Persisting randomness in randomly growing discrete structures: - - PowerPoint PPT Presentation

Persisting randomness in randomly growing discrete structures: graphs and search trees

• R. Gr¨

ubel Leibniz Universit¨ at Hannover

Paris, AofA 2014

Examples: Coin tossing vs. P´

• lya urn

Examples: Coin tossing vs. P´

• lya urn

Examples: Coin tossing vs. P´

• lya urn

Examples: Coin tossing vs. P´

• lya urn

Examples: Coin tossing vs. P´

• lya urn

Examples: Coin tossing vs. P´

• lya urn

Examples: Coin tossing vs. P´

• lya urn

Persistence of randomness: What is it?

Persistence of randomness: What is it?

In words: The influence of early values may or may not go away in the long run.

Persistence of randomness: What is it?

In words: The influence of early values may or may not go away in the long run. Formally, we have a sequence X = (Xn)n∈N of random variables on some probability space (Ω, A, P), and the tail σ-field T (X) :=

∞

• n=1

σ

• {Xm : m ≥ n}
• may or may not be P-trivial in the sense of

P(A) = 0 or P(A) = 1 for all A ∈ T (X).

Persistence of randomness: What is it?

In words: The influence of early values may or may not go away in the long run. Formally, we have a sequence X = (Xn)n∈N of random variables on some probability space (Ω, A, P), and the tail σ-field T (X) :=

∞

• n=1

σ

• {Xm : m ≥ n}
• may or may not be P-trivial in the sense of

P(A) = 0 or P(A) = 1 for all A ∈ T (X). Kolmogorov’s zero-one law: No persisting randomness in i.i.d. sequences.

Tail σ-fields (and topologies) via boundary theory

Tail σ-fields (and topologies) via boundary theory

• Let F be a combinatorial family, Fn: objects of size n.

Tail σ-fields (and topologies) via boundary theory

• Let F be a combinatorial family, Fn: objects of size n.
• Let X be a Markov chain with P(Xn ∈ Fn) = 1, n ∈ N.

Tail σ-fields (and topologies) via boundary theory

• Let F be a combinatorial family, Fn: objects of size n.
• Let X be a Markov chain with P(Xn ∈ Fn) = 1, n ∈ N.
• Say that (yn)n∈N with yn ∈ Fn, n ∈ N, converges iff
• P(X1 = x1, . . . , Xl = xl|Xn = yn)
• n∈N

converges for all fixed l ∈ N, x1 ∈ F1, . . . , xl ∈ Fl.

Tail σ-fields (and topologies) via boundary theory

• Let F be a combinatorial family, Fn: objects of size n.
• Let X be a Markov chain with P(Xn ∈ Fn) = 1, n ∈ N.
• Say that (yn)n∈N with yn ∈ Fn, n ∈ N, converges iff
• P(X1 = x1, . . . , Xl = xl|Xn = yn)
• n∈N

converges for all fixed l ∈ N, x1 ∈ F1, . . . , xl ∈ Fl.

• This leads to a compactification of F with boundary ∂F.

Tail σ-fields (and topologies) via boundary theory

• Let F be a combinatorial family, Fn: objects of size n.
• Let X be a Markov chain with P(Xn ∈ Fn) = 1, n ∈ N.
• Say that (yn)n∈N with yn ∈ Fn, n ∈ N, converges iff
• P(X1 = x1, . . . , Xl = xl|Xn = yn)
• n∈N

converges for all fixed l ∈ N, x1 ∈ F1, . . . , xl ∈ Fl.

• This leads to a compactification of F with boundary ∂F.

Theorem (Doob, Dynkin,. . .)

(a) Xn → X∞ almost surely with P(X∞ ∈ ∂F) = 1. (b) X∞ generates T (X).

Graph limits

Graph limits

• Let F = G be the family of finite simple graphs.
• Let V (G) and E(G) be the vertices and edges of G ∈ G.

Graph limits

• Let F = G be the family of finite simple graphs.
• Let V (G) and E(G) be the vertices and edges of G ∈ G.
• For G, H ∈ G let Γ(H, G) be the set of injective functions

φ : V (H) → V (G).

• Let T(H, G) be the set of all φ ∈ Γ(H, G) that satisfy

{i, j} ∈ E(H) ⇐ ⇒ {φ(i), φ(j)} ∈ E(G).

• Let

t(H, G) := #T(H, G)/#ΓH

G.

Graph limits

• Let F = G be the family of finite simple graphs.
• Let V (G) and E(G) be the vertices and edges of G ∈ G.
• For G, H ∈ G let Γ(H, G) be the set of injective functions

φ : V (H) → V (G).

• Let T(H, G) be the set of all φ ∈ Γ(H, G) that satisfy

{i, j} ∈ E(H) ⇐ ⇒ {φ(i), φ(j)} ∈ E(G).

• Let

t(H, G) := #T(H, G)/#ΓH

G.

In words: Sample #V (H) vertices from V (G) without replacement. Then t(H, G) is the probability that the induced subgraph is isomorphic to H.

Graph limits

• Let F = G be the family of finite simple graphs.
• Let V (G) and E(G) be the vertices and edges of G ∈ G.
• For G, H ∈ G let Γ(H, G) be the set of injective functions

φ : V (H) → V (G).

• Let T(H, G) be the set of all φ ∈ Γ(H, G) that satisfy

{i, j} ∈ E(H) ⇐ ⇒ {φ(i), φ(j)} ∈ E(G).

• Let

t(H, G) := #T(H, G)/#ΓH

G.

In words: Sample #V (H) vertices from V (G) without replacement. Then t(H, G) is the probability that the induced subgraph is isomorphic to H.

The subgraph sampling topology: (Gn)n∈N ⊂ G of converges iff (t(H, Gn))n∈N converges for all H ∈ G. (Aldous, Lovasz, . . .).

A structural view

A structural view

• Let F be a family of bounded functions f : F → R.
• Embed F into RF via evaluation, y →
• f → f (y)
• .

A structural view

• Let F be a family of bounded functions f : F → R.
• Embed F into RF via evaluation, y →
• f → f (y)
• .
• Doob-Martin: F = {K(x, ·) : x ∈ F} with

K(x, y) = P(Xn = y|Xm = x) P(Xn = y) . This is the Martin kernel.

A structural view

• Let F be a family of bounded functions f : F → R.
• Embed F into RF via evaluation, y →
• f → f (y)
• .
• Doob-Martin: F = {K(x, ·) : x ∈ F} with

K(x, y) = P(Xn = y|Xm = x) P(Xn = y) . This is the Martin kernel.

• Subgraph sampling: F = {t(H, ·) : H ∈ G}.

A structural view

• Let F be a family of bounded functions f : F → R.
• Embed F into RF via evaluation, y →
• f → f (y)
• .
• Doob-Martin: F = {K(x, ·) : x ∈ F} with

K(x, y) = P(Xn = y|Xm = x) P(Xn = y) . This is the Martin kernel.

• Subgraph sampling: F = {t(H, ·) : H ∈ G}.

General questions for a given graph model of the Markovian type, growing one node at a time: (a) Are these topologies the same? (b) What is the respective tail σ-field?

Graphs: The Erd˝

• s-R´

enyi model

Graphs: The Erd˝

• s-R´

enyi model

Model description, as a Markov chain that grows by one node at a time, with parameter p, 0 < p < 1:

• X ER

1

is the single element of G1.

Graphs: The Erd˝

• s-R´

enyi model

Model description, as a Markov chain that grows by one node at a time, with parameter p, 0 < p < 1:

• X ER

1

is the single element of G1.

• To move from X ER

n

to X ER

n+1,

– add the node n + 1, – add the edges {i, n + 1}, i ∈ {1, . . . , n}, independently and with probability p. – randomly relabel V (Xn+1),

Graphs: The Erd˝

• s-R´

enyi model

Model description, as a Markov chain that grows by one node at a time, with parameter p, 0 < p < 1:

• X ER

1

is the single element of G1.

• To move from X ER

n

to X ER

n+1,

– add the node n + 1, – add the edges {i, n + 1}, i ∈ {1, . . . , n}, independently and with probability p. – randomly relabel V (Xn+1),

Theorem

For X ER = (X ER

n )n∈N the Doob-Martin and the graph testing

topology coincide. In particular, T (X ER) is trivial.

Graphs: The Erd˝

• s-R´

enyi model

Model description, as a Markov chain that grows by one node at a time, with parameter p, 0 < p < 1:

• X ER

1

is the single element of G1.

• To move from X ER

n

to X ER

n+1,

– add the node n + 1, – add the edges {i, n + 1}, i ∈ {1, . . . , n}, independently and with probability p. – randomly relabel V (Xn+1),

Theorem

For X ER = (X ER

n )n∈N the Doob-Martin and the graph testing

topology coincide. In particular, T (X ER) is trivial.

– If we omit the relabelling then the Markov chain is of the complete memory type, and the Doob-Martin boundary is the projective limit (‘the sequence is the limit’).

Graphs: The uniform attachment model

Graphs: The uniform attachment model

• Again, X ua

1 is the single element of G1.

• Construct X ua

n+1 from X ua n by adding all edges {i, j} ⊂ [n + 1]

not yet in Xn, independently and with probability 1/(n + 1).

Graphs: The uniform attachment model

• Again, X ua

1 is the single element of G1.

• Construct X ua

n+1 from X ua n by adding all edges {i, j} ⊂ [n + 1]

not yet in Xn, independently and with probability 1/(n + 1). Let 1{i,j}, 1 ≤ i < j, be the edge indicator functions.

Graphs: The uniform attachment model

• Again, X ua

1 is the single element of G1.

• Construct X ua

n+1 from X ua n by adding all edges {i, j} ⊂ [n + 1]

not yet in Xn, independently and with probability 1/(n + 1). Let 1{i,j}, 1 ≤ i < j, be the edge indicator functions.

Theorem

In the Doob-Martin topology associated with X ua convergence of a sequence of graphs is equivalent to the pointwise convergence of all edge indicator functions. Further, T (X ua) is trivial.

Graphs: The uniform attachment model

• Again, X ua

1 is the single element of G1.

• Construct X ua

n+1 from X ua n by adding all edges {i, j} ⊂ [n + 1]

not yet in Xn, independently and with probability 1/(n + 1). Let 1{i,j}, 1 ≤ i < j, be the edge indicator functions.

Theorem

In the Doob-Martin topology associated with X ua convergence of a sequence of graphs is equivalent to the pointwise convergence of all edge indicator functions. Further, T (X ua) is trivial.

– The result fits the structural view, with F the set of edge indicators. In essence, a graph is identified with its adjacency matrix.

Graphs: The uniform attachment model

• Again, X ua

1 is the single element of G1.

• Construct X ua

n+1 from X ua n by adding all edges {i, j} ⊂ [n + 1]

not yet in Xn, independently and with probability 1/(n + 1). Let 1{i,j}, 1 ≤ i < j, be the edge indicator functions.

Theorem

In the Doob-Martin topology associated with X ua convergence of a sequence of graphs is equivalent to the pointwise convergence of all edge indicator functions. Further, T (X ua) is trivial.

– The result fits the structural view, with F the set of edge indicators. In essence, a graph is identified with its adjacency matrix. – This differs from the graph testing topology: The Doob-Martin topology depends on the transition mechanism.

Graphs: The uniform attachment model

• Again, X ua

1 is the single element of G1.

• Construct X ua

n+1 from X ua n by adding all edges {i, j} ⊂ [n + 1]

not yet in Xn, independently and with probability 1/(n + 1). Let 1{i,j}, 1 ≤ i < j, be the edge indicator functions.

Theorem

In the Doob-Martin topology associated with X ua convergence of a sequence of graphs is equivalent to the pointwise convergence of all edge indicator functions. Further, T (X ua) is trivial.

– The result fits the structural view, with F the set of edge indicators. In essence, a graph is identified with its adjacency matrix. – This differs from the graph testing topology: The Doob-Martin topology depends on the transition mechanism.

Work in progress: Other graph models, especially cases with non-trivial tail σ-fields. P´

• lya urns play a prominent role.

Binary search trees

• V = {0, 1}⋆ (0-1 words) is the set of nodes,

Binary search trees

• V = {0, 1}⋆ (0-1 words) is the set of nodes,
• a prefix-stable x ⊂ V is a binary tree,

Bn is the family of binary trees with n nodes.

Binary search trees

• V = {0, 1}⋆ (0-1 words) is the set of nodes,
• a prefix-stable x ⊂ V is a binary tree,

Bn is the family of binary trees with n nodes. ⋆ BST chain: L(Xn+1|Xn = x) = unif

• {y ∈ Bn+1 : x ⊂ y}
• .

BST: The boundary

– Let F = B be the family of binary trees. – For x ∈ B and u ∈ V let #x(u) be the number of nodes in x with prefix u (fringe subtree size).

BST: The boundary

– Let F = B be the family of binary trees. – For x ∈ B and u ∈ V let #x(u) be the number of nodes in x with prefix u (fringe subtree size).

Theorem (Evans, Gr., Wakolbinger 2012)

Let X = (Xn)n∈N be the BST chain. (a) Doob-Martin convergence of (xn)n∈N ⊂ B is equivalent to convergence of #xn(u)/#xn for all u ∈ V.

BST: The boundary

– Let F = B be the family of binary trees. – For x ∈ B and u ∈ V let #x(u) be the number of nodes in x with prefix u (fringe subtree size).

Theorem (Evans, Gr., Wakolbinger 2012)

Let X = (Xn)n∈N be the BST chain. (a) Doob-Martin convergence of (xn)n∈N ⊂ B is equivalent to convergence of #xn(u)/#xn for all u ∈ V. (b) The Martin boundary ∂B may be represented by the set of all probability measures on {0, 1}∞, endowed with weak convergence.

BST: The boundary

– Let F = B be the family of binary trees. – For x ∈ B and u ∈ V let #x(u) be the number of nodes in x with prefix u (fringe subtree size).

Theorem (Evans, Gr., Wakolbinger 2012)

Let X = (Xn)n∈N be the BST chain. (a) Doob-Martin convergence of (xn)n∈N ⊂ B is equivalent to convergence of #xn(u)/#xn for all u ∈ V. (b) The Martin boundary ∂B may be represented by the set of all probability measures on {0, 1}∞, endowed with weak convergence. (c) The support of X∞ is the whole of ∂B. In particular, T (X) is not trivial.

BST: The boundary

– Let F = B be the family of binary trees. – For x ∈ B and u ∈ V let #x(u) be the number of nodes in x with prefix u (fringe subtree size).

Theorem (Evans, Gr., Wakolbinger 2012)

Let X = (Xn)n∈N be the BST chain. (a) Doob-Martin convergence of (xn)n∈N ⊂ B is equivalent to convergence of #xn(u)/#xn for all u ∈ V. (b) The Martin boundary ∂B may be represented by the set of all probability measures on {0, 1}∞, endowed with weak convergence. (c) The support of X∞ is the whole of ∂B. In particular, T (X) is not trivial. Plan: Use this to improve results on convergence in distribution to convergence almost surely.

BST: the IDLA view

(Internal diffusion limited aggregation, Diaconis and Fulton 1991)

BST: the IDLA view

(Internal diffusion limited aggregation, Diaconis and Fulton 1991)

• An infinite graph (here: V) is gradually filled up by randomly

growing contiguous subsets.

BST: the IDLA view

(Internal diffusion limited aggregation, Diaconis and Fulton 1991)

• An infinite graph (here: V) is gradually filled up by randomly

growing contiguous subsets.

• Describe these by their boundary or frontier function, here

Bx : ∂V → N, Bx(v) := min

• k ∈ N : (v1, . . . , vk) /

∈ x

• ,

for all v = (vi)i∈N ∈ ∂V.

BST: the IDLA view

(Internal diffusion limited aggregation, Diaconis and Fulton 1991)

• An infinite graph (here: V) is gradually filled up by randomly

growing contiguous subsets.

• Describe these by their boundary or frontier function, here

Bx : ∂V → N, Bx(v) := min

• k ∈ N : (v1, . . . , vk) /

∈ x

• ,

for all v = (vi)i∈N ∈ ∂V.

• ∂V = {0, 1}∞ is the boundary of the infinite binary tree V.

BST: the IDLA view

(Internal diffusion limited aggregation, Diaconis and Fulton 1991)

• An infinite graph (here: V) is gradually filled up by randomly

growing contiguous subsets.

• Describe these by their boundary or frontier function, here

Bx : ∂V → N, Bx(v) := min

• k ∈ N : (v1, . . . , vk) /

∈ x

• ,

for all v = (vi)i∈N ∈ ∂V.

• ∂V = {0, 1}∞ is the boundary of the infinite binary tree V.
• The tree boundary ∂V can be mapped to the unit interval via

v = (vi)i∈N → 1 2 +

∞

• i=1

2vi − 1 2i+1 .

(useful for visualization)

The frontier function

1 {0, 1}∞

The frontier function

1 {0, 1}∞

The frontier function

1 {0, 1}∞

The frontier function

1 {0, 1}∞

Dynamics: An example with n = 50, 100, 200

Dynamics: An example with n = 50, 100, 200

Dynamics: An example with n = 50, 100, 200

Picture suggests that we need to shift and to smooth.

Picture suggests that we need to shift and to smooth.

• Let µ be the uniform distribution on ∂V (coin tossing),
• let ‘’ be the lexicographic order on ∂V,
• let Hn = n

k=1 1/k be the harmonic numbers.

Picture suggests that we need to shift and to smooth.

• Let µ be the uniform distribution on ∂V (coin tossing),
• let ‘’ be the lexicographic order on ∂V,
• let Hn = n

k=1 1/k be the harmonic numbers.

With (Xn)n∈N be the BST chain define Yn = (Yn(v))v∈∂V by Yn(v) :=

• wv
• BXn(w) − Hn
• µ(dw), v ∈ ∂V.

Picture suggests that we need to shift and to smooth.

• Let µ be the uniform distribution on ∂V (coin tossing),
• let ‘’ be the lexicographic order on ∂V,
• let Hn = n

k=1 1/k be the harmonic numbers.

With (Xn)n∈N be the BST chain define Yn = (Yn(v))v∈∂V by Yn(v) :=

• wv
• BXn(w) − Hn
• µ(dw), v ∈ ∂V.
• For u, v ∈ ∂V, u = v, let d(u, v) := 2−|u∧v|, with u ∧ v the

common prefix of u and v.

• (∂V, d) is a compact, totally disconnected metric space.
• With the supremum norm · ∞, the space C(∂V) of

continuous functions is a separable Banach space.

• The Yn’s are random element of this Banach space.

Theorem (2009, 2014)

(a) Yn → Y∞ almost surely as n → ∞. (b) Almost all paths of the limit process Y∞ are not Lipschitz continuous. (c) The paths of Y∞ are H¨

• lder continuous with exponent α for

all α < 1, again with probability 1.

Theorem (2009, 2014)

(a) Yn → Y∞ almost surely as n → ∞. (b) Almost all paths of the limit process Y∞ are not Lipschitz continuous. (c) The paths of Y∞ are H¨

• lder continuous with exponent α for

all α < 1, again with probability 1.

Yn(ω) for two different ω’s, with n = 500 (blue) and n = 1000 (red).

1 1 1 1