group walk random graphs - university of bristolmb13434/pr_talks/... · agelos georgakopoulos...

Group Walk Random Graphs

Agelos Georgakopoulos

Bristol, 13/2/17

Partly joint work with J. Haslegraveand with V. Kaimanovich

Geometric Random Graphs Literature

[Remco Van Der Hofstad. Random graphs and complex networks. LectureNotes, 2013.]

[Mathew Penrose. Random Geometric Graphs. Oxford University Press,2003.]

Random planar graphs ...

Percolation theory ...

Geometric Random Graphs Literature

[Remco Van Der Hofstad. Random graphs and complex networks. LectureNotes, 2013.]

[Mathew Penrose. Random Geometric Graphs. Oxford University Press,2003.]

Random planar graphs ...

Percolation theory ...

How Mafia’s grow

A network evolves in (continuous or discrete) time with thefollowing rules:

When a (Poisson) clock ticks, nodes split into two;When a node x splits into two nodes x ′, x ′′, each of itsexisting edges gets inherited by x ′ or x ′′ independentlywith probability 1/2;Moreover, a Poisson(k)-distributed number of new edgesare added between x ′ and x ′′.

As time goes to infinity, the distribution of the component(mafia) of a designated vertex converges.

Is the component in the limit distribution finite or infinite?If it is finite, is its expected size finite or infinite?

If finite, how does it depend on k?

How Mafia’s grow

Is the component in the limit distribution finite or infinite?

If it is finite, is its expected size finite or infinite?If finite, how does it depend on k?

How Mafia’s grow

Random Graphs from trees

R13(T )

Simulations by C. Moniz.

Further examples. Here are some more random graphs Rk,n outputted by my algo-

rithm for di↵erent values of k and n:

910 11

4950 5152

53 545556

Figure 3: Rk,n obtained from binary trees, for di↵erent values of k and n. From top-leftto bottom-right, the values are k = 3 and n = 6, k = 3 and n = 8, k = 6 and n = 10,k = 7 and n = 10, respectively. Drawn using Python igraph [20].

Notice that while the graph is connected for k = 3 walks on a tree of depth n = 6, it is

disconnected for k = 3 walks on a tree of depth n = 8. The example of R3,8 illustrated in

figure 3 has 15 connected components.

One may be interested in how often the random graph obtained from this construction is

connected, for di↵erent values of k and n. In the case of n = 10, I observed (informally)

that more often than not, a graph with k = 6 would be disconnected and a graph with

k = 7 walkers would be connected. Some examples of this are illustrated in figure 3.

910 11

4950 5152

53 545556

Simulations by C. Moniz.

910 11

4950 5152

53 545556

910 11

4950 5152

53 545556

A nice property

Proposition

E(] edges xy in Rn

with x in X and y in Y )

converges.

A nice property

Proposition

E(] edges xy in Rn

converges.

A nice property

Proposition

E(] edges xy in Rn

converges.

?Agelos Georgakopoulos

The Poisson integral representation formula

The classical Poisson formula

h(z) =∫ 1

0h(θ)P(z, θ)dθ

0h(θ)dνz(θ)

where P(z, θ) := 1−|z |2

|e2πiθ−z |2 ,recovers every continuous harmonicfunction h on D from its boundaryvalues h on the circle ∂D.

The Poisson integral representation formula

The classical Poisson formula

h(z) =∫ 1

0h(θ)P(z, θ)dθ =

0h(θ)dνz(θ)

where P(z, θ) := 1−|z |2

|e2πiθ−z |2 ,recovers every continuous harmonicfunction h on D from its boundaryvalues h on the circle ∂D.

The Poisson-Furstenberg boundary

The Poisson boundary of an (infinite) graph G consists of- a (Lebesgue-Rohlin) measurable space (PG,Σ), and

- a family of probability measures {νz , z ∈ VG},such that

every bounded harmonic function h can be obtained by

h(z) =∫PG

h(η)dνz(η)

this h ∈ L∞(PG) is unique up to modification on a null-set;conversely, for every h ∈ L∞(PG) the functionz 7→

h(η)dνz(η) is bounded and harmonic.

i.e. there is Poisson-like formula establishing an isometrybetween the Banach spaces H∞(G) and L∞(PG).

The Poisson boundary of an (infinite) graph G consists of- a (Lebesgue-Rohlin) measurable space (PG,Σ), and- a family of probability measures {νz , z ∈ VG},

such that

h(z) =∫PG

h(η)dνz(η)

such that

h(z) =∫PG

h(η)dνz(η)

such that

h(z) =∫PG

h(η)dνz(η)

this h ∈ L∞(PG) is unique up to modification on a null-set;

conversely, for every h ∈ L∞(PG) the functionz 7→

such that

h(z) =∫PG

h(η)dνz(η)

Selected work on the Poisson boundary

Introduced by Furstenberg to study semi-simpleLie groups [Annals of Math. ’63]Kaimanovich & Vershik give a general criterion using theentropy of random walk [Annals of Probability ’83]Kaimanovich identifies the Poisson boundary of hyperbolicgroups, and gives general criteria [Annals of Math. ’00]

A nice property

PropositionFor every two measurable

subsets X ,Y of the Poisson (orMartin) boundary ∂G,

E(] edges xy in Rn

with x ‘close to’ Xand y ‘close to’ Y )

converges.

We use the limit to define a measure on ∂G × ∂G via

C(X ,Y ) := limE(] edges ...)

A nice property

PropositionFor every two measurable

subsets X ,Y of the Poisson (orMartin) boundary ∂G,

E(] edges xy in Rn

with x ‘close to’ Xand y ‘close to’ Y )

converges.

We use the limit to define a measure on ∂G × ∂G via

C(X ,Y ) := limE(] edges ...)

Energy and Douglas’ formula

The classical Douglas formula [Douglas ’31]

E(h) =∫ 2π

∫ 2π

0(h(η) − h(ζ))2Θ(ζ, η)dηdζ

calculates the (Dirichlet) energy of aharmonic function h on D from itsboundary values h on the circle ∂D.

Energy in finite electrical networks

E(h) =∑

a,b∈B (h(a) − h(b))2 Cab,

Compare with Douglas: E(h) =∫ 2π0

∫ 2π0 (h(η) − h(ζ))2Θ(ζ, η)dηdζ

How can we generalise this to an arbitrary domain?To an infinite graph?

E(h) =∑

∫ 2π0 (h(η) − h(ζ))2Θ(ζ, η)dηdζ

E(h) =∑

∫ 2π0 (h(η) − h(ζ))2Θ(ζ, η)dηdζ

How can we generalise this to an arbitrary domain?

To an infinite graph?

E(h) =∑

∫ 2π0 (h(η) − h(ζ))2Θ(ζ, η)dηdζ

Effective conductance

We call C the effective conductance measure, because

Theorem (G & V. Kaimanovich ’12-’17+)For every locally finite network G, and every harmonicfunction h, we have

E(h) =∫∂G×∂G

(h(η) − h(ζ)

)2dC(η, ζ).

History: Douglas ’31, Naim ’57, Doob ’62, Silverstein ’74

Finite version: E(h) =∑

a,b∈B (h(a) − h(b))2 Cab

The Naim Kernel

Doob’s formula:

E(h) = q∫M2

(h(η) − h(ζ))2Θ(ζ, η)dµoηdµoζ,

for h fine-continuous quasi-everywhere [Doob ’62]. Analogousformula for graphs: [Silverstein ’74].

where the Naim Kernel Θ is defined as

Θ(ζ, η) := limzn→ζ,yn→η

G(zn, yn)G(zn,o)G(o, yn)

... in the fine topology [Naim ’57].

The Naim Kernel

Doob’s formula:

E(h) = q∫M2

for h fine-continuous quasi-everywhere [Doob ’62].

Analogousformula for graphs: [Silverstein ’74].

The Naim Kernel

Doob’s formula:

E(h) = q∫M2

The Naim Kernel

Doob’s formula:

E(h) = q∫M2

The Naim Kernel

Doob’s formula:

E(h) = q∫M2

Convergence of the Naim Kernel

Problem: Let (zi )i∈N and (yi )i∈N be independent simple randomwalks from o. Then limn,m→∞Θ(zn, ym) exists almost surely.

Convergence of the Naim Kernel

Problem: Let (zi )i∈N and (yi )i∈N be independent simple randomwalks from o. Then limn,m→∞Θ(zn, ym) exists almost surely.

Random Interlacements and C

Random Interlacements I [Sznitman]:

• A Poisson point process whose ‘points’ are 2-way infinitetrajectories

• applied to study the vacanct set on the discrete 3D-torus• governed by a certain σ-finite measure ν

Theorem (G & Kaimanovich ’17+)For every transient, locally finite graph G,

C(X ,Y ) = ν(1XY W ∗).

Random Interlacements I [Sznitman]:• A Poisson point process whose ‘points’ are 2-way infinite

trajectories

• applied to study the vacanct set on the discrete 3D-torus• governed by a certain σ-finite measure ν

C(X ,Y ) = ν(1XY W ∗).

trajectories• applied to study the vacanct set on the discrete 3D-torus

• governed by a certain σ-finite measure ν

C(X ,Y ) = ν(1XY W ∗).

trajectories• applied to study the vacanct set on the discrete 3D-torus• governed by a certain σ-finite measure ν

C(X ,Y ) = ν(1XY W ∗).

trajectories• applied to study the vacanct set on the discrete 3D-torus• governed by a certain σ-finite measure ν

C(X ,Y ) = ν(1XY W ∗).

Long range percolation

(Joint work in progress with J. Haslegrave.Thanks to O. Angel and G. Ray for important ideas)

Theorem (Newman & Schulman, Aizenman & Newman ’86)

In long range percolation on Z, with parameters e−λ/|x−y |s ,percolation occurs for large enough λ if s ≤ 2.

Rλn(Z2) converges (a la Benjamini-Schramm) to an instance Rλ∞of this (with s = 2) as n → ∞.

But Rλ∞(Tree) does not percolate for any λ!

How large is Rλ∞(T )?

The expected size of the TWRG

Let Cλo denote the component of a uniformly random vertex ofRλn(T ) (or Rλ∞(T )).

Theorem (G & Haslegrave, state of the art 2/17)

Aeaλ ≤ E(|Cλo |) ≤ Beebλ.

Conjecture:E(|Cλo |) ∼ λ

(backed by simulations)

The expected size of the TWRG

Let Cλo denote the component of a uniformly random vertex ofRλn(T ) (or Rλ∞(T )).

Theorem (G & Haslegrave, state of the art 2/17)

Aeaλ ≤ E(|Cλo |) ≤ Beebλ.

Conjecture:E(|Cλo |) ∼ λ

(backed by simulations)

Outlook

Understand TWRGsInterplay between the host group Γ and its GWRGsLet Γ act on C and see what happens

Thank you!

These slides are on-line.

Outlook

Understand TWRGs

Interplay between the host group Γ and its GWRGsLet Γ act on C and see what happens

Thank you!

Outlook

Understand TWRGsInterplay between the host group Γ and its GWRGs

Let Γ act on C and see what happens

Thank you!

Outlook

Thank you!

Outlook

Thank you!

Outlook

Thank you!

group walk random graphs - university of bristolmb13434/pr_talks/... · agelos georgakopoulos...

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