Condensation in reinforced branching processes Anna Senkevich - - PowerPoint PPT Presentation

Condensation in reinforced branching processes Anna Senkevich

as2945@bath.ac.uk Supervised by Peter Mörters and Cécile Mailler University of Bath

June 19, 2017

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 1 / 17

Overview

1

Preferential Attachment Trees

2

Model definition

3

Growth of the system

4

Simulations

5

Open problems

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 2 / 17

Preferential Attachment Tree: Barabasi and Albert

time

Figure: Scale-free network (such that P(k) ∼ k−3).

Preferential Attachment Tree: Barabasi and Albert

time

Figure: Scale-free network (such that P(k) ∼ k−3).

Preferential Attachment Tree: Barabasi and Albert

time

Figure: Scale-free network (such that P(k) ∼ k−3).

Preferential Attachment Tree: Barabasi and Albert

time

Figure: Scale-free network (such that P(k) ∼ k−3).

Preferential Attachment Tree: Barabasi and Albert

time

Figure: Scale-free network (such that P(k) ∼ k−3).

Preferential Attachment Tree: Barabasi and Albert

time 5 5 3 1 1 1 1 1 1 1 1

Figure: Scale-free network (such that P(k) ∼ k−3).

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 3 / 17

Preferential Attachment Tree: Bianconi and Barabasi

time

Figure: Probability density function of a given distribution µ.

Preferential Attachment Tree: Bianconi and Barabasi

time

Figure: Probability density function of a given distribution µ.

Preferential Attachment Tree: Bianconi and Barabasi

time

Figure: Probability density function of a given distribution µ.

Preferential Attachment Tree: Bianconi and Barabasi

time

Figure: Probability density function of a given distribution µ.

Preferential Attachment Tree: Bianconi and Barabasi

time

Figure: Probability density function of a given distribution µ.

Preferential Attachment Tree: Bianconi and Barabasi

time 4F1 F2 5F3 F4 F5 4F6 F7 F8 F9 F10 F11

Figure: Probability density function of a given distribution µ.

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 4 / 17

Model Definition

At time t we have N(t) particles (= half-edges); M(t) families (= set of particles sharing the same fitness = nodes); Zn(t) the size of the nth family (= degree); Fn fitness of the nth family; τn the time of the foundation of the nth family. At time t, each family reproduces at rate FnZn(t).

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 5 / 17

Model Parameters

Model Parameters

0 ≤ β, γ ≤ 1 mutation and selection probability; µ the fitness distribution on (0, 1);

Mutation/Selection probability

When a birth event happens in a family n with probability γ a new particle is added to family n; with probability β a mutant having fitness FM(t)+1 is born.

Specific models

Bianconi and Barabasi model: β = 1 = γ. Kingman model : γ = 1 − β.

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 6 / 17

Yule process with rate η (= Growth of nth family)

time t = 0 Y (0) = 1 exp(η) exp(η) exp(η) exp(η) exp(η) t Y (t) = # particles at t The size of a family with fitness Fn grows like a Yule process, Y (t), with rate γFn. So that Y (t) ∼t→∞ eηtξ, where ξ is an exponentially distributed random variable.

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 7 / 17

Yule process with rate η (= Growth of nth family)

time t = 0 Y (0) = 1 exp(η) exp(η) exp(η) exp(η) exp(η) t Y (t) = # particles at t The size of a family with fitness Fn grows like a Yule process, Y (t), with rate γFn. So that Y (t) ∼t→∞ eηtξ, where ξ is an exponentially distributed random variable.

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 7 / 17

Yule process with rate η (= Growth of nth family)

time t = 0 Y (0) = 1 exp(η) exp(η) exp(η) exp(η) exp(η) t Y (t) = # particles at t The size of a family with fitness Fn grows like a Yule process, Y (t), with rate γFn. So that Y (t) ∼t→∞ eηtξ, where ξ is an exponentially distributed random variable.

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 7 / 17

Yule process with rate η (= Growth of nth family)

time t = 0 Y (0) = 1 exp(η) exp(η) exp(η) exp(η) exp(η) t Y (t) = # particles at t The size of a family with fitness Fn grows like a Yule process, Y (t), with rate γFn. So that Y (t) ∼t→∞ eηtξ, where ξ is an exponentially distributed random variable.

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 7 / 17

Yule process with rate η (= Growth of nth family)

time t = 0 Y (0) = 1 exp(η) exp(η) exp(η) exp(η) exp(η) t Y (t) = # particles at t The size of a family with fitness Fn grows like a Yule process, Y (t), with rate γFn. So that Y (t) ∼t→∞ eηtξ, where ξ is an exponentially distributed random variable.

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 7 / 17

Population Growth and Empirical Fitness Distribution

time fitness 1 F1 t Ξt Empirical Fitness Distribution at time t: Ξt =

1 N(t)

M(t)

n=1 Zn(t)δFn.

Population Growth and Empirical Fitness Distribution

time fitness 1 F1 t Ξt t Ξt Empirical Fitness Distribution at time t: Ξt =

1 N(t)

M(t)

n=1 Zn(t)δFn.

Population Growth and Empirical Fitness Distribution

time fitness 1 F1 t Ξt t Ξt τ2 Empirical Fitness Distribution at time t: Ξt =

1 N(t)

M(t)

n=1 Zn(t)δFn.

Population Growth and Empirical Fitness Distribution

time fitness 1 F1 t Ξt t Ξt τ2 F2 t Ξt Empirical Fitness Distribution at time t: Ξt =

1 N(t)

M(t)

n=1 Zn(t)δFn.

Population Growth and Empirical Fitness Distribution

time fitness 1 F1 t Ξt t Ξt τ2 F2 t Ξt τ3 F3 t Ξt Empirical Fitness Distribution at time t: Ξt =

1 N(t)

M(t)

n=1 Zn(t)δFn.

Population Growth and Empirical Fitness Distribution

time fitness 1 F1 t Ξt t Ξt τ2 F2 t Ξt τ3 F3 t Ξt t Ξt Empirical Fitness Distribution at time t: Ξt =

1 N(t)

M(t)

n=1 Zn(t)δFn.

Population Growth and Empirical Fitness Distribution

time fitness 1 F1 t Ξt t Ξt τ2 F2 t Ξt τ3 F3 t Ξt t Ξt t Ξt Empirical Fitness Distribution at time t: Ξt =

1 N(t)

M(t)

n=1 Zn(t)δFn.

Population Growth and Empirical Fitness Distribution

time fitness 1 F1 t Ξt t Ξt τ2 F2 t Ξt τ3 F3 t Ξt t Ξt t Ξt t Ξt Empirical Fitness Distribution at time t: Ξt =

1 N(t)

M(t)

n=1 Zn(t)δFn.

Population Growth and Empirical Fitness Distribution

time fitness 1 F1 t Ξt t Ξt τ2 F2 t Ξt τ3 F3 t Ξt t Ξt t Ξt t Ξt τ4 F4 t Ξt Empirical Fitness Distribution at time t: Ξt =

1 N(t)

M(t)

n=1 Zn(t)δFn.

Population Growth and Empirical Fitness Distribution

time fitness 1 F1 t Ξt t Ξt τ2 F2 t Ξt τ3 F3 t Ξt t Ξt t Ξt t Ξt τ4 F4 t Ξt t Ξt Empirical Fitness Distribution at time t: Ξt =

1 N(t)

M(t)

n=1 Zn(t)δFn.

Population Growth and Empirical Fitness Distribution

time fitness 1 F1 t Ξt t Ξt τ2 F2 t Ξt τ3 F3 t Ξt t Ξt t Ξt t Ξt τ4 F4 t Ξt t Ξt τ5 F5 t Ξt Empirical Fitness Distribution at time t: Ξt =

1 N(t)

M(t)

n=1 Zn(t)δFn.

Population Growth and Empirical Fitness Distribution

time fitness 1 F1 t Ξt t Ξt τ2 F2 t Ξt τ3 F3 t Ξt t Ξt t Ξt t Ξt τ4 F4 t Ξt t Ξt τ5 F5 t Ξt t Ξt Empirical Fitness Distribution at time t: Ξt =

1 N(t)

M(t)

n=1 Zn(t)δFn.

Population Growth and Empirical Fitness Distribution

time fitness 1 F1 t Ξt t Ξt τ2 F2 t Ξt τ3 F3 t Ξt t Ξt t Ξt t Ξt τ4 F4 t Ξt t Ξt τ5 F5 t Ξt t Ξt t Ξt Empirical Fitness Distribution at time t: Ξt =

1 N(t)

M(t)

n=1 Zn(t)δFn.

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 8 / 17

Population Growth: possible scenarios

Scenarios of growth of the system:

1 growth driven by bulk

behaviour;

2 growth driven by extremal

behaviour (condensation):

non-extensive occupancy; macroscopic occupancy.

Condition for condensation

β β + γ

1

dµ(x) 1 − x < 1. (cond)

Definition of Macroscopic Occupancy

lim inf

n→∞

max degree at time n n > 0.

1

Figure: Ξt=∞, growth driven by bulk behaviour.

1

Figure: Ξt=∞, growth driven by extremal behaviour.

Definition of Macroscopic Occupancy

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 9 / 17

Condensation

Condition for condensation

β β + γ

1

dµ(x) 1 − x < 1. (cond)

Theorem

If (cond) fails then there exists λ∗ ∈ [γ, β + γ) such that

β β+γ

1

λ∗ λ∗−γx dµ(x) = 1, otherwise, we let λ∗ = γ. In both cases:

1

0 xdΞt(x) → λ∗ β+γ almost surely when t → ∞;

Ξt → π almost surely weakly when t → ∞, where

1 if (cond) fails then dπ(x) =

β β+γ λ∗ λ∗−γx dµ(x);

2 if (cond) holds then dπ(x) =

β β+γ dµ(x) 1−x + ̟(β, γ)δ1.

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 10 / 17

An example without condesnation

Figure: Empirical fitness distribution, for µ(x, 1) = (1 − x)α+1, for α = 2, β = 0.8, γ = 0.2.

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 11 / 17

An example with condensation

Figure: Empirical fitness distribution, for µ(x, 1) = (1 − x)α+1, for α = 2.

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 12 / 17

Window for the emergence of the largest family at time t

We introduce n(t) :=

• 1

µ(1− 1

t ,1)

• ≈ tα and the random times

T(t) = inf{s > 0 : M(s) ≥ n(t)} ≈ log t ≈ first time when there exists a fitness at least 1 − 1/t.

✻ ✲

1

✲ ✛ ❄ ✻

O(1)

1 t

t s = T(t)

time to grow family determines growth rate

• ✒

fitness

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 13 / 17

An example with condensation

Figure: Time of introduction of nodes of different fitnesses, with a relative degree

• f a node indicated by the area of the bubble, for µ(x, 1) = (1 − x)α+1, α = 2.

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 14 / 17

Results for a Class of Regularly Varying Functions

Regular variation assumption on µ

µ(1 − xε, 1) µ(1 − ε, 1) → xα, α > 1, ∀x > 0 as ε ↓ 0.

Theorem [2]

Size S(t) of the largest family: e−λ∗(t−T(t))S(t) ⇒ Γ(λ∗, α). Fitness V (t) of the largest family: t(1 − V (t)) ⇒ W (explicit). Time of birth Θ(t) of the largest family: Θ(t) − T(t) ⇒ Z.

The winner does not take it all [2]

In probability when t → ∞, S(t)

N(t) = maxn∈{1...M(t)} Zn(t) N(t)

→ 0.

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 15 / 17

Open Problems

Precise growth of the system: log N(t) = λ∗t + o(t); More general branching, and Bianconi and Barabasi networks; Different classes of fitness distributions: whether there exist bounded fitness distributions where we experience condensation by macroscopic

• ccupancy.

Figure: Not this condensation.

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 16 / 17

Bibliography

[1] Athreya, Krishna B. and Ney, Peter E. Branching Processes. Springer-Verlag, 1972. [2] Dereich, Steffen and Mailler, Cécile, and Mörters, Peter. Non-extensive condensation in reinforced branching processes. arXiv:1601.08128 Preprint. [3] Dereich, Steffen. Preferential attachment with fitness: Unfolding the

• condensate. Electronic Journal of Probability, Vol. 21, 2016.

Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 17 / 17