Interacting diffusions on random sparse graphs EBP July 2019 - PowerPoint PPT Presentation

Interacting diffusions on random sparse graphs EBP July 2019 Guilherme Reis - Federal University of Bahia ghreis@impa.br Joint work with Roberto Imbuzeiro Oliveira - IMPA - rimfo@impa.br and Lucas Stolerman - UCSD - lstolerman@eng.ucsd.edu

Outline I’m going to present a mean-field model of particle systems. 2 / 73

Outline I’m going to present a mean-field model of particle systems. Mean-field means every particle has a direct interaction with all other particles. 3 / 73

Outline I’m going to present a mean-field model of particle systems. Mean-field means every particle has a direct interaction with all other particles. My purpose in this talk is to show how we go beyond the mean-field case. 4 / 73

Outline I’m going to present a mean-field model of particle systems. Mean-field means every particle has a direct interaction with all other particles. My purpose in this talk is to show how we go beyond the mean-field case. Remark: everything is loosely stated. I’m also going to present everything in a particular example. 5 / 73

Outline I’m going to present a mean-field model of particle systems. Mean-field means every particle has a direct interaction with all other particles. My purpose in this talk is to show how we go beyond the mean-field case. Remark: everything is loosely stated. I’m also going to present everything in a particular example. First I will introduce the Kuramoto Model. 6 / 73

The Kuramoto model The Kuramoto model is the following system of ODEs ( t ∈ [0 , T ]) : N d θ i ( t ) = 1 � sin( θ j ( t ) − θ i ( t )) d t + ω i d t , i ∈ { 1 , · · · , N } N j =1 7 / 73

The Kuramoto model The Kuramoto model is the following system of ODEs ( t ∈ [0 , T ]) : N d θ i ( t ) = 1 � sin( θ j ( t ) − θ i ( t )) d t + ω i d t , i ∈ { 1 , · · · , N } N j =1 Describes the path of the particle i in the time interval [0 , T ] . 8 / 73

The Kuramoto model The Kuramoto model is the following system of ODEs ( t ∈ [0 , T ]) : N d θ i ( t ) = 1 � sin( θ j ( t ) − θ i ( t )) d t + ω i d t , i ∈ { 1 , · · · , N } N j =1 Describes the path of the particle i in the time interval [0 , T ] . Particle i interacts with all the other particles (Mean-Field). We normalize by N . 9 / 73

The Kuramoto model The Kuramoto model is the following system of ODEs ( t ∈ [0 , T ]) : N d θ i ( t ) = 1 � sin( θ j ( t ) − θ i ( t )) d t + ω i d t , i ∈ { 1 , · · · , N } N j =1 Describes the path of the particle i in the time interval [0 , T ] . Particle i interacts with all the other particles (Mean-Field). We normalize by N . Interaction between the particles. 10 / 73

The Kuramoto model The Kuramoto model is the following system of ODEs ( t ∈ [0 , T ]) : N d θ i ( t ) = 1 � sin( θ j ( t ) − θ i ( t )) d t + ω i d t , i ∈ { 1 , · · · , N } N j =1 Describes the path of the particle i in the time interval [0 , T ] . Particle i interacts with all the other particles (Mean-Field). We normalize by N . Interaction between the particles. Natural frequencies. In the absence of interaction the particles will evolve independently with velocities ω i . 11 / 73

The Kuramoto model The Kuramoto model is the following system of ODEs ( t ∈ [0 , T ]) : N d θ i ( t ) = 1 � sin( θ j ( t ) − θ i ( t )) d t + ω i d t , i ∈ { 1 , · · · , N } N j =1 Describes the path of the particle i in the time interval [0 , T ] . Particle i interacts with all the other particles (Mean-Field). We normalize by N . Interaction between the particles. Natural frequencies. In the absence of interaction the particles will evolve independently with velocities ω i . Mean-field equals complete graph. 12 / 73

Motivation Lucas and Roberto were very interested in the Kuramoto Model. They were trying to understand the behaviour of neuronal populations under epileptic phenomenon. 13 / 73

Motivation Lucas and Roberto were very interested in the Kuramoto Model. They were trying to understand the behaviour of neuronal populations under epileptic phenomenon. The thing is that the Kuramoto Model is famous to understand synchronization: https://www.youtube.com/watch?v=T58lGKREubo 14 / 73

Motivation Lucas and Roberto were very interested in the Kuramoto Model. They were trying to understand the behaviour of neuronal populations under epileptic phenomenon. The thing is that the Kuramoto Model is famous to understand synchronization: https://www.youtube.com/watch?v=T58lGKREubo In the works I’m going to present we go in other direction. We do NOT have any theorems about synchronization here (despite we have some simulations). 15 / 73

Motivation Lucas and Roberto were very interested in the Kuramoto Model. They were trying to understand the behaviour of neuronal populations under epileptic phenomenon. The thing is that the Kuramoto Model is famous to understand synchronization: https://www.youtube.com/watch?v=T58lGKREubo In the works I’m going to present we go in other direction. We do NOT have any theorems about synchronization here (despite we have some simulations). Since our spins live on R (and not in S 1 ) we need that t ∈ [0 , T ] . Our contribution is to remove the mean-field assumption. 16 / 73

Interacting particles on graphs Start with a graph G . 1 2 5 3 4

Interacting particles on graphs R Start with a graph G . 1 θ 1 ( t ) 2 Think that each particle is on a vertex. It is like a spin in R evolving in time. 5 3 4

Interacting particles on graphs R R θ 2 ( t ) Start with a graph G . 1 θ 1 ( t ) 2 Think that each particle is on a vertex. It is like a spin in R evolving in time. 5 3 4

Interacting particles on graphs R R θ 2 ( t ) Start with a graph G . 1 θ 1 ( t ) 2 Think that each particle R is on a vertex. It is like a spin in R evolving in θ 3 ( t ) time. 5 3 4

Interacting particles on graphs R R θ 2 ( t ) Start with a graph G . 1 θ 1 ( t ) 2 Think that each particle R is on a vertex. It is like a spin in R evolving in θ 3 ( t ) time. 5 3 4 θ 4 ( t ) R

Interacting particles on graphs R R θ 2 ( t ) Start with a graph G . 1 θ 1 ( t ) 2 Think that each particle R is on a vertex. It is like a spin in R evolving in θ 3 ( t ) time. 5 3 θ 5 ( t ) R 4 θ 4 ( t ) R 22 / 73

Interacting particles on graphs R R θ 2 ( t ) Start with a graph G . 1 θ 1 ( t ) 2 Think that each particle R is on a vertex. It is like a spin in R evolving in θ 3 ( t ) time. 5 3 θ 5 ( t ) We want the Kuramoto model with interactions given by G . R 4 θ 4 ( t ) R 23 / 73

The stochastic Kuramoto model on graphs The particles follow the following system of SDEs ( t ∈ [0 , T ]): d θ i ( t ) = 1 � sin( θ j ( t ) − θ i ( t )) d t + ω i d t + d B i ( t ) , ∀ i ∈ G . d i j ∼ G i 24 / 73

The stochastic Kuramoto model on graphs The particles follow the following system of SDEs ( t ∈ [0 , T ]): d θ i ( t ) = 1 � sin( θ j ( t ) − θ i ( t )) d t + ω i d t + d B i ( t ) , ∀ i ∈ G . d i j ∼ G i Describes the evolution of the random paths of each particle. 25 / 73

The stochastic Kuramoto model on graphs The particles follow the following system of SDEs ( t ∈ [0 , T ]): d θ i ( t ) = 1 � sin( θ j ( t ) − θ i ( t )) d t + ω i d t + d B i ( t ) , ∀ i ∈ G . d i j ∼ G i Describes the evolution of the random paths of each particle. Particle i interacts with its neighbours. We normalize by d i . 26 / 73

The stochastic Kuramoto model on graphs The particles follow the following system of SDEs ( t ∈ [0 , T ]): d θ i ( t ) = 1 � sin( θ j ( t ) − θ i ( t )) d t + ω i d t + d B i ( t ) , ∀ i ∈ G . d i j ∼ G i Describes the evolution of the random paths of each particle. Particle i interacts with its neighbours. We normalize by d i . The same interaction function of the mean-field case. 27 / 73

The stochastic Kuramoto model on graphs The particles follow the following system of SDEs ( t ∈ [0 , T ]): d θ i ( t ) = 1 � sin( θ j ( t ) − θ i ( t )) d t + ω i d t + d B i ( t ) , ∀ i ∈ G . d i j ∼ G i Describes the evolution of the random paths of each particle. Particle i interacts with its neighbours. We normalize by d i . The same interaction function of the mean-field case. Natural frequencies. 28 / 73

The stochastic Kuramoto model on graphs The particles follow the following system of SDEs ( t ∈ [0 , T ]): d θ i ( t ) = 1 � sin( θ j ( t ) − θ i ( t )) d t + ω i d t + d B i ( t ) , ∀ i ∈ G . d i j ∼ G i Describes the evolution of the random paths of each particle. Particle i interacts with its neighbours. We normalize by d i . The same interaction function of the mean-field case. Natural frequencies. Independent standard Brownian motions. 29 / 73

The stochastic Kuramoto model on graphs The particles follow the following system of SDEs ( t ∈ [0 , T ]): d θ i ( t ) = 1 � sin( θ j ( t ) − θ i ( t )) d t + ω i d t + d B i ( t ) , ∀ i ∈ G . d i j ∼ G i Describes the evolution of the random paths of each particle. Particle i interacts with its neighbours. We normalize by d i . The same interaction function of the mean-field case. Natural frequencies. Independent standard Brownian motions. What do we want to prove? 30 / 73

Law of Large Numbers 31 / 73

Law of Large Numbers Take a sequence of graphs ( G N ) N ≥ 1 . Each G N has N ∈ N vertices. 32 / 73