Synchronisation Properties of Trees in the Kuramoto Model

0.61 MB·English

by  Anthony H. Dekker

SYNCHRONISATION PROPERTIES OF TREES IN THE
KURAMOTO MODEL
ANTHONY H. DEKKER∗ AND RICHARD TAYLOR†
Abstract. WeconsidertheKuramotomodel ofcoupledoscillators,specificallythecaseoftree
networks, for which we prove a simple closed-form expression for the critical coupling. For several
2 classes of tree, and for both uniform and Gaussian vertex frequency distributions, weprovide tight
1 closedformboundsandempiricalexpressionsfortheexpectedvalueofthecriticalcoupling. Wealso
0 provideseveral bounds on theexpected value ofthe criticalcoupling foralltrees. Finally, weshow
2 thatforagivensetofvertexfrequencies,thereisarearrangementofoscillatorfrequenciesforwhich
thecriticalcouplingisboundedbythespreadoffrequencies.
v
o
Key words. tree,synchronisation,Kuramotomodel
N
9 AMS subject classifications. 05C05,34C25,34C28,37C25, 82B26
1

1. Introduction. The Kuramoto model [16, 15, 7, 10] was originally motivated
]
S bythephenomenonofcollectivesynchronisationwherebyasystemofcoupledoscillat-
D ingvertices(nodes)willsometimeslockontoacommonfrequencydespitedifferences
. inthenaturalfrequenciesoftheindividualvertices. Biologicalexamplesincludeoscil-
h
lations of the heart [21] and of chemical systems [12]. Systems of coupled oscillators
t
a can also be used as an abstract model for synchronisation in organisations [3, 9, 4].
m
While Kuramoto studied the infinite complete network it is natural to consider finite
[ networks of any given topology. This would correspond to a notion of coupling that

is not universal across all vertex (node) pairs, but rather applies to a subset of all
2
v possible edges (links). For example the work patterns of human individuals in an
3 organisationmightenjoya couplingeffectinrelationtopairsofindividualsthathave
7 a working relationship.
2
A typical networkof coupled oscillatorsis as shown in Figure 1. Eachvertex has
1
an associated phase angle θ , as well as its own natural frequency ω . In this paper
. i i
0 we are primarily concerned with the case where the network is a tree. The basic
1
governing equation is the differential equation:
2
1
n
v: (1) θ˙ =ω +k A sin(θ θ )
i i ij j i
−
i
X Xi=1
r where A is the adjacency matrix of the network and k is a coupling constant which
a
determinesthestrengthofthecoupling. Werefertoanetwork(graph)withpreferred
oscillator frequencies ω attached to the vertices as a Kuramoto graph.
i
Ithas beenobserved[7,17]thatmanyKuramotographssynchronise, inthat the
actual vertex frequencies θ˙ converge to a common value. That is, the vertex phases
i
θ rotate at the same rate, with a constant phase difference between each pair of
i
vertices. Moreover this phenomenon appears at a critical coupling constant k , and
c
as k increases,this phenomenon applies to a greater range of initial phases (the θ at
i
t = 0). Thus the graph has a frequency fixed point characterised by all the θ˙ being
i
equal.
∗Joint Operations Division, Defence Science and Technology Organisation (DSTO), Canberra,
ACT,Australia([email protected])
†JointOperationsDivision,DSTO,Canberra,ACT([email protected])
1
2 A.H.DEKKERANDR.TAYLOR
Fig. 1. An example Kuramoto tree.
At the frequency fixed point, the θ˙ are equal to ω¯, the mean of the frequencies
i
ω , and it is convenient to apply a rotating frame of reference, with:
i
(2) φ (t)=θ (t) ω¯t
i i
−
We then obtain a system of differential equations equivalent to the original:
n
(3) φ˙ =ω ω¯+k A sin(φ φ )
i i ij j i
− −
i=1
X
At the frequency fixed point, we then have φ˙ = 0 for all i. While the literature
i
contains several definitions of “critical coupling,” here we define the critical coupling
k as that number for which a frequency fixed point exists exactly when k k (in
c c
≥
[8], this critical coupling is called K ).
L
In general the value of the critical coupling is the solution of simultaneous tran-
scendentalequationsandwouldnotbeexpectedtohaveaclosedformsolution. How-
ever, for complete graphs and complete bipartite graphs the critical coupling can be
computed efficiently as the solution of non-linear equations [18, 19]. For the graph
withonlytwoverticesν andν andasingleedgebetweenthem,itiseasytoseethat
1 2
solutions exist precisely when ω ω 2k, i.e.
1 2
| − |≤
ω ω
1 2
(4) k = | − |
c
2
Our first theorem generalises this to any tree. We then use that result to calculate
the expected value of the critical coupling for severalclasses of Kuramoto tree having
randomlychosenfrequencies,andto findbounds onthe expected valueofthe critical
coupling for Kuramoto trees in general. Finally, we use the result to show that
the critical coupling for a specific tree topology can be reduced to a value which is
independent of the number of vertices, by reshuffling the frequencies ω .
i
2. Main Lemma and Corollaries. The first step to our initial theorem is a
lemma which simplifies the calculation of critical couplings by breaking a Kuramoto
SYNCHRONISATIONPROPERTIESOFTREES 3
Fig.2. Construction for Lemma 1.
graphintocomponents. Inparticular,weconsiderKuramotographswithacut-vertex,
i.e. one whose removal disconnects the graph. Figure 2 shows an example.
Lemma 1. Let G be a Kuramoto graph with n vertices ν ,ν ,...,ν with nat-
1 2 n
ural frequencies ω ,ω ,...,ω in which vertex ν is a cut-vertex (whose removal
1 2 n 1
disconnects the graph) with vertices ν ,...,ν on one side of the cut and vertices
2 m
ν ,...,ν on the other. As shown in Figure 2, let G and G be the two graphs
m+1 n 1 2
formed by deleting all the vertices on one side of the cut vertex and replacing the cut
vertex by new vertices x (for G ) and y (for G ) having frequencies ω and ω :
1 2 x y
m n
(5) ω =ω¯ (ω ω¯)=ω + (ω ω¯)
x i 1 i
− − −
i=2 i=m+1
X X
n m
(6) ω =ω¯ (ω ω¯)=ω + (ω ω¯)
y i 1 i
− − −
i=m+1 i=2
X X
Then G has a frequency fixed point if and only if both G and G have frequency fixed
1 2
points, and hence k (G)=max(k (G ),k (G )).
c c 1 c 2
Proof. Firstwe note thatthe frequencies ω andω arechosenin5 and6so that
x y
the average frequencies of both G and G are the same as that of G. Let G have a
1 2
frequency fixed point so that we have k and φ ,i = 1,...,n satisfying the system of
i
equations for i=1,...,n:
n
(7) 0=ω ω¯+k A sin(φ φ )
i ij j i
− −
j=1
X
If we sum these equations from i = 1,...,m, then each term A sin(φ φ ) is
ij j i
−
cancelled by the term A sin(φ φ ) with the exception of the pairs (1,j),j =
ji i j
−
m+1,...,n and we obtain:
m n
(8) 0= (ω ω¯)+k A sin(φ φ )
i 1i i 1
− −
i=1 i=m+1
X X
4 A.H.DEKKERANDR.TAYLOR
Subtracting equation (8) from (7), for the case i=1, gives an equation for vertex x:
m m
0= (ω ω¯)+k A sin(φ φ )
i 1i i 1
− − −
i=2 i=2
X X
m
(9) =ω¯ ω +k A sin(φ φ )
x 1i i 1
− −
i=2
X
Similarly by summing (7) from i=m+1,...,n we obtain an equation for vertex y:
n
(10) 0=ω¯ ω +k A sin(φ φ )
y 1i i 1
− −
i=m+1
X
It follows that a solution to the system of equations (7) is a solution to the system of
equationsforG andG inwhichtheverticesxofG andyofG havethesamephase
1 2 1 2
φ . If onthe other handwe havea solutionto the systemoffixed pointequations for
1
G and G , then we can shift all the phase angles of the vertices of G by the same
1 2 2
amountsothatφ =φ ,without changingthe phase angledifferences. This provides
y x
a fixed point solution to the system of equations (7) and completes the proof.
Repeated application of this lemma can reduce the calculation of the critical
coupling to 2-connected graphs (in which any two vertices lie on a cycle) and tree
edges. For example, we have the following corollary for graphs with a cut-edge, i.e.
an edge whose removal disconnects the graph:
Corollary2. LetGbeaKuramotographwithnvertices1,2,...,nwithnatural
frequencies ω ,ω ,...,ω inwhich theedgefrom ν toν is acut-edgewithvertices ν
1 2 n 1 2 1
and ν ,...,ν on one side of the cut and vertices ν and ν ,...,ν on the other.
3 m 2 m+1 n
Let G′ be the one-edge graph containing only the vertices x and y, where:
m
(11) ω =ω + (ω ω¯)
x 1 i
−
i=3
X
n
(12) ω =ω + (ω ω¯)
y 2 i
−
i=m+1
X
Then G has a frequency fixed point only if G′ has a frequency fixed point, i.e.
m n
(13) ω ω + (ω ω¯) (ω ω¯) 2k
1 2 i i
(cid:12) − − − − (cid:12)≤
(cid:12) Xi=3 i=Xm+1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof. Applyin(cid:12)g Lemma 1 twice yields three compon(cid:12)ents: one based on the
vertices ν ,ν ,...,ν , one based on ν ,ν ,...,ν , and the one-edge graph
1 3 m 2 m+1 n
{ } { }
G′ = x,y based on ν ,ν . The condition on k follows by applying (4) to G′.
1 2
{ } { }
Forgraphswhicharetrees,repeatedapplicationofLemma1producescomponents
which are exactly the tree edges, with a similar adjustment of frequencies. We can
thereforeobtainan“ifandonlyif”condition,andhenceobtainaprecisevaluefork :
c
Theorem 3. Let T beany treewith at least two vertices. For any edge e of T, let
T and T′ be the two subtrees formed when e is deleted from T. These two partitions
e e
have T and T′ vertices respectively. Define the partition sum:
| e| | e|
(14) Ω(T )= ω ω¯ = T ω¯ ω
e i e i
| − | (cid:12)| | − (cid:12)
iX∈Te (cid:12) iX∈Te (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
SYNCHRONISATIONPROPERTIESOFTREES 5
be the sum of the frequency deviations from ω¯. Then Ω(T ) = Ω(T′) and the critical
e e
coupling k for T is given by:
c
(15) k =maxΩ(T )
c e
e∈T
Proof. We repeatedly apply Lemma 1 to obtain n 1 graphs each consisting of
−
a single edge. The condition on k follows from (4), applied to each of the n 1 edge
−
graphs obtained.
We note that, for trees, the condition in this theorem is equivalent to the more
general sufficient conditions given in Remark 10 of [8] and Statement G1 of [6]. In
the remainder of the paper we consider various implications of the result, facilitated
by the form in which our result is expressed. We begin by considering the expected
value of k , when frequencies are chosen randomly.
c
3. Expected Critical Couplings. It may be the case that we do not know
the precise frequencies for a Kuramoto tree, but we do know the distribution of the
frequencies. In this case, we can apply Theorem 3 to determine the expected value of
the critical coupling k for various kinds of tree. Figure 3 shows some of the trees
c
considered. In this section we combine theoretical derivations, using Theorem 3 and
statistical theory, with Monte Carlo experiments. In these experiments, we consid-
ered a variety of trees, and made 1,000,000frequency assignments with ω uniformly
i
distributed over the interval [0,1]. The frequencies ω therefore have mean 1/2 and
i
variance σ2 = 1/12, and the expected critical coupling E(k ) is proportional to σ,
c
the square root of the variance. We used Theorem 3 to determine the critical cou-
plingk foreachassignment,andhence theexpectedvalue ofthe criticalcouplingfor
c
that size and shape of tree. We also performed the same experiments using normally
distributed frequencies with the same mean and variance, calculating over 8,000,000
frequency assignments in that case, to allow adequate sampling of the tail.
3.1. Chains. Forthecaseofachain(atreewithallverticesconnectedinaline)
with n vertices ν ,...,ν , by Theorem 3, the critical coupling kchain is:
1 n c
j
(16) kchain =max (ω ω¯)
c j (cid:12) i− (cid:12)
(cid:12)Xi=1 (cid:12)
(cid:12) (cid:12)
The partition sums j (ω ω¯) for the(cid:12)first j verti(cid:12)ces form a random walk as j
i=1 i − (cid:12) (cid:12)
increases. Figure 4 illustrates an example, for n = 60. The walk must return to
zero, since ω¯ = n Pω , and so the maximum translation distance is likely to occur
i=1 i
in the vicinity of the centre of the chain. This random walk has step variance σ2,
anda crude estimPate for the expectedcriticalcoupling E(kchain) is the expectedfinal
c
displacement ∆ for a random walk with this step variance and τ steps, where τ is
some value between n/2 and n. However, this takes into account neither the return
to zero nor the fact that the maximum in (16) is being taken over all values of j. An
asymptotic expression for ∆ was given by Coffman et al. [1, 2] which, for the case
τ =n/2, gives:
2n n n
(17) ∆=σ σc=σ σc= 0.148976
π 2 − π − 12π −
r r r
where c = 0.516068, and we have ignored terms in n−1/2. Although this expected
translation distance under-estimates the critical coupling, we will still have a result
of the form a√n+b, and we can determine the values of a and b empirically.
6 A.H.DEKKERANDR.TAYLOR
(a) Star (b) Dumb-bell
(c) Binary tree (d) “Tadpole”
Fig.3. Some example trees with n=60.
2
) 1
-wi
(wof
m
u 0
s
e
v
ati
ul
m
Cu −1
2
−
0 10 20 30 40 50 60
Position in chain
increFasiign.g4j.. DOontetedralnindeosmshwowalkthoefempapritriitcioanlesxupmecstedPvjia=lu1e(ωEi(−kcωh¯a)inf)or=aχ(cnh)a,iwnhwicihthisnth=ee6xp0ecatnedd
c
peak deviation from 0.
For our empirical Monte Carlo studies, we considered chains with the number
of vertices n ranging from 2 to 300, with 1,000,000 different uniformly distributed
frequencyassignments(and8,000,000differentnormallydistributedfrequencyassign-
ments) on each. We used Theorem 3 to calculate critical couplings in each case, and
fitted a curve of the form a√n+b to the expected values. Empirically, as illustrated
SYNCHRONISATIONPROPERTIESOFTREES 7
in Figure 5, the expected critical coupling for a chain of n vertices is:
(18) E(kchain)=χ(n) 0.252√n 0.168=0.873σ√n 0.581σ
c ≈ − −
We express the result both in terms of σ, and for our special case of σ = 1/12. As
wewillseein 3.6,theexpectedcriticalcouplingforthechainactsasanupperbound
§ p
onE(k )fortreesingeneral. Tofacilitateuseinthisway,weabbreviatethisfunction
c
as χ(n). An upper bound on χ(n) itself is the expected maximum displacementfor n
steps given by Weiss [20, p. 192]. This is too high because it ignores the constraint
of returning to zero:
πn πn
(19) E(kchain)=χ(n) σ =
c ≤ 4 48
r r
Figure 5 shows this bound, together with the corresponding lower bound, which is
the expectedmaximumdisplacementforn/2steps. Thisis toolowbecauseitignores
the fact the peak deviation from zero can occur at either end of the chain:
πn πn
(20) E(kchain)=χ(n) σ =
c ≥ 8 96
r r
4
g
plin 3
u
o
c
al
c
d criti 2
e
ct
e
p
x
E
1
0 50 100 150 200 250 300
Order n
Fig. 5. Expected critical couplings for the chain, as a function of the number of vertices n.
ThesolidlineshowstheempiricalcurveE(kchain)=χ(n) 0.252√n 0.168, whilethedashed and
c ≈ −
dotted linesshows the random-walk bounds pπn/48and pπn/96. Results fornormally distributed
frequenciesare, as expected, virtually identical, and cannot be distinguished on thisplot.
3.2. Stars. Forstartrees,suchastheoneinFigure3(a),whereν isthecentral
1
vertex:
(21) kstar = max ω ω¯
c i=2...n| i− |
For our choice of frequencies, taken from a distribution with mean 1/2, the critical
coupling, which is the expected value of this maximum, is given approximately by:
1 1
(22) E(kstar) E max ω +E ω¯
c ≈ i i− 2 2 −
(cid:18) (cid:12) (cid:12)(cid:19) (cid:18)(cid:12) (cid:12)(cid:19)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
8 A.H.DEKKERANDR.TAYLOR
where the second term is small. The expected maximum of ω 1 , which is uni-
i− 2
formly distributed, is given approximately by:
(cid:12) (cid:12)
(cid:12) (cid:12)
1 n 2 n 2
(23) E max ω σ√3 − = −
i
i − 2 ≈ n 2n
(cid:18) (cid:12) (cid:12)(cid:19) (cid:18) (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
Thisusesthefactthattheex(cid:12)pectedm(cid:12) aximumofnindependentuniformdistributions
is [11]:
n 1
σ√3 −
n+1
(cid:18) (cid:19)
and the fact that the distributions are partially correlated, so that one of the ω can
i
be inferredfromω¯ and the other ω . In addition, the expected value ofω¯ differs from
i
1/2 by approximately:
1 2 1
(24) E ω¯ σ =
2 − ≈ nπ 6nπ
(cid:18)(cid:12) (cid:12)(cid:19) r r
(cid:12) (cid:12)
(cid:12) (cid:12)
By (22), (23), and (24), a goo(cid:12)d estim(cid:12) ate for the expected critical coupling E(kstar)
c
will be:
n 2 2 n 2 1
(25) E(kstar) σ√3 − +σ = − +
c ≈ n nπ 2n 6nπ
(cid:18) (cid:19) r r
Figure 6 shows that, for n 40, this is an excellent estimate of the actual critical
≥
couplings, which are shown by solid triangles. It can also be seen that, as n tends
to infinity, the value of E(kstar) approaches 1/2. The expected critical coupling for
c
the star acts as a lower bound for trees in general, since star trees have the smallest
possible partitions.
Forcomparison,withnormallydistributedfrequencies,theexpectedmaximumof
ω ω¯ can be very closely approximated using the expected value of the maximum
i
| − |
oftheabsolutevaluesofxindependentnormallydistributedrandomvariablesofunit
variance, which we write as µ(x). For x = 1, µ(1) = 2/π. For x 2, the value of
≥
µ(x) is slightlyhigher than (andconvergesto)the expected value ofthe maximumof
p
xindependentnormallydistributedrandomvariablesofunitvarianceandzeromean,
which can be roughly approximated by 2log(x) [11]. The function µ(x) can be
expressed more precisely using the inverse complementary error function:
p
1 1
(26) µ(x)=√2 (1 γ) erfc−1 +γ erfc−1
− x xe
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
where γ = 0.57721566490... is the Euler-Mascheroni Constant. However, we set
µ(0)=0 by definition.
The expected critical coupling for normally distributed frequencies, shown by
open triangles in Figure 6, is very closely approximated by:
µ(n 2)
σµ(n 2)= −
− √12
SYNCHRONISATIONPROPERTIESOFTREES 9
0
1.
8
0.
g
n
pli
u
o 6
al c 0.
c
criti
cted 0.4
e
p
x
E
2
0.
0
0.
0 50 100 150 200 250 300
Order n
Fig.6. Expected critical couplings for the star, as a functionof the numberof verticesn. The
thickersolidlineshowsthecurve(n 2)/(2n)+p1/(6nπ),whilethethinnerlineshowstheinfinite-
−
size limit of 1/2 for uniform distributions. For comparison, open triangles show expected critical
couplings for normally distributed frequencies, and the dashed line shows the estimator σµ(n 2).
−
s
al
m 5
or 3.
n
d
ndar 3.0
a
st
of x 2.5
s
e
valu 2.0
e
ut
ol 5
bs 1.
a
of
ax 1.0
m
d
ecte 0.5
p
x
e
) = 0.0
x
(
m 0 50 100 150 200 250 300
Number of standard normals x
Fig. 7. The function µ(x), the expected value of the maximum of the absolute values of x
independent normally distributed random variables of unit variance (solid line). The dashed line
shows the approximation p2log(x), while the dotted line shows µ(1)=p2/π.
3.3. Dumb-bells. For “dumb-bell” trees, such as the one in Figure 3 (b), the
critical coupling is the result of the combined effects of the n 2 leaves and the fact
−
that the tree canbe partitioned into two halves. The latter has the greaterinfluence,
10 A.H.DEKKERANDR.TAYLOR
so by Theorem 3:
n/2 n
(27) kdb (ω ω¯) = (ω ω¯)
c ≈(cid:12) i− (cid:12) (cid:12) i− (cid:12)
(cid:12)(cid:12)Xi=1 (cid:12)(cid:12) (cid:12)(cid:12)i=X1+n/2 (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
That is, the criticalcoupling(cid:12)(cid:12)kcdb is the de(cid:12)(cid:12)viat(cid:12)(cid:12)ion from ω¯ for th(cid:12)(cid:12)e averageof the ωi for
one half of the dumb-bell (without loss of generality, we will use ω ,...,ω ). To
1 n/2
calculate E(kdb), the expected value of this deviation, we note that:
c
n/2 n
1
(28) ω¯ = ω + ω
i i
n 
Xi=1 i=X1+n/2
 
Thus:
n/2 n/2 n n/2 n
n 1 1 1
(29) ω ω¯ = ω ω = ω ω
i i i i i
− 2 − 2 2 − 2
Xi=1 Xi=1 Xi=1 Xi=1 i=X1+n/2
The variance for this difference of sums is nσ2/4, and hence the expected critical
coupling will be:
2 nσ2 n n
(30) E(kdb) =σ =
c ≈ π 4 2π 24π
r r r r
The dashed line in Figure 8 shows this approximation. A better approximation can
be obtained by noting that when calculating the critical coupling from ω ω¯ for
i
| − |
half the dumb-bell gives a result of less than 1/2 = σ√3, then the critical coupling
will be determined by the effect of the leaves, in a similar way to the expected value
for the star. Since the variance for (29) is nσ2/4, the standard deviation is:
σ
(31) σ′ = √n
2
Using a first-order approximation, the probability that a normal distribution with
that standard deviation is within 1/2=σ√3 of the mean will be approximately:
σ√3 6
2 =2
σ′√2π nπ
r
The impact of the leaves is, with this probability, to replace an expected value of
about 1/4 = σ(√3)/2 by one of about 1/2 = σ√3, i.e. to increase the expected
critical coupling to:
n 6 √3
(32) E(kdb)=σ +2 σ
c 2π nπ 2
r r !
n 18 n 3
=σ +σ = +
2π nπ 24π 2nπ
r r r r
Figure 8 shows that this correction ensures an excellent fit for n 20. Note that as
≥
n increases, the correction factor tends to zero, and E(kdb) tends to the value given
c
by (30).