T ) > 0.

18 PA heuristics: maximum degree Degree evolution governed by Pólya urns ( 2n 2 dpa(n,s) (i), d PA(n,S) (i) ) Replacement matrix: ( ) ; initial condition: If i S then (2 S 2 ds (i), d S (i)); If i / S then (2i 3, 1).

19 PA heuristics: maximum degree Degree evolution governed by Pólya urns ( 2n 2 dpa(n,s) (i), d PA(n,S) (i) ) Replacement matrix: ( ) ; initial condition: If i S then (2 S 2 ds (i), d S (i)); If i / S then (2i 3, 1). Rescaled degrees converge almost surely: d PA(n,S) (i) / n n D i (S) (PA (n, S)) / n n D max (S) D max (S) = max i 1 D i (S) See Móri (2005), Janson (2006), Peköz, Röllin, Ross (2013, 2014).

20 Influence of the seed on the maximum degree Lemma (Tail behavior of the maximum degree) Let S be a finite tree and let m := {i {1,..., S } : d S (i) = (S)}. Then P (D max (S) > t) m c ( S, (S)) t 1 2 S +2 (S) exp ( t 2 /4 ) as t, where the constant c is explicit.

21 Influence of the seed on the maximum degree Lemma (Tail behavior of the maximum degree) Let S be a finite tree and let m := {i {1,..., S } : d S (i) = (S)}. Then P (D max (S) > t) m c ( S, (S)) t 1 2 S +2 (S) exp ( t 2 /4 ) as t, where the constant c is explicit. the seed influences the polynomial factor!

22 Influence of the seed on the maximum degree Lemma (Tail behavior of the maximum degree) Let S be a finite tree and let m := {i {1,..., S } : d S (i) = (S)}. Then P (D max (S) > t) m c ( S, (S)) t 1 2 S +2 (S) exp ( t 2 /4 ) as t, where the constant c is explicit. the seed influences the polynomial factor! Corollary (Distinguishing seeds) If S (S) T (T ), then δ PA (S, T ) > 0.

23 Influence of the seed on the maximum degree Lemma (Tail behavior of the maximum degree) Let S be a finite tree and let m := {i {1,..., S } : d S (i) = (S)}. Then P (D max (S) > t) m c ( S, (S)) t 1 2 S +2 (S) exp ( t 2 /4 ) as t, where the constant c is explicit. the seed influences the polynomial factor! Corollary (Distinguishing seeds) If S (S) T (T ), then δ PA (S, T ) > 0. Two trees with the same degree profile:

24 The approach of Curien et al. D τ (T ) := ϕ [d T (ϕ (u))] l(u) u τ Combinatorial interpretation: D τ (T ) = # decorated embeddings Heuristic: large degree nodes contribute the most; captures geometric structure of large degree nodes.

25 The approach of Curien et al. General framework: Construct a family of martingales using decorated embeddings: M τ (S) (n) = ( c n τ, τ ) D τ (PA (n, S)). τ τ For any S and T, there exists τ and n such that [ ] [ ] E (n) E (n). M (S) τ M (T ) τ Prove that the martingales are bounded in L 2. Conclude using the Cauchy-Schwarz inequality that δ PA (S, T ) > 0.

26 The Brownian looptree Theorem (Curien, Duquesne, Kortchemski, Manolescu) For any S there exists a random compact metric space L (S) such that the following convergence holds a.s. in the Gromov-Hausdorff topology: n 1/2 Loop (PA (n, S)) n 2 2 L (S).

27 The Brownian looptree The metric space L is constructed as a quotient of Aldous s Brownian Continuum Random Tree. Conjecture (Curien, Duquesne, Kortchemski, Manolescu) For any pair of seeds S and T, δ PA (S, T ) = TV ( L (S), L (T )).

28 Uniform attachment Uniform attachment: the subtree sizes under v l and v r are unbalanced in S but balanced in T, and this likely remains so throughout the process. Preferential attachment: the degrees of v l and v r are unbalanced in S but balanced in T, and this likely remains so throughout the process.

29 An example: distinguishing P 4 and S 4 Measuring balancedness: g (T, e) := T 1 2 T 2 2 T 4 G (T ) := e g (T, e) In order to show that δ UA (P 4, S 4 ) > 0, it suffices to show that lim inf E [G (UA (n, P))] E [G (UA (n, S))] > 0 n lim sup (Var[G(UA(n, P))] + Var[G(UA(n, S))]) < n

30 An example: distinguishing P 4 and S 4 Let {ej P } and {ej S } denote the edges. For every j 4: ( ) ( ) g UA (n, P), ej P d= g UA (n, S), ej S. We also have this for j = 1 and j = 3, so E [G (UA (n, P))] E [G (UA (n, S))] = E[g(UA(n, P), e P 2 )] E[g(UA(n, S), e S 2 )] = 2n3 + 5n 2 + 8n n

31 An example: distinguishing P 4 and S 4 Let {ej P } and {ej S } denote the edges. For every j 4: ( ) ( ) g UA (n, P), ej P d= g UA (n, S), ej S. We also have this for j = 1 and j = 3, so E [G (UA (n, P))] E [G (UA (n, S))] = E[g(UA(n, P), e P 2 )] E[g(UA(n, S), e S 2 )] For the variance we use Cauchy-Schwarz: j=1 = 2n3 + 5n 2 + 8n n n 1 Var[G(UA (n, S))] Var[g(UA (n, S), e j )], and estimates on moments of the beta-binomial distribution to give E[g(UA (n, S), e j ) 2 ] C/j 4.

32 General statistics [ fϕ(u) (T ) ] l(u) F τ (T ) := ϕ u τ Combinatorial interpretation: F τ (T ) = # decorated embeddings Heuristic: embeddings that are central contribute the most; captures global balancedness properties of the tree.

33 General framework Construct a family of martingales using decorated embeddings: M τ (S) (n) = ( c n τ, τ ) F τ (UA (n, S)). τ τ For any S and T, there exists τ and n such that [ ] [ ] E (n) E (n). M (S) τ M (T ) τ Prove that the martingales are bounded in L 2. Conclude using the Cauchy-Schwarz inequality that δ UA (S, T ) > 0.

34 Main technical issue: second moment Lemma (First moment) Let τ D + be a decorated tree with positive labels and τ 2, and let S be a seed tree. Then where w (τ) = u τ l (u). Lemma (Second moment) n w(τ) < E [ F τ (UA (n, S)) ] < n w(τ), Let τ D + be a decorated tree with positive labels and τ 2, and let S be a seed tree. Then [ (a) E F τ (UA (n, S)) 2] < n 2w(τ), (b) E [ (Fτ (UA (n + 1, S)) F τ (UA (n, S)) ) 2 ] < n 2w(τ) 2.

35 Main technical issue: second moment Top row: a decorated tree τ and two decorated embeddings, ϕ 1 and ϕ 2, of it into a larger tree T. Bottom row: an associated decorated tree σ and the decorated embedding ψ of it into T. Note: w (σ) 2w (τ).

36 Main technical issue: second moment Top row: a decorated tree τ and two decorated embeddings, ϕ 1 and ϕ 2, of it into a larger tree T. Bottom row: an associated decorated tree σ and the decorated embedding ψ of it into T. Note: w (σ) 2w (τ), but no a priori bound on σ. use the fact that diam (UA (n, S)) = O (log n) whp.

37 Main technical issue: second moment Top row: There are two types of decorated embeddings that use the new vertex. Bottom row: associated decorated trees and decorated embeddings. Roughly speaking, the two arrows associated with the new vertex give the extra factor of n 2 required in the bound of (b).

38 Takeaways: Summary and open questions Every seed has an influence, both in PA and in UA Degrees (PA) and balancedness (UA) are key statistics Open questions: Multiple edges added at each time step? Is δ α (S, T ) > 0 for α (0, 1)? Is it monotone in α? Is it convex? Other models of randomly growing graphs. Estimation. Finding the seed. (Bubeck, Devroye, Lugosi) Hiding the seed. Rumor source obfuscation.

39 Takeaways: Summary and open questions Every seed has an influence, both in PA and in UA Degrees (PA) and balancedness (UA) are key statistics Open questions: Multiple edges added at each time step? Is δ α (S, T ) > 0 for α (0, 1)? Is it monotone in α? Is it convex? Other models of randomly growing graphs. Estimation. Finding the seed. (Bubeck, Devroye, Lugosi) Hiding the seed. Rumor source obfuscation. Thank you!