Dynamics on Random Graphs and Point Processes

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F. Baccelli

Simons Center on Network Mathematics UT Austin

Joint work with M.-O. Haji-Mirsadeghi and A. Khezeli

Stochastic Networks, June 24, 2016

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Structure of the talk

Dynamics on discrete random structures – Point-shifts on stationary point processes – Vertex-shifts on unimodular networks Foliation Classiﬁcation Eternal Family Trees Quantitative results Asymptotic results (P.P.)

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Dynamics on Discrete Random Structures

Discrete random structures considered:

– point processes: countable collection of

points of the Euclidean space – random graphs/networks: countable

collection of vertices and edges Associated topological spaces: – counting measures with a point at 0 – rooted graphs/networks

✫ ✪ ✬ ✩ Dynamics on Discrete Random Structure

Dynamics: point-shifts, vertex-shifts 1. Select one node (point/vertex) – in the discrete rooted structure – as a function of the discrete rooted structure 2. Move the origin/root there Point-shifts (H. Thorisson 00) in the literature – Palm calculus ∗ J. Mecke 75: mass stationarity ∗ H. Thorisson & P. Ferrari 05 ∗ A. Holroyd & Y. Peres 05: allocation rule – Navigation on a point process F.B. & C. Bordenave 07

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Point Process

Underlying topological spaces – N: space of all counting measures φ – N0: space of all counting measures with a mass at 0 A point process Φ is a (N, N )-valued random variable on (Ω, F, IP)

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Stationary Point Process

{θt}t∈IRd: measure preserving ﬂow on (Ω, F, IP) A point process Φ is stationary if the translations of Φ are a factor of the ﬂow θt:

Φ ◦ θt(B)= StΦ(B)= Φ(B + t) ∀t, ∀B

λ intensity of the P.P. assumed ﬁnite Φ IP0 = IP0 Palm probability Example: homogeneous Poisson Point Process in IRd

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Factor Point-Shifts

Point-shift: f = f(Φ, ·): supp(Φ) → supp(Φ) f f(Φ, t)= s t −→ s θt ↓ ↑ θ−t Factor point-shift: 0 −→ s − t There exists g (point-map) g which associates to each φ ∈ N0 a point of φ and s.t. f is translation co-variant

f(Φ, t)= t + g ◦ θt, ∀t

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Examples of Point-Shifts on Poisson Point Processes

Strip Routing PS P. Ferrari, C. Landim & H. Thorisson 05

Directional PS F.B. & C. Bordenave 07

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Networks

Network Benjamini & Schramm 01, Aldous & Lyons 11 Graph G =(V(G), E(G)) locally ﬁnite and connected, with – a mark on each vertex – two marks on each edge (one for each end of the edge) Marks are in a good space G: isomorphism-classes of locally ﬁnite connected networks

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Rooted Networks, Random Rooted Networks

G∗ and G∗∗: isomorphism classes of – rooted networks (G, o) – doubly rooted networks (G, o, v) Topological spaces (in contrast to G) A random rooted network (G, o): (G∗, N )-valued random variable on (Ω, F, IP)

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Unimodular Networks

Unimodular network: + a random network (G, o) s.th. for all measurable g : G∗∗→ IR

E  g[G, o,v] = E  g[G,v, o] v∈XV (G) v∈XV (G)     Examples: – Deterministic ﬁnite networks with a uniform random root – Weak limits of ﬁnite networks – Factor graphs of a stationary P.P. under its Palm prob.

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Factor Vertex-Shifts

Factor Vertex-shift: map that associates to each network G a function fG fG : V(G) → V(G), s.t. a −→ v – Covariant: ρ ↓ ↑ ρ−1 a′ −→ v′ fG′ ◦ ρ = ρ ◦ fG fG′ ∀ isomorphisms ρ : G → G′

– The map [G, o, s] → 1s=fG(o) is measurable on G∗∗

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Example of Factor Vertex Shift

3 2 f(v) On the set of one-ended trees 1 (e.g. on the d-canopy tree) 0 Vertex-shift: f(v): the neighbor of v in the unique one ended- Father vertex shift inﬁnite simple path starting at v. on the 3-canopy tree Unimodular under some geometric level of root

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Questions on such dynamics

Properties of dynamical system? Quantitative analysis What happens asymptotically?

✫ ✪ ✬ ✩ Discrete Foliation

Analogue of the stable manifold of f f-Graph of (point/vertex)-shift f – Nodes: the atoms of Φ; directed edges: (a, f(a)) – Nodes: the vertices of G; directed edges: (v, f(v)) Two partitions of the set of nodes 1. Connected components of the f-Graph 2. Foliation: equivalence relation n n x ∼f y ⇔∃n ∈ N; f (x)= f (y)

f-foliation: equivalence classes of the set of nodes w.r.t. ∼f

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Illustration: Father Vertex-Shift on One Ended Tree

One ended Tree Father vertex-shift Foil of root

✫ ✪ ✬ ✩ Illustration: Foliation of the Support of a P.P.P.

Φ Poisson P.P. in IR2 Palm Law Strip PS a.s. one component Foil of origin

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Classiﬁcation

Theorem 1 Let f be – a factor point-shift on a stationary point process Φ – a vertex-shift on a unimodular random network [G, o] Almost surely each connected component C of f-Graph has at most two ends belongs to one of the three classes: 1. Class F/F: C is finite, each of its f-foils is finite 2. Class I/F: C is infinite and each of its f-foils is finite 3. Class I/I: C is infinite and all its f-foils are infinite

✫ ✪ ✬ ✩ Evaporation

Descendants of order n : of x ∈ Φ or x ∈ V(G) −n Dn(x) := f (x), dn(x) := card Dn(x)

Descendants of x : ∞ D(x) := Dn(x), d(x) := card D(x) n[=1

Deﬁnition Let C be a connected component of f-Graph. The point/vertex-shift f evaporates C if f ∞(C)= ∅, i.e., for all x ∈ C, there exists some n > 0 s.t. Dn(x)= ∅

✫ ✪ ✬ ✩ F/F Class

Class F/F: – C is ﬁnite (non inﬁnite end);

– each of its f-foils is ﬁnite

# foils: 1 ≤ n = n(C) < ∞:

C has a unique cycle of length n; Vertices of this cycle: f ∞(C);

Each foil contains 1 vertex of cycle; f does not evaporate C Example nearest neighbor point-shift on the P.P.P.

✫ ✪ ✬ ✩ I/F Class

Class I/F: – C is infinite – Each of its f-foils is finite C is a tree; Each foil has a junior foil; f ∞(C): unique 2 end path Infinite descendancy f does not evaporate C. Finite foil Example: Royal Line of Succession on the strip point-shift graph on a 2 dim. P.P.P.

✫ ✪ ✬ ✩ I/I Class Class I/I: – C is infinite – All its f-foils are infinite C is a one-ended tree; f ∞(C)= ∅, namely f evaporates C Examples: Finite descendancy – Strip point-shift on a two dim. P.P.P. Infinite foil – Father Vertex-shift on Unimodular Eternal Galton Watson tree

The point/vertex-shift f evaporates C if and only if C is of Class I/I

✫ ✪ ✬ ✩ Eternal Family Trees

Eternal Family Tree (EFT) Directed rooted tree where each node, including the root, has a father Examples GW' – Canopy tree – Eternal Galton Watson tree GW ∗ GW: Galton Watson ∗ GW’: Galton Watson with ﬁrst generation sized biased -1 Eternal GW Tree

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Unimodular EFTs

Theorem 2 A random EFT [T, o] is unimodular if and only if

1. its distribution is invariant under σ1 with 1 σ1Q[A]= 1A[T, v]dQ([T, o]) d1(T, o)dQ([T, o]) Z T∗ v∈DX1(T,o) R T∗ Moving the root to a typical child

2. it is critical: E [d1(o)] = 1 Generalization of critical branching processes

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Unimodular EFTs and Vertex-Shift Partitions

Theorem 3 For all point-shifts f on a stationary P.P. Φ, when it is inﬁnite, the connected component of f-Graph containing the origin is a unimodular EFT Theorem 3’ For all vertex-shifts f on a unimodular network [G, o], when it is inﬁnite, the connected component of f-Graph containing the root is a unimodular EFT

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Properties of Unimodular EFTs

Same classification as infinite components of vertex shifts I/I Construction theorem: Each unimodular EFT of type I/I type is the weak limit of some random finite tree when moving the root to a typical far descendant I/F Construction theorem: Each unimodular EFT of type I/F is the joining of a stationary sequence of finite mean cardinality trees

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Order on Foils, I/I Case

Order on brothers Order on ﬁrst cousins Order on a foil Order preserved by f Linear order as Z

f⊥ next point in this order

✫ ✪ ✬ ✩ Foil Relative Intensities

P0: the Palm prob. of Φ

f a point-shift with associated f⊥ Consider an I/I component Theorem 4 (Point processes) for P0 almost-all φ n d (fφ(0), fφ ◦ f⊥(0)) λ+(φ) := lim n→∞ n d (0, f⊥(0)) d (f (0), f ◦ f n(0)) = lim φ φ ⊥ n→∞ n All foils are of the

exists, is positive and in L1(P0) same ”dimension” Extension to the network case

✫ ✪ ✬ ✩ f⊥

A bijective 60

point-shift 50

preserving 40 Palm 30 from a 20 non-bijective 10 point-shift 0 not preserving -10 Palm 0 20 40 60 80 100

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Mecke’s Invariance for Point Processes

Theorem Mecke (1975) The Palm probability of a P.P. is invariant under the action of all bijective Point-Shifts Corollary

– θf⊥ is measure preserving even when θf is non-measure preserving – Birkhoff’s pointwise ergodic theorem applies on a foil! The foil of the origin is not always the Palm version of a point process

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Mecke’s Invariance for Unimodular Networks

Theorem 5 The distribution of a unimodular network is invariant under the action of all bijective vertex-shifts

One of several theorem that can be imported from the theory of P.Ps to that of Unimodular networks (e.g. Neveu’s exchange formula)

✫ ✪ ✬ ✩ Examples of General Quantitative Results

f Point-shift f f Foil L (x): the limit of the increasing sets Ln(x), where f n n Ln(x) := {y ∈ X; f (y)= f (x)}

cardinality of Lf (x): l(x) f cardinality of Ln(x): ln(x) Proposition for all n ≥ 0,

n 1 IP0 [0 ∈ f (Φ)] = E 0 ln(0) In addition ∞ 1 IP0 [0 ∈ f (Φ)] = E 0 l∞(0)

✫ ✪ ✬ ✩ Asymptotic Behavior of Point-Shifts

First g-Palm probability of Φ: g,1 IP0 =(θg)∗IP0

Interpretation: distribution of Φ given that the origin is in f 1(Φ), considering multipl. n-th g-Palm probability of Φ: distribution of Φ given that 0 is in f n(Φ) g,n g,n−1 IP0 =(θg)∗IP0 , n ≥ 1

Interpretation: distribution of Φ seen from a typical point of f n(Φ) with mult.

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Point-Map Probabilities

Let – g be a point-map

– Φ be a stationary point process with Palm distribution P0 Deﬁnition

– Each accumulation point of {(θgn)∗P0} is called a g-probability of P0 – If the limit of the sequence P ∞ Pg,n ∞ {(θgn)∗ 0}n=1 = { 0 }n=1 P Pg exists, it is called the g-probability of 0 and denoted by 0

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