network limits and graphons
TRANSCRIPT
![Page 1: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/1.jpg)
Network Limits and Graphons
Nikolaj Takata Mücke
TUM
7/2 - 2018
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 1 / 58
![Page 2: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/2.jpg)
Overview
1 Introduction2 A Little bit About Graphs and Networks
K-Nearest-Neighbour GraphsSmall-World GraphsNetwork Limits and Graphons
3 Dynamics on NetworksNetwork Limits and GraphonsApproximation Properties
4 The Kuramoto Modelq-Twisted StatesContinuum Limit of The Kuramoto ModelStability AnalysisSynchronizationContinuation
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 2 / 58
![Page 3: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/3.jpg)
A Little bit About Graphs and Networks
A Little bit About Graphs and Networks
What is a graph?An ordered pair, G = (V ,E )
V : The set of vertices (nodes)E : The of edges
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 3 / 58
![Page 4: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/4.jpg)
A Little bit About Graphs and Networks
Graph RepresentationGraph picture
Works well to get an overview of the structureBecomes very messy for large graphs!
Adjacency MatrixGood when doing computationsDifficult to get an intuitive understanding of the graph
Pixel PictureGood to get an overview of large graphs
Figure 1: The Petersen graph, its adjacency matrix, and its pixel pictureNikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 4 / 58
![Page 5: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/5.jpg)
A Little bit About Graphs and Networks K-Nearest-Neighbour Graphs
K-Nearest-Neighbour Graphs
Definition (k-Nearest-Neighbour Graph)Let Cn,k be a graph. If
V (Cn,k) = [n] and E (Cn,k) = {(i , j) ∈ [n]× [n] | 0 < dn(i , j) ≤ k}
for sufficiently large n ∈ N and k ∈ N such that 2k < n, wheredn(i , j) = min{|i − j |, n − |i − j |}. Then we say that Cn,k is a k-NN graph.
Intuition: A graph where every node is only connected to the k nearestnodes. Where nearest is defined by some metric.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 5 / 58
![Page 6: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/6.jpg)
A Little bit About Graphs and Networks K-Nearest-Neighbour Graphs
(a) Network representation
0 10 20 30 40 50 60 70 80 90 100
nz = 5000
0
10
20
30
40
50
60
70
80
90
100
(b) Pixel picture
Figure 2: k-NN graph with n = 100 and k = 25
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 6 / 58
![Page 7: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/7.jpg)
A Little bit About Graphs and Networks Small-World Graphs
Small-World Graphs
Definition (Small-World Graph)Let L be the distance between two arbitrary nodes in a graph Gn, i.e. thenumber of steps required to go from one node to the other. Then we saythat Gn is a Small-World graph if L ∝ log n.
Intuition: A graph where you can come from an arbitrary chosen node toany node by a small number of steps.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 7 / 58
![Page 8: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/8.jpg)
A Little bit About Graphs and Networks Small-World Graphs
W-Graphs
Definition (W-Graphs)Let Xn = {x1, x2, . . . , xn} ⊂ I = [0, 1], W : I2 → I be a symmetricmeasurable function and Gn = 〈[n],E (Gn)〉. Then we call Gn a W-randomgraph if
P ((i , j) ∈ E (Gn)) ={
W (xi , xj), i 6= j0, Otherwise
and we denote it Gn = G(W ,Xn).
Intuition: A graph where the probability of two nodes are connected isgiven by some probability function, W .
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 8 / 58
![Page 9: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/9.jpg)
A Little bit About Graphs and Networks Small-World Graphs
SW-Graphs
Definition (SW-Graphs)
Let Xn ={0, 1
n ,2n , . . . ,
n−1n
}and W be defined as
W (x , y) ={
1, d(x , y) ≤ r0, Otherwise , (1)
and
Wp = (1− p)W + p(1−W ), p ∈ [0, 0.5], (2)
then Gn,p = G(Wp,Xn) is an SW-graph.
Intution: A brnc-NN graph with certain edges made into randomconnections with any other node.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 9 / 58
![Page 10: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/10.jpg)
A Little bit About Graphs and Networks Small-World Graphs
W (x , y) ={
1, d(x , y) ≤ r0, Otherwise
Wp = (1− p)W + p(1−W ), p ∈ [0, 0.5]
p = 0: W0 = Wp = 0.5: W0.5 = 0.5p = 1: W1 = 1−W
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 10 / 58
![Page 11: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/11.jpg)
A Little bit About Graphs and Networks Small-World Graphs
K-NN Random Graphs
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 11 / 58
![Page 12: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/12.jpg)
A Little bit About Graphs and Networks Small-World Graphs
K-NN Random Graphs
0 10 20 30 40 50 60 70 80 90 100
nz = 5000
0
10
20
30
40
50
60
70
80
90
100
(a) p = 0
0 10 20 30 40 50 60 70 80 90 100
nz = 5000
0
10
20
30
40
50
60
70
80
90
100
(b) p = 0.25
0 10 20 30 40 50 60 70 80 90 100
nz = 5000
0
10
20
30
40
50
60
70
80
90
100
(c) p = 0.5
Figure 3: SW-graphs n = 100, k = 25 and varying randomness parameter p.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 12 / 58
![Page 13: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/13.jpg)
A Little bit About Graphs and Networks Network Limits and Graphons
Network Limits and Graphons
What happens when we increase the number of nodes and edges?What happens at the limit, n→∞?
Definition (Graphon)A graphon is a measurable function W : I2 → I.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 13 / 58
![Page 14: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/14.jpg)
A Little bit About Graphs and Networks Network Limits and Graphons
a) Adjacency matrix of a k-NN graphb) The support of the corresponding graphon
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 14 / 58
![Page 15: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/15.jpg)
A Little bit About Graphs and Networks Network Limits and Graphons
Graph Limit for W-graphs
TheoremLet {Gn,p}n∈N be a sequnce of W-random graphs. Then the seqeunce isconverging with probability one and it’s limit is the graphon W .
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 15 / 58
![Page 16: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/16.jpg)
Dynamics on Networks
The General Form
Every node in a graph is in some state. This state evolves with timeaccording to some dynamicsIn a network with n node we have a system of n ODE’s:
ddt u(n)
i =n∑
j=1a(n)
ij K (u(n)i , u(n)
j ), (3)
a(n)ij is the entries of the adjacency matrix.
K : R2 → R, is some function.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 16 / 58
![Page 17: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/17.jpg)
Dynamics on Networks
Non-Linear Heat Equation
The non-linear heat equation on graphs Gn. This dynamical system isgiven by
ddt u(n)
i = 1n
n∑j=1
w (n)ij D(u(n)
i − u(n)j ), (4)
where w (n)ij is only non-zero if (i , j) ∈ E (Gn). we assume that D : R→ R
is Lipschitz continuous.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 17 / 58
![Page 18: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/18.jpg)
Dynamics on Networks Network Limits and Graphons
Network Limits and Graphons
Why consider limits?If n gets large we run into troubles
Very difficult to assess behaviour analytically (fixpoints, etc.)Very time consuming to compute
We get an infinite dimensional dynamical system, i.e. a PDEThese are (sometimes) easier to analyse
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 18 / 58
![Page 19: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/19.jpg)
Dynamics on Networks Network Limits and Graphons
How does the limit PDE look?
We define the n-dimensional vector
u(n)(t) = (u(n)1 (t), u(n)
2 (t), . . . , u(n)n (t))
If we let n→∞ one will obtain an infinite dimensional vector or simply afunction u(x , t) where x ∈ I.
u(n)(t)→ u(x , t), n→∞
This will be denoted the continuum limit of u(n)(t).
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 19 / 58
![Page 20: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/20.jpg)
Dynamics on Networks Network Limits and Graphons
How does the limit PDE look?
The heat equation:
ddt u(n)
i = 1n
n∑j=1
w (n)ij D(u(n)
i − u(n)j ) (5)
Riemann sum:n∑
j=1f (ti)(xi − xi−1), ti ∈ [xi , xi−1]
Riemann integral:
limn→∞
n∑j=1
f (ti)(xi − xi−1) =∫ b
af (x) dx
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 20 / 58
![Page 21: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/21.jpg)
Dynamics on Networks Network Limits and Graphons
How does the limit PDE look?
The heat equation for n→∞:
u(n)(t)→ u(x , t)
1n
n∑j=1
w (n)ij D(u(n)
i − u(n)j )→
∫ 1
0W (x , y)D(u(x , t)− u(y , t)) dy
where W (x , y) is the limit graphon.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 21 / 58
![Page 22: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/22.jpg)
Dynamics on Networks Network Limits and Graphons
The Continuum Limit PDE
ddt u(x , t) =
∫ 1
0W (x , y)D(u(x , t)− u(y , t)) dy
u(x , 0) = g(x)
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 22 / 58
![Page 23: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/23.jpg)
Dynamics on Networks Approximation Properties
How Good is This Approximation?
So far we have only provided the intuitive arguments:The right hand side of the Dynamical System resembles a RiemannsumWe therefore "guess" that a Riemann integral approximates the sumin the limit n→∞
Can we provide rigourous arguments for that?
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 23 / 58
![Page 24: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/24.jpg)
Dynamics on Networks Approximation Properties
YES WE CAN!
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 24 / 58
![Page 25: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/25.jpg)
Dynamics on Networks Approximation Properties
Approximation on Deterministic Network
TheoremSuppose g ∈ L∞(I), W : I2 → {0, 1} is a symmetric measurable functionand
γ := dimB∂W + ∈ [0, 2).
Let u and un denote the vector-valued functions corresponding to thesolutions of the continuum limit PDE and the original system of ODE’s,respectively. Then for any ε > 0 there exists N(ε) ∈ N such that forn ≥ N(ε) :
||u − u(n)||C(0,T ;L2(I)) ≤ C1(||g − gn||L2(I) + nγ/2+ε−1
)where constant C1 is independent of n.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 25 / 58
![Page 26: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/26.jpg)
Dynamics on Networks Approximation Properties
||u − u(n)||C(0,T ;L2(I)) ≤ C1(||g − gn||L2(I) + nγ/2+ε−1
)
Small box dimension of the support of ∂W =⇒ fast convergencegn → g fast =⇒ fast convergence
Note: This is only for W a binary function!
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 26 / 58
![Page 27: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/27.jpg)
Dynamics on Networks Approximation Properties
Approximation of Random Network
TheoremSuppose W is almost everywhere continuous on I2, D : R→ R is Lipschitzcontinuous, and g ∈ L∞(I). Let u(x , t) denote the solution of thecontinuum limit PDE. Suppose further
mint∈[0,T ]
∫I2
D(u(y , t)− u(x , t))W (x , y)(1−W (x , y)) dx dy > 0
for some T > 0. Then
||u(n) − u||C(0,T ;L2(I)) → 0
in probability.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 27 / 58
![Page 28: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/28.jpg)
Dynamics on Networks Approximation Properties
Convergence in probability means
limn→∞
P(||un − u||C(0,T ;L2(I)) > ε
)= 0
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 28 / 58
![Page 29: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/29.jpg)
Dynamics on Networks Approximation Properties
Approximation Properties
With only the following assumptionsD is Lipschitz continuousThe initial condition g ∈ L∞
W is measurable and binary... we can analyse dynamics on the following graphs
k-NN graphsSmall-worls graphsRandom graphs
W-graphsSW-graphs
... by their continuum limit!
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 29 / 58
![Page 30: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/30.jpg)
The Kuramoto Model
The Kuramoto Model
Models a network of oscillators on some graph G , by the set of ODE’s;
ddt u(n)
i = ωi + σ
n∑
j:(i ,j)∈E(Gn)sin(2π(u(n)
i − u(n)j )
), i ∈ [n]. (6)
Models coupled oscillators such asJosephson JunctionNeural NetworksCoupled lasersMuch more...!
Proposed by Japanese mathematician Yoshiki Kuramoto.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 30 / 58
![Page 31: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/31.jpg)
The Kuramoto Model
The Kuramoto Model
ddt u(n)
i = ωi + σ
n∑
j:(i ,j)∈E(Gn)sin(2π(u(n)
i − u(n)j )
), i ∈ [n]. (7)
i denotes the oscillatorui denotes the phase of the ith oscillatorωi is the natural frequency of oscillator iσ is the coupling between oscillatorsn is the number of oscillators
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 31 / 58
![Page 32: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/32.jpg)
The Kuramoto Model
The Kuramoto Model
We will only study the case withAll intrinsic frequencies are the same, ωi = ωj for all i and j .Attractive coupling, σ = 1
This gives the system of ODE’s:
ddt u(n)
i = 1n
∑j:(i ,j)∈E(Gn)
sin(2π(u(n)
i − u(n)j )
), i ∈ [n]. (8)
These restrictions simplify the problem quite a lot, but makes it easier forus to convey the important points of this talk.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 32 / 58
![Page 33: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/33.jpg)
The Kuramoto Model
The Kuramoto Model
ddt u(5)
1 = 15(sin(u(5)
1 − u(5)4
)+ sin
(u(5)
1 − u(5)5
))ddt u(5)
2 = 15(sin(u(5)
2 − u(5)5
))ddt u(5)
3 = 15(sin(u(5)
3 − u(5)5
))ddt u(5)
4 = 15(sin(u(5)
4 − u(5)1
)+ sin
(u(5)
4 − u(5)5
))ddt u(5)
5 = 15(sin(u(5)
5 − u(5)1
)+ sin
(u(5)
5 − u(5)2
)+ sin
(u(5)
5 − u(5)3
)+ sin
(u(5)
5 − u(5)4
))
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 33 / 58
![Page 34: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/34.jpg)
The Kuramoto Model
The Kuramoto Model on SW-graphs
We will study the Kuramoto model on small world SW-graph, Gn,p, on acircle.
0 10 20 30 40 50 60 70 80 90 100
nz = 5000
0
10
20
30
40
50
60
70
80
90
100
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 34 / 58
![Page 35: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/35.jpg)
The Kuramoto Model
The Kuramoto Model on SW-graphs
Remember, connections between nodes are given by
Wp = (1− p)W + p(1−W ), p ∈ [0, 0.5], (9)
with
W (x , y) ={
1, d(x , y) ≤ r0, Otherwise . (10)
Then the Kuramoto model can be written as
ddt u(n)
i = 1n
n∑j=1
Wp(i , j) sin(2π(u(n)
i − u(n)j )
), i ∈ [n]. (11)
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 35 / 58
![Page 36: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/36.jpg)
The Kuramoto Model q-Twisted States
q-Twisted States
An important class of steady state solutions on k-NN graphs is theso-called q-Twisted States:
u(n)i ,q = q(i − 1)
n + c mod 1, c ∈ [0, 1), i ∈ [n], (12)
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 36 / 58
![Page 37: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/37.jpg)
The Kuramoto Model q-Twisted States
q-Twisted States
0 10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 5: q = 1, q = 2, q = 3, q = 4
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 37 / 58
![Page 38: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/38.jpg)
The Kuramoto Model Continuum Limit of The Kuramoto Model
Continuum Limit of The Kuramoto Model
We want to derive the continuum limit PDE of the Kuramoto system sowe can study
Stability of the q-twisted statesSynchronization
Both in terms of r and the randomness parameter, p.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 38 / 58
![Page 39: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/39.jpg)
The Kuramoto Model Continuum Limit of The Kuramoto Model
Continuum Limit of The Kuramoto Model
As a reminder, the discrete Kuramoto model on SW-graphs is given by
ddt u(n)
i = 1n
n∑j=1
Wp(i , j) sin(2π(u(n)
i − u(n)j )
), i ∈ [n]. (13)
From our theorems earlier we have, in the limit n→∞, the continuumlimit PDE:
∂
∂t u(x , t) =∫
IWp(x , y) sin (2π(u(x , t)− u(y , t))) dy . (14)
Note: We assume that the assumptions are fulfilled.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 39 / 58
![Page 40: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/40.jpg)
The Kuramoto Model Continuum Limit of The Kuramoto Model
Continuous q-twisted state
uq(x) = qx + c mod 1, c ∈ [0, 1), q ∈ Z. (15)
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 40 / 58
![Page 41: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/41.jpg)
The Kuramoto Model Continuum Limit of The Kuramoto Model
LemmaLet
R(n)i (u(n)) = 1
n∑
j:(i ,j)∈E(Gn)sin(2π(u(n)
i − u(n)j )
). (16)
Then for any q ∈ N ∪ {0} and i ∈ N
limn→∞
R(n)i (u(n)
q ) = 0 (17)
almost surely.
Basically: The continuous q-twisted state is also a steady state solutionfor the discrete system in the limit n→∞.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 41 / 58
![Page 42: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/42.jpg)
The Kuramoto Model Stability Analysis
Rewriting the Continuum Limit
∂
∂t u(x , t) =∫
IWp(x , y) sin (2π(u(y , t)− u(x , t))) dy
Becomes
∂
∂t u(x , t) =∫
IKp(y − x) sin (2π(u(y , t)− u(x , t))) dy
where
Kp(x) = pG1/2(x) + (1− 2p)Gr (x), Gr ={
1, d(x) ≤ r0, Otherwise .
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 42 / 58
![Page 43: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/43.jpg)
The Kuramoto Model Stability Analysis
TheoremFor q ∈ Z and p ∈ [0, 0.5], the q-twisted state is a steady state of (14).Moreover, it is linearly stable with respect to perturbations from L∞(I)provided
λp(q,m) := K̃p(m + q)− 2K̃p(q) + K̃ (q −m) < 0, ∀m ∈ N, (18)
where
K̃p(m) =∫
IKp(x) cos(2πmx) dx , m ∈ Z. (19)
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 43 / 58
![Page 44: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/44.jpg)
The Kuramoto Model Stability Analysis
Claim:
K̃p(m) ={
p 1πm sin(2πmr) + (1− 2p) 1
πm sin(2πmr), m 6= 0p + (1− 2p)2r , m = 0 .
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 44 / 58
![Page 45: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/45.jpg)
The Kuramoto Model Stability Analysis
... Which gives us the following criteria for stability of the q-twisted state
λp(q,m) = K̃p(m + q)− 2K̃p(q) + K̃ (q −m)= p(−2δq0 + δqm) + (1− 2p)λ0(q,m) < 0
where
λ0(q,m) = 1π
[ 1q + m sin(2πr(q + m))− 2
q sin(2πrq)
+ 1q −m sin(2πr(q −m))
]for q 6= 0.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 45 / 58
![Page 46: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/46.jpg)
The Kuramoto Model Stability Analysis
Solving
λp(q,m) = p(−2δq0 + δqm) + (1− 2p)λ0(q,m) < 0
for p or r is quite dificult! But it is done in [9] for r in the case with p = 0:
0 ≤ qr ≤ µ ≈ 0.66 (20)
and in [7] for p
0 < p < −λ0(q, q)1− 2λ0(q, q) (21)
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 46 / 58
![Page 47: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/47.jpg)
The Kuramoto Model Synchronization
Synchronization
What is Synchronization?When all phases are the same!u1 = u2 = . . . = un
Corresponds to q = 0 in q-twisted states
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 47 / 58
![Page 48: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/48.jpg)
The Kuramoto Model Synchronization
Synchronization
Using that q = 0 corresponds to synchronization, we can use the conditionfor stability of the q-twisted states;
λp(0,m) = −2p + (1− 2p)λ0(0,m) < 0
λ0(0,m) = 2πm sin(2πmr)− 4r
It is stable for all r !
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 48 / 58
![Page 49: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/49.jpg)
The Kuramoto Model Synchronization
Synchronization Rate
The rate of the synchronization is the time it takes for the system toreach the synchronized state. The rate is determined by
supm∈N
λp(0,m) = supm∈N
(−2p + (1− 2p)λ0(0,m)) = λp(0, 1) (22)
Larger |λp(0, 1)| → faster synchronization!
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 49 / 58
![Page 50: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/50.jpg)
The Kuramoto Model Synchronization
Synchronization Rate in Terms of r
How does r affect the rate of synchronization? We set p = 0 and take acloser look at λ(0, 1)
λ0(0, 1) = λ0(0, 1) = 2πsin(2πr)− 4r (23)
By Taylor expansion in r
λ0(0, 1) = −8π2
3 r3 +O(r5) (24)
Hence, larger r gives faster synchronization!
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 50 / 58
![Page 51: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/51.jpg)
The Kuramoto Model Synchronization
Synchronization Rate in Terms of rN = 100, p = 0.2, initial condition q = 1.
5 10 15 20 25 30 35 40
k
0
50
100
150
200
250
Tim
e it ta
kes to s
ynchro
niz
e
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 51 / 58
![Page 52: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/52.jpg)
The Kuramoto Model Synchronization
Synchronization Rate in Terms of p
How does p affect the rate of synchronization?
λp(0, 1) = −2p + (1− 2p)λ0(0,m) (25)
ddpλp(0, 1) = −2− 2λ0(0, 1) < 0 (26)
Thus, increse in p → larger |λp(0, 1)| → faster synchronization!
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 52 / 58
![Page 53: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/53.jpg)
The Kuramoto Model Synchronization
N = 100, k = 10, initial condition q = 1.
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
p-value
40
50
60
70
80
90
100
110
120
130
Tim
e it ta
kes to s
ynchro
niz
e
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 53 / 58
![Page 54: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/54.jpg)
The Kuramoto Model Continuation
Continuation
What is continuation?Changing a parameter (or more parameters) continuously to seehow the solution changes
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 54 / 58
![Page 55: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/55.jpg)
The Kuramoto Model Continuation
Continuation of 1-twisted stateN = 100, and p = 0 → p = 3.5 · 10−3
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 55 / 58
![Page 56: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/56.jpg)
The Kuramoto Model Continuation
Continuation of 1-twisted stateN = 100, and p = 4.9 · 10−3 → p = 3.9 · 10−3
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 56 / 58
![Page 57: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/57.jpg)
The Kuramoto Model Continuation
Continuation of 2-twisted stateN = 100, and p = 0 → p = 6 · 10−4
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 57 / 58
![Page 58: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/58.jpg)
The Kuramoto Model Continuation
Continuation of 2-twisted stateN = 100, and p = 1.1 · 10−3 → p = 1.63 · 10−3
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 58 / 58
![Page 59: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/59.jpg)
The Kuramoto Model Continuation
Christian Borgs, Jennifer T Chayes, László Lovász, Vera T Sós, andKatalin Vesztergombi.Convergent sequences of dense graphs i: Subgraph frequencies, metricproperties and testing.Advances in Mathematics, 219(6):1801–1851, 2008.
Daniel Glasscock.a graphon?Notices of the AMS, 62(1), 2015.
László Lovász.Large networks and graph limits, volume 60.American Mathematical Society Providence, 2012.
László Lovász and Balázs Szegedy.Limits of dense graph sequences.Journal of Combinatorial Theory, Series B, 96(6):933–957, 2006.
Georgi S Medvedev.The nonlinear heat equation on dense graphs and graph limits.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 58 / 58
![Page 60: Network Limits and Graphons](https://reader033.vdocuments.us/reader033/viewer/2022052023/6286b5125d955c00aa66f08c/html5/thumbnails/60.jpg)
The Kuramoto Model Continuation
SIAM Journal on Mathematical Analysis, 46(4):2743–2766, 2014.
Georgi S Medvedev.The nonlinear heat equation on w-random graphs.Archive for Rational Mechanics and Analysis, 212(3):781–803, 2014.
Georgi S Medvedev.Small-world networks of kuramoto oscillators.Physica D: Nonlinear Phenomena, 266:13–22, 2014.
Duncan J Watts and Steven H Strogatz.Collective dynamics of ‘small-world’networks.nature, 393(6684):440–442, 1998.
Daniel A Wiley, Steven H Strogatz, and Michelle Girvan.The size of the sync basin.Chaos: An Interdisciplinary Journal of Nonlinear Science,16(1):015103, 2006.
Nikolaj Takata Mücke (TUM) Network Limits and Graphons 7/2 - 2018 58 / 58