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 Preferential Attachment Trees 1 Model definition 2 Growth of the system 3 Simulations 4 Open problems 5 Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 2 / 17

Preferential Attachment Tree: Barabasi and Albert 0 Figure: Scale-free network (such that P ( k ) ∼ k − 3 ). time

Preferential Attachment Tree: Barabasi and Albert 0 Figure: Scale-free network (such that P ( k ) ∼ k − 3 ). time

Preferential Attachment Tree: Barabasi and Albert 0 Figure: Scale-free network (such that P ( k ) ∼ k − 3 ). time

Preferential Attachment Tree: Barabasi and Albert 0 Figure: Scale-free network (such that P ( k ) ∼ k − 3 ). time

Preferential Attachment Tree: Barabasi and Albert 0 Figure: Scale-free network (such that P ( k ) ∼ k − 3 ). time

Preferential Attachment Tree: Barabasi and Albert 0 5 5 3 1 1 1 1 1 1 Figure: Scale-free network (such 1 that P ( k ) ∼ k − 3 ). 1 time Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 3 / 17

Preferential Attachment Tree: Bianconi and Barabasi 0 Figure: Probability density function of a given distribution µ . time

Preferential Attachment Tree: Bianconi and Barabasi 0 Figure: Probability density function of a given distribution µ . time

Preferential Attachment Tree: Bianconi and Barabasi 0 Figure: Probability density function of a given distribution µ . time

Preferential Attachment Tree: Bianconi and Barabasi 0 Figure: Probability density function of a given distribution µ . time

Preferential Attachment Tree: Bianconi and Barabasi 0 Figure: Probability density function of a given distribution µ . time

Preferential Attachment Tree: Bianconi and Barabasi 0 4 F 1 5 F 3 F 2 F 4 4 F 6 F 5 Figure: Probability density F 7 function of a given F 8 distribution µ . F 9 F 10 F 11 time 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); Z n ( t ) the size of the n th family (= degree); F n fitness of the n th family; τ n the time of the foundation of the n th family. At time t , each family reproduces at rate F n Z n ( 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 F M ( 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) Y (0) = 1 Y ( t ) = # particles at t exp( η ) exp( η ) exp( η ) exp( η ) exp( η ) time t = 0 t The size of a family with fitness F n grows like a Yule process , Y ( t ), with rate γ F n . So that Y ( t ) ∼ t →∞ e η t ξ , 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) Y (0) = 1 Y ( t ) = # particles at t exp( η ) exp( η ) exp( η ) exp( η ) exp( η ) time t = 0 t The size of a family with fitness F n grows like a Yule process , Y ( t ), with rate γ F n . So that Y ( t ) ∼ t →∞ e η t ξ , 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) Y (0) = 1 Y ( t ) = # particles at t exp( η ) exp( η ) exp( η ) exp( η ) exp( η ) time t = 0 t The size of a family with fitness F n grows like a Yule process , Y ( t ), with rate γ F n . So that Y ( t ) ∼ t →∞ e η t ξ , 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) Y (0) = 1 Y ( t ) = # particles at t exp( η ) exp( η ) exp( η ) exp( η ) exp( η ) time t = 0 t The size of a family with fitness F n grows like a Yule process , Y ( t ), with rate γ F n . So that Y ( t ) ∼ t →∞ e η t ξ , 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) Y (0) = 1 Y ( t ) = # particles at t exp( η ) exp( η ) exp( η ) exp( η ) exp( η ) time t = 0 t The size of a family with fitness F n grows like a Yule process , Y ( t ), with rate γ F n . So that Y ( t ) ∼ t →∞ e η t ξ , 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 fitness Ξ t 1 F 1 0 t time � M ( t ) 1 Empirical Fitness Distribution at time t : Ξ t = n =1 Z n ( t ) δ F n . N ( t )

Population Growth and Empirical Fitness Distribution fitness Ξ t Ξ t 1 F 1 0 t t time � M ( t ) 1 Empirical Fitness Distribution at time t : Ξ t = n =1 Z n ( t ) δ F n . N ( t )

Population Growth and Empirical Fitness Distribution fitness Ξ t Ξ t 1 F 1 0 τ 2 t t time � M ( t ) 1 Empirical Fitness Distribution at time t : Ξ t = n =1 Z n ( t ) δ F n . N ( t )

Population Growth and Empirical Fitness Distribution fitness Ξ t Ξ t Ξ t 1 F 2 F 1 0 τ 2 t t t time � M ( t ) 1 Empirical Fitness Distribution at time t : Ξ t = n =1 Z n ( t ) δ F n . N ( t )

Population Growth and Empirical Fitness Distribution fitness Ξ t Ξ t Ξ t Ξ t 1 F 2 F 1 F 3 0 τ 2 τ 3 t t t t time � M ( t ) 1 Empirical Fitness Distribution at time t : Ξ t = n =1 Z n ( t ) δ F n . N ( t )

Population Growth and Empirical Fitness Distribution fitness Ξ t Ξ t Ξ t Ξ t Ξ t 1 F 2 F 1 F 3 0 τ 2 τ 3 t t t t t time � M ( t ) 1 Empirical Fitness Distribution at time t : Ξ t = n =1 Z n ( t ) δ F n . N ( t )

Population Growth and Empirical Fitness Distribution fitness Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t 1 F 2 F 1 F 3 0 τ 2 τ 3 t t t t t t time � M ( t ) 1 Empirical Fitness Distribution at time t : Ξ t = n =1 Z n ( t ) δ F n . N ( t )

Population Growth and Empirical Fitness Distribution fitness Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t 1 F 2 F 1 F 3 0 τ 2 τ 3 t t t t t t t time � M ( t ) 1 Empirical Fitness Distribution at time t : Ξ t = n =1 Z n ( t ) δ F n . N ( t )

Population Growth and Empirical Fitness Distribution fitness Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t 1 F 4 F 2 F 1 F 3 0 τ 2 τ 3 τ 4 t t t t t t t t time � M ( t ) 1 Empirical Fitness Distribution at time t : Ξ t = n =1 Z n ( t ) δ F n . N ( t )

Population Growth and Empirical Fitness Distribution fitness Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t 1 F 4 F 2 F 1 F 3 0 τ 2 τ 3 τ 4 t t t t t t t t t time � M ( t ) 1 Empirical Fitness Distribution at time t : Ξ t = n =1 Z n ( t ) δ F n . N ( t )

Population Growth and Empirical Fitness Distribution fitness Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t 1 F 4 F 2 F 5 F 1 F 3 0 τ 2 τ 3 τ 4 τ 5 t t t t t t t t t t time � M ( t ) 1 Empirical Fitness Distribution at time t : Ξ t = n =1 Z n ( t ) δ F n . N ( t )

Population Growth and Empirical Fitness Distribution fitness Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t 1 F 4 F 2 F 5 F 1 F 3 0 τ 2 τ 3 τ 4 τ 5 t t t t t t t t t t t time � M ( t ) 1 Empirical Fitness Distribution at time t : Ξ t = n =1 Z n ( t ) δ F n . N ( t )

Population Growth and Empirical Fitness Distribution fitness Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t Ξ t 1 F 4 F 2 F 5 F 1 F 3 0 τ 2 τ 3 τ 4 τ 5 t t t t t t t t t t t t time � M ( t ) 1 Empirical Fitness Distribution at time t : Ξ t = n =1 Z n ( t ) δ F n . N ( t ) Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 8 / 17

Population Growth: possible scenarios Scenarios of growth of the system: Condition for condensation � 1 1 growth driven by bulk β d µ ( x ) 1 − x < 1 . (cond) behaviour ; β + γ 0 2 growth driven by extremal behaviour (condensation): Definition of Macroscopic Occupancy non-extensive occupancy; macroscopic occupancy. max degree at time n lim inf > 0 . n n →∞ 0 1 0 1 Figure: Ξ t = ∞ , growth driven by Figure: Ξ t = ∞ , growth driven by bulk behaviour . extremal behaviour . Anna Senkevich (University of Bath) Condensation in branching processes June 19, 2017 9 / 17 Definition of Macroscopic Occupancy