text: ∗ The motion of a neutral atom endowed with a magnetic moment interacting with the magnetic ﬁeld is determined from the Ehrenfest-like equations of motion. These equations for the average values of the translational and spin degrees of freedom are derived from the Schro¨dinger-Pauli wave equation and they form a set of nine coupled nonlinear evolution equations. The numerical and analytic solutions of these equations are obtained for the combination of the rotating magnetic ﬁeld of a wave carrying orbital angular momentum and a static magnetic ﬁeld. The running wave traps the atom only in the transverse direction while the standing wave traps the atom also in the direction of the beam. I. INTRODUCTION ratio is e/m . The classical evolution equations are uni- el versal; they do not depend on the value of the spin. There are three methods of trapping neutral atoms We shall study the case when the magnetic ﬁeld is a that have been in the past been described theoretically combination of the wave with the vortex line and a static and applied in many experiments (see, for example, the component. We consider two cases: the running wave reviews in [1–3]). These are: magnetic traps, radiation- and the standing wave. The magnetic component of the pressure traps, and optical dipole traps. In the present electromagnetic ﬁeld in these two cases is given by the paper we analyze a diﬀerent trap produced by the elec- formulas:   tromagnetic wave carrying orbital angular momentum.   The atomic magnetic moment interacts with the mag- − B⊥k(y cos ζ x sin ζ) netic ﬁeld of such waves as in standard magnetic traps. B (r, t) = B⊥k(x cos ζ + y sin ζ) , (2a) However, in contrast to the traps in which the magnetic run B ﬁeld is static, our trapping mechanism exploits in an es-  z −  sential way the rotation of the magnetic ﬁeld. Rotation of B⊥k cos(kz)[y cos(ωt) x sin(ωt)] the magnetic ﬁeld is necessary since owing to the Earn- Bst(r, t) = B⊥k cos(kz)[x cos(ωt) + y sin(ωt)] , shaw theorem static magnetic ﬁelds cannot trap parti- Bz cles with permanent magnetic moments. One method to (2b) overcome the limitations imposed by the Earnshaw theo- − rem is to use diamagnetic levitating objects [4]. Another where ζ = ωt kz, k = ω/c is the wave number, B⊥ method is(cid:114)to employ the rotation of the magnets as in measures the strength of the vortex wave, and Bz is the Levitrons [5, 6]. Our method may be viewed as an ap- constant ﬁeld. In order to preserve the correct dimension plication of this last idea to atomic objects. The rotation of B⊥ we inserted a factor of k in these formulas. The vor- of the atomic magnetic moment is achieved by placing the tex part can be viewed as the paraxial approximation of atom in the magnetic ﬁeld of an electromagnetic wave either a Bessel beam or a Laguerre-Gauss beam with or- endowed with orbital angular momentum. The rotating bital angular momentum quantum number equal to 1. Of magnetic ﬁeld of such a wave plays a similar role to the course, the magnetic ﬁelds (2), together with their elec- rotating electric ﬁeld in the Paul trap [7]. tric counterparts, are exact solutions of Maxwell’s equa- Our theoretical tool is the set of coupled Ehrenfest tions but the question is whether it is a good approxima- equations for the translational and spin degrees of free- tion of a realistic beam. One may explain this approxi- dom [8, 9]. They are derived from the Schr¨odinger-Pauli mation by starting from the exact formulas for the Bessel equation for a neutral particle endowed with the mag- or the Laguerre-Gauss beams. For Bessel beams the size netic moment µ = gs, (cid:18) (cid:19) of the waist is determined by the inverse of the trans- verse wave vector 1/k⊥ and for Laguerre-Gauss beams it (cid:126) is determined by the waist size parameter w (see, for ex- (cid:126) ∂ − 2∆ − · 0 i Ψ(r, t) = gs B Ψ(r, t), (1) ample [10]). The approximation leading to the formulas ∂t 2M (2) is essentially the paraxial approximation. It simply consists (cf. [11, 12]) of the replacement of the exact so- where g is the gyromagnetic ratio and the spin vector lutions by the ﬁrst term of the expansion in the following s is built from the appropriate spin matrices. The sign dimensionless parameter: the ratio of the distance from of g is positive or negative depending on whether the the beam center to the size of the beam waist. Therefore, magnetic moment is parallel or antiparallel to the spin our approximate formulas are valid for distances smaller angular momentum. For the electron the spin operator that the beam waist. (cid:126) contains Pauli matrices, s = /2 σ and the gyromagnetic Similar solutions of Maxwell equations have appeared before in our study of the trapping of charged particles by electromagnetic vortices [11–14]. However, this time in addition to a running wave, we consider also a standing ∗ wave. We show that the standing wave can trap particles ##LTLine## ##LTLine## ##LTLine##

text: The

label: abstract

text: motion

text: of

text: a

text: neutral

text: atom

text: endowed

text: with

text: a

text: magnetic

text: moment

text: interacting

text: with

text: the

text: magnetic

text: ﬁeld

text: is

text: determined

text: from

text: the

text: Ehrenfest-like

text: equations

text: of

text: motion.

text: These

text: equations

text: for

text: the

text: average

text: values

text: of

text: the

text: translational

text: and

text: spin

text: degrees

text: of

text: freedom

text: are

text: derived

text: from

text: the

text: Schro¨dinger-Pauli

text: wave

text: equation

text: and

text: they

text: form

text: a

text: set

text: of

text: nine

text: coupled

text: nonlinear

text: evolution

text: equations.

text: The

text: numerical

text: and

text: analytic

text: solutions

text: of

text: these

text: equations

text: are

text: obtained

text: for

text: the

text: combination

text: of

text: the

text: rotating

text: magnetic

text: ﬁeld

text: of

text: a

text: wave

text: carrying

text: orbital

text: angular

text: momentum

text: and

text: a

text: static

text: magnetic

text: ﬁeld.

text: The

text: running

text: wave

text: traps

text: the

text: atom

text: only

text: in

text: the

text: transverse

text: direction

text: while

text: the

text: standing

text: wave

text: traps

text: the

text: atom

text: also

text: in

text: the

text: direction

text: of

text: the

text: beam.

text: I.

text: INTRODUCTION

text: ratio

text: is

text: e/m

text: The

text: classical

text: evolution

text: equations

text: are

text: uni-

text: el

text: versal;

text: they

text: do

text: not

text: depend

text: on

text: the

text: value

text: of

text: the

text: spin.

text: There

text: are

text: three

text: methods

text: of

text: trapping

text: neutral

text: atoms

text: We

text: shall

text: study

text: the

text: case

text: when

text: the

text: magnetic

text: ﬁeld

text: is

text: a

text: that

text: have

text: been

text: in

text: the

text: past

text: been

text: described

text: theoretically

text: combination

text: of

text: the

text: wave

text: with

text: the

text: vortex

text: line

text: and

text: a

text: static

text: and

text: applied

text: in

text: many

text: experiments

text: (see,

text: for

text: example,

text: the

text: component.

text: We

text: consider

text: two

text: cases:

text: the

text: running

text: wave

text: reviews

text: in

text: These

text: are:

text: magnetic

text: traps,

text: radiation-

text: and

text: the

text: standing

text: wave.

text: The

text: magnetic

text: component

text: of

text: the

text: pressure

text: traps,

text: and

text: optical

text: dipole

text: traps.

text: In

text: the

text: present

text: electromagnetic

text: ﬁeld

text: in

text: these

text: two

text: cases

text: is

text: given

text: by

text: the

text: paper

text: we

text: analyze

text: a

text: diﬀerent

text: trap

text: produced

text: by

text: the

text: elec-

text: formulas:

text: tromagnetic

text: wave

text: carrying

text: orbital

text: angular

text: momentum.

text: The

text: atomic

text: magnetic

text: moment

text: interacts

text: with

text: the

text: mag-

text: B⊥k(y

text: cos

text: ζ

text: x

text: sin

text: ζ)

text: netic

text: ﬁeld

text: of

text: such

text: waves

text: as

text: in

text: standard

text: magnetic

text: traps.

text: B

text: (r,

text: t)

text: B⊥k(x

text: cos

text: ζ

text: y

text: sin

text: ζ)

text: However,

text: in

text: contrast

text: to

text: the

text: traps

text: in

text: which

text: the

text: magnetic

text: run

text: B

text: ﬁeld

text: is

text: static,

text: our

text: trapping

text: mechanism

text: exploits

text: in

text: an

text: es-

text: z

text: sential

text: way

text: the

text: rotation

text: of

text: the

text: magnetic

text: ﬁeld.

text: Rotation

text: of

text: B⊥k

text: cos(kz)[y

text: cos(ωt)

text: x

text: sin(ωt)]

text: the

text: magnetic

text: ﬁeld

text: is

text: necessary

text: since

text: owing

text: to

text: the

text: Earn-

text: Bst(r,

text: t)

text: B⊥k

text: cos(kz)[x

text: cos(ωt)

text: y

text: sin(ωt)]

text: shaw

text: theorem

text: static

text: magnetic

text: ﬁelds

text: cannot

text: trap

text: parti-

text: Bz

text: cles

text: with

text: permanent

text: magnetic

text: moments.

text: One

text: method

text: to

text: overcome

text: the

text: limitations

text: imposed

text: by

text: the

text: Earnshaw

text: theo-

text: rem

text: is

text: to

text: use

text: diamagnetic

text: levitating

text: objects

text: Another

text: where

text: ζ

text: ωt

text: kz,

text: k

text: ω/c

text: is

text: the

text: wave

text: number,

text: B⊥

text: method

text: employ

text: the

text: rotation

text: of

text: the

text: magnets

text: as

text: in

text: measures

text: the

text: strength

text: of

text: the

text: vortex

text: wave,

text: and

text: Bz

text: is

text: the

text: Levitrons

text: Our

text: method

text: may

text: be

text: viewed

text: as

text: an

text: ap-

text: constant

text: ﬁeld.

text: In

text: order

text: to

text: preserve

text: the

text: correct

text: dimension

text: plication

text: of

text: this

text: last

text: idea

text: to

text: atomic

text: objects.

text: The

text: rotation

text: of

text: B⊥

text: we

text: inserted

text: a

text: factor

text: of

text: k

text: in

text: these

text: formulas.

text: The

text: vor-

text: of

text: the

text: atomic

text: magnetic

text: moment

text: is

text: achieved

text: by

text: placing

text: the

text: tex

text: part

text: can

text: be

text: viewed

text: as

text: the

text: paraxial

text: approximation

text: of

text: atom

text: in

text: the

text: magnetic

text: ﬁeld

text: of

text: an

text: electromagnetic

text: wave

text: either

text: a

text: Bessel

text: beam

text: or

text: a

text: Laguerre-Gauss

text: beam

text: with

text: or-

text: endowed

text: with

text: orbital

text: angular

text: momentum.

text: The

text: rotating

text: bital

text: angular

text: momentum

text: quantum

text: number

text: equal

text: to

text: Of

text: magnetic

text: ﬁeld

text: of

text: such

text: a

text: wave

text: plays

text: a

text: similar

text: role

text: to

text: the

text: course,

text: the

text: magnetic

text: ﬁelds

text: together

text: with

text: their

text: elec-

text: rotating

text: electric

text: ﬁeld

text: in

text: the

text: Paul

text: trap

text: tric

text: counterparts,

text: are

text: exact

text: solutions

text: of

text: Maxwell’s

text: equa-

text: Our

text: theoretical

text: tool

text: is

text: the

text: set

text: of

text: coupled

text: Ehrenfest

text: tions

text: but

text: the

text: question

text: is

text: whether

text: it

text: is

text: a

text: good

text: approxima-

text: equations

text: for

text: the

text: translational

text: and

text: spin

text: degrees

text: of

text: free-

text: tion

text: of

text: a

text: realistic

text: beam.

text: One

text: may

text: explain

text: this

text: approxi-

text: dom

text: They

text: are

text: derived

text: from

text: the

text: Schr¨odinger-Pauli

text: mation

text: by

text: starting

text: from

text: the

text: exact

text: formulas

text: for

text: the

text: Bessel

text: equation

text: for

text: a

text: neutral

text: particle

text: endowed

text: with

text: the

text: mag-

text: or

text: the

text: Laguerre-Gauss

text: beams.

text: For

text: Bessel

text: beams

text: the

text: size

text: netic

text: moment

text: µ

text: gs,

text: of

text: the

text: waist

text: is

text: determined

text: by

text: the

text: inverse

text: of

text: the

text: trans-

text: verse

text: wave

text: vector

text: and

text: for

text: Laguerre-Gauss

text: beams

text: it

text: is

text: determined

text: by

text: the

text: waist

text: size

text: parameter

text: w

text: (see,

text: for

text: ex-

text: 2∆

text: i

text: Ψ(r,

text: t)

text: gs

text: B

text: Ψ(r,

text: t),

text: ample

text: The

text: approximation

text: leading

text: to

text: the

text: formulas

text: ∂t

text: is

text: essentially

text: the

text: paraxial

text: approximation.

text: It

text: simply

text: consists

text: (cf.

text: of

text: the

text: replacement

text: of

text: the

text: exact

text: so-

text: where

text: g

text: is

text: the

text: gyromagnetic

text: ratio

text: and

text: the

text: spin

text: vector

text: lutions

text: by

text: the

text: ﬁrst

text: term

text: of

text: the

text: expansion

text: in

text: the

text: following

text: s

text: is

text: built

text: from

text: the

text: appropriate

text: spin

text: matrices.

text: The

text: sign

text: dimensionless

text: parameter:

text: the

text: ratio

text: of

text: the

text: distance

text: from

text: of

text: g

text: is

text: positive

text: or

text: negative

text: depending

text: on

text: whether

text: the

text: the

text: beam

text: center

text: to

text: the

text: size

text: of

text: the

text: beam

text: waist.

text: Therefore,

text: magnetic

text: moment

text: is

text: parallel

text: or

text: antiparallel

text: to

text: the

text: spin

text: our

text: approximate

text: formulas

text: are

text: valid

text: for

text: distances

text: smaller

text: angular

text: momentum.

text: For

text: the

text: electron

text: the

text: spin

text: operator

text: that

text: the

text: beam

text: waist.

text: contains

text: Pauli

text: matrices,

text: s

text: σ

text: and

text: the

text: gyromagnetic

text: Similar

text: solutions

text: of

text: Maxwell

text: equations

text: have

text: appeared

text: before

text: in

text: our

text: study

text: of

text: the

text: trapping

text: of

text: charged

text: particles

text: by

text: electromagnetic

text: vortices

text: However,

text: this

text: time

text: in

text: addition

text: to

text: a

text: running

text: wave,

text: we

text: consider

text: also

text: a

text: standing

text: wave.

text: We

text: show

text: that

text: the

text: standing

text: wave

text: can

text: trap

text: particles

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: cavity (inner radius R1 = 35 nm and outer radius R2 = 70 nm) where the MD resonates at point F (λF = 411 nm). However at this point the scattering magnitude of ED is much smaller, rendering signiﬁcant backward scattering, as can be observed in the scattering patterns shown in Fig. 4(b). At the same time, with a low index dielectric layer, the EDs can be efﬁciently excited at small wavelength, while under such circumstance the MD can not be efﬁciently excited, making the scattering pattern a typical ED type with equal forward and backward scattering. This is demonstrated in Fig. 4(c) where we show the scattering spectra of a silver-dielectric (ε1 = 1.22) core-shell resonator (R1 = 42 nm, R2 = 80 nm). Here the ED resonates at point G (λG = 411 nm), where however the MD excitation is negligible. As a result, the scattering is contributed only by ED and thus not unidirectional [see the scattering patterns shown in Fig. 4(d)]. The challenge of overlapping EDs and MDs at small wavelengths close to the plasmon fre- quency is not insurmountable for multilayered plasmonic cavities of effective radial anisotropy. we employ a multilayered plasmonic cavity consisting of 16 units: each unit is made up of a silver layer of width 1 nm and a isotropic dielectric layer (ε1 = 32) of width 4 nm, and thus f = 0.2 and overall radius of the cavity R = 80 nm. In Fig. 4(e) we show the scattering spectra of such a plasmonic multilayered cavity. It is clear that ED and MD can be made to resonantly overlap at point H (λH = 411 nm), which induces unidirectional forward superscattering [see the scattering patterns shown in Fig. 4(f)]. 5. Conclusions and discussions To conclude, in this work we study the plane wave scattering by multilayered cavities that possess large effective radial anisotropy. In such cavities, relying on the effective radial anisotropy, the EDs and MDs can be tuned to resonantly overlap, which thus satisﬁes the Kerker’s condition and produces unidirectional forward superscattering. It is demonstrated that such scattering shaping can be realized in both all-dielectric and plasmonic multilayered cav- ities. Moreover it is shown that in plasmonic cavities of effective radial anisotropy, the EDs and MDs can be made to resonantly overlap at small wavelengths in the violet spectral regime, which is not accessible for simple two layered metal-dielectric core-shell resonators. We note here that in this work we have conﬁned our studies to dipolar modes, and based on the same approach higher order modes can be made to overlap, which can produced more collimated forward superscattering [28]. Also, we have studied only the case of η > 0, and quite naturally hyperbolic cavities of η < 0 can be studied using the same method [31,33]. Moreover, the prin- ciple we have revealed is general, which is also applicable to resonators of other shapes and to other kinds of anionotropy such as magnetic anisotropy or the intrinsic huge anisotropy of 2D materials. Such mechanism of resonance control based on effective anisotropy can shed new light to many particle scattering problems and is quite promising for various applications in the ﬁelds of nanoantennas, solar cells, bio-sensing and so on. Acknowledgments We thank Andrey E. Miroshnichenko and Yuri S. Kivshar for valuable discussions. We also acknowledge the ﬁnancial support from the National Natural Science Foundation of China (Grant number: 11404403) and the Basic Research Scheme of College of Optoelectronic Sci- ence and Engineering, National University of Defense Technology.

text: cavity

label: paragraph

text: (inner

text: radius

text: nm

text: and

text: outer

text: radius

text: nm)

text: where

text: the

text: MD

text: resonates

text: at

text: point

text: F

text: (λF

text: nm).

text: However

text: at

text: this

text: point

text: the

text: scattering

text: magnitude

text: of

text: ED

text: is

text: much

text: smaller,

text: rendering

text: signiﬁcant

text: backward

text: scattering,

text: as

text: can

text: be

text: observed

text: in

text: the

text: scattering

text: patterns

text: shown

text: in

text: Fig.

text: At

text: the

text: same

text: time,

text: with

text: a

text: low

text: index

text: dielectric

text: layer,

text: the

text: EDs

text: can

text: be

text: efﬁciently

text: excited

text: at

text: small

text: wavelength,

text: while

text: under

text: such

text: circumstance

text: the

text: MD

text: can

text: not

text: be

text: efﬁciently

text: excited,

text: making

text: the

text: scattering

text: pattern

text: a

text: typical

text: ED

text: type

text: with

text: equal

text: forward

text: and

text: backward

text: scattering.

text: This

text: is

text: demonstrated

text: in

text: Fig.

text: where

text: we

text: show

text: the

text: scattering

text: spectra

text: of

text: a

text: silver-dielectric

text: core-shell

text: resonator

text: nm,

text: nm).

text: Here

text: the

text: ED

text: resonates

text: at

text: point

text: G

text: (λG

text: nm),

text: where

text: however

text: the

text: MD

text: excitation

text: is

text: negligible.

text: As

text: a

text: result,

text: the

text: scattering

text: is

text: contributed

text: only

text: by

text: ED

text: and

text: thus

text: not

text: unidirectional

text: [see

text: the

text: scattering

text: patterns

text: shown

text: in

text: Fig.

text: The

text: challenge

text: of

text: overlapping

text: EDs

text: and

text: MDs

text: at

text: small

text: wavelengths

text: close

text: to

text: the

text: plasmon

text: fre-

text: quency

text: is

text: not

text: insurmountable

text: for

text: multilayered

text: plasmonic

text: cavities

text: of

text: effective

text: radial

text: anisotropy.

text: we

text: employ

text: a

text: multilayered

text: plasmonic

text: cavity

text: consisting

text: of

text: units:

text: each

text: unit

text: is

text: made

text: up

text: of

text: a

text: silver

text: layer

text: of

text: width

text: nm

text: and

text: a

text: isotropic

text: dielectric

text: layer

text: of

text: width

text: nm,

text: and

text: thus

text: f

text: and

text: overall

text: radius

text: of

text: the

text: cavity

text: R

text: nm.

text: In

text: Fig.

text: we

text: show

text: the

text: scattering

text: spectra

text: of

text: such

text: a

text: plasmonic

text: multilayered

text: cavity.

text: It

text: is

text: clear

text: that

text: ED

text: and

text: MD

text: can

text: be

text: made

text: to

text: resonantly

text: overlap

text: at

text: point

text: H

text: (λH

text: nm),

text: which

text: induces

text: unidirectional

text: forward

text: superscattering

text: [see

text: the

text: scattering

text: patterns

text: shown

text: in

text: Fig.

text: Conclusions

text: and

text: discussions

text: To

text: conclude,

text: in

text: this

text: work

text: we

text: study

text: the

text: plane

text: wave

text: scattering

text: by

text: multilayered

text: cavities

text: that

text: possess

text: large

text: effective

text: radial

text: anisotropy.

text: In

text: such

text: cavities,

text: relying

text: on

text: the

text: effective

text: radial

text: anisotropy,

text: the

text: EDs

text: and

text: MDs

text: can

text: be

text: tuned

text: to

text: resonantly

text: overlap,

text: which

text: thus

text: satisﬁes

text: the

text: Kerker’s

text: condition

text: and

text: produces

text: unidirectional

text: forward

text: superscattering.

text: It

text: is

text: demonstrated

text: that

text: such

text: scattering

text: shaping

text: can

text: be

text: realized

text: in

text: both

text: all-dielectric

text: and

text: plasmonic

text: multilayered

text: cav-

text: ities.

text: Moreover

text: it

text: is

text: shown

text: that

text: in

text: plasmonic

text: cavities

text: of

text: effective

text: radial

text: anisotropy,

text: the

text: EDs

text: and

text: MDs

text: can

text: be

text: made

text: to

text: resonantly

text: overlap

text: at

text: small

text: wavelengths

text: in

text: the

text: violet

text: spectral

text: regime,

text: which

text: is

text: not

text: accessible

text: for

text: simple

text: two

text: layered

text: metal-dielectric

text: core-shell

text: resonators.

text: We

text: note

text: here

text: that

text: in

text: this

text: work

text: we

text: have

text: conﬁned

text: our

text: studies

text: to

text: dipolar

text: modes,

text: and

text: based

text: on

text: the

text: same

text: approach

text: higher

text: order

text: modes

text: can

text: be

text: made

text: to

text: overlap,

text: which

text: can

text: produced

text: more

text: collimated

text: forward

text: superscattering

text: Also,

text: we

text: have

text: studied

text: only

text: the

text: case

text: of

text: η

text: and

text: quite

text: naturally

text: hyperbolic

text: cavities

text: of

text: η

text: can

text: be

text: studied

text: using

text: the

text: same

text: method

text: Moreover,

text: the

text: prin-

text: ciple

text: we

text: have

text: revealed

text: is

text: general,

text: which

text: is

text: also

text: applicable

text: to

text: resonators

text: of

text: other

text: shapes

text: and

text: to

text: other

text: kinds

text: of

text: anionotropy

text: such

text: as

text: magnetic

text: anisotropy

text: or

text: the

text: intrinsic

text: huge

text: anisotropy

text: of

text: materials.

text: Such

text: mechanism

text: of

text: resonance

text: control

text: based

text: on

text: effective

text: anisotropy

text: can

text: shed

text: new

text: light

text: to

text: many

text: particle

text: scattering

text: problems

text: and

text: is

text: quite

text: promising

text: for

text: various

text: applications

text: in

text: the

text: ﬁelds

text: of

text: nanoantennas,

text: solar

text: cells,

text: bio-sensing

text: and

text: so

text: on.

text: Acknowledgments

text: We

text: thank

text: Andrey

text: E.

text: Miroshnichenko

text: and

text: Yuri

text: S.

text: Kivshar

text: for

text: valuable

text: discussions.

text: We

text: also

text: acknowledge

text: the

text: ﬁnancial

text: support

text: from

text: the

text: National

text: Natural

text: Science

text: Foundation

text: of

text: China

text: (Grant

text: number:

text: and

text: the

text: Basic

text: Research

text: Scheme

text: of

text: College

text: of

text: Optoelectronic

text: Sci-

text: ence

text: and

text: Engineering,

text: National

text: University

text: of

text: Defense

text: Technology.


text: 8 t =t t =0 v 3 30 ∆=0.1t ∆=0.5t 25 20 15 10 ∆=10 t 5 6 0 5.95 5.9 5 5.85 0 1 0 1 0 1 φ/φ φ/φ φ/φ 0 0 0 FIG. 7. Self similar structure of the entanglement spectrum of fermionic BLH lattice versus reduced magnetic ﬂux, at t = 0, ≤ ≤ 3 in the presence of on-site energy diﬀerence, for 0 p q = 51. A transition from butterﬂy to a tree-like picture is shown to be occurred for non-zero ∆. A. Butterﬂy entanglement spectrum (cid:113) (cid:113) − ∗ ∗ (ξ /v )α (ξ+ /v )β In order to obtain the entanglement spectrum of the a = m(cid:126)k m(cid:126)k m(cid:126)k + m(cid:126)k m(cid:126)k m(cid:126)k , non-interacting free fermions model on BLH lattice in the m(cid:126)k (cid:113)1 + (ξm−(cid:126)k/|vm(cid:126)k|)2 (cid:113)1 + (ξm+(cid:126)k/|vm(cid:126)k|)2 presence of a uniform magnetic ﬁeld we calculate again α β the single particle correlations. The non-zero single par- b = m(cid:126)k + m(cid:126)k ,(21) ticle correlations read m(cid:126)k 1 + (ξ− /|v |)2 1 + (ξ+ /|v |)2 m(cid:126)k m(cid:126)k m(cid:126)k m(cid:126)k the Hamilton(cid:88)ian (19) is diagonalized as (cid:104) | † | (cid:105) (cid:104) | † | (cid:105) ψ a a ψ , ψ b b ψ , m m(cid:126)k m(cid:126)k m m m(cid:126)k m(cid:126)k m (cid:104) | † | (cid:105) (cid:104) | † | (cid:105) H † − † ψm am(cid:126)kbm(cid:126)k ψm , ψm bm(cid:126)kam(cid:126)k ψm , (18) md = (ξm+(cid:126)kαm(cid:126)kαm(cid:126)k + ξm(cid:126)kβm(cid:126)kβm(cid:126)k). (22) (cid:126)k ± † † Here, ξ are the entanglement spectra which are given where am(cid:126)k and bm(cid:126)k (am(cid:126)k and bm(cid:126)k) are creation (an- by m(cid:126)k nihilation) operators of fermions on the lower layer and | (cid:105) ψ is the ground state of H . Using these non-zero m m(cid:126)k correlations one can build the entanglement Hamiltonian ± ± ξ = 2arctanh(2n ), (23) of each structural unit cell as m(cid:126)k m(cid:126)k (cid:88) where (cid:18) H † † † = ua a a +ub b b +(v a b +h.c.), m m(cid:126)k m(cid:126)k m(cid:126)k m(cid:126)k m(cid:126)k m(cid:126)k m(cid:126)k m(cid:126)k m(cid:126)k (cid:126)k ± (cid:113)1 (cid:104) † (cid:105) (cid:104) † (cid:105) (cid:19) (19) n = a a + b b m(cid:126)k 2 m(cid:126)k m(cid:126)k m(cid:126)k m(cid:126)k where the coeﬃcients ua , ub , and v could be ob- ± (cid:104) † (cid:105) − (cid:104) † (cid:105) |(cid:104) † (cid:105)| tained in terms of the nonmz(cid:126)kero msi(cid:126)nkgle parmti(cid:126)ckle correlation ( am(cid:126)kam(cid:126)k bm(cid:126)kbm(cid:126)k )2 + 4 am(cid:126)kbm(cid:126)k 2 . functions (18). By deﬁning (cid:16) (cid:113) (cid:17) In the absence of on-site energy diﬀerence, at zero skew hopping the ES versus reduced magnetic ﬂux φ/φ , is 0 ± 1 − ± − | | perfectly symmetric with respect to ξ = 0, with a self ξ = ua ub (ua ub )2 + 4 v 2 , m(cid:126)k 2 m(cid:126)k m(cid:126)k m(cid:126)k m(cid:126)k m(cid:126)k similar structure like Hofstadter butterﬂy. Increasing in- − (20) terlayer vertical hopping causes ξ+ and ξ get closer to each other so that at large values of t∼v they collapse to and using the following unitary transformation: ξ = 0. The butterﬂy-like ES at t = 2.82t is exactly v ξ ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: t

font: default

text: =t

text: t

text: v

text: t

text: φ/φ

text: φ/φ

text: φ/φ

text: FIG.

text: Self

label: caption

text: similar

text: structure

text: of

text: the

text: entanglement

text: spectrum

text: of

text: fermionic

text: BLH

text: lattice

text: versus

text: reduced

text: magnetic

text: ﬂux,

text: at

text: t

text: in

text: the

text: presence

text: of

text: on-site

text: energy

text: diﬀerence,

text: for

text: p

text: q

text: A

text: transition

text: from

text: butterﬂy

text: to

text: a

text: tree-like

text: picture

text: is

text: shown

text: to

text: be

text: occurred

text: for

text: non-zero

text: A.

text: Butterﬂy

text: entanglement

text: spectrum

text: (ξ

text: /v

text: )α

text: (ξ+

text: /v

text: )β

text: In

text: order

text: to

text: obtain

text: the

text: entanglement

text: spectrum

text: of

text: the

text: a

text: non-interacting

text: free

text: fermions

text: model

text: on

text: BLH

text: lattice

text: in

text: the

text: presence

text: of

text: a

text: uniform

text: magnetic

text: ﬁeld

text: we

text: calculate

text: again

text: α

text: β

text: the

text: single

text: particle

text: correlations.

text: The

text: non-zero

text: single

text: par-

text: b

text: ticle

text: correlations

text: read

text: (ξ−

text: /|v

text: (ξ+

text: /|v

text: the

text: is

text: diagonalized

text: as

text: ψ

text: a

text: a

text: ψ

text: ψ

text: b

text: b

text: ψ

text: m

text: m

text: m

text: m

text: H

text: ψm

text: ψm

text: ψm

text: ψm

text: md

text: Here,

text: ξ

text: are

text: the

text: entanglement

text: spectra

text: which

text: are

text: given

text: where

text: and

text: and

text: are

text: creation

text: (an-

text: by

text: nihilation)

text: operators

text: of

text: fermions

text: on

text: the

text: lower

text: layer

text: and

text: ψ

text: is

text: the

text: ground

text: state

text: of

text: H

text: Using

text: these

text: non-zero

text: m

text: correlations

text: one

text: can

text: build

text: the

text: entanglement

text: Hamiltonian

text: ξ

text: of

text: each

text: structural

text: unit

text: cell

text: as

text: where

text: H

text: ua

text: a

text: a

text: +ub

text: b

text: b

text: +(v

text: a

text: b

text: +h.c.),

text: m

text: n

text: a

text: a

text: b

text: b

text: where

text: the

text: coeﬃcients

text: ua

text: ub

text: and

text: v

text: could

text: be

text: ob-

text: tained

text: in

text: terms

text: of

text: the

text: correlation

text: functions

text: By

text: deﬁning

text: In

text: the

text: absence

text: of

text: on-site

text: energy

text: diﬀerence,

text: at

text: zero

text: skew

text: hopping

text: the

text: ES

text: versus

text: reduced

text: magnetic

text: ﬂux

text: φ/φ

text: is

text: perfectly

text: symmetric

text: with

text: respect

text: to

text: ξ

text: with

text: a

text: self

text: ξ

text: ua

text: ub

text: (ua

text: ub

text: v

text: similar

text: structure

text: like

text: Hofstadter

text: butterﬂy.

text: Increasing

text: in-

text: terlayer

text: vertical

text: hopping

text: causes

text: ξ+

text: and

text: ξ

text: get

text: closer

text: to

text: each

text: other

text: so

text: that

text: at

text: large

text: values

text: of

text: t∼v

text: they

text: collapse

text: to

text: and

text: using

text: the

text: following

text: unitary

text: transformation:

text: ξ

text: The

text: butterﬂy-like

text: ES

text: at

text: t

text: is

text: exactly

text: v

text: ξ

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 Numerical study 3.1 Design setup § In the numerical study in this section and the following simulation study in , we focus on irregularly spaced data with an unstructured covariance matrix (Sun and Stein 2014). The obser- − − − ∈ { } vations are generated at the locations n 1/2(r 0.5 + X , (cid:96) 0.5 + Y ) for r, (cid:96) 1, . . . , n1/2 , r(cid:96) r(cid:96) where n is the number of locations, and X s and Y s are independent and identically distributed, r(cid:96) r(cid:96) − uniform on ( 0.4, 0.4). The advantage of this design is that it is irregular, and we can guarantee that no two locations are too close. § § Here, we study the performances of different approximation methods proposed in and in different settings. We consider a zero-mean Gaussian process model with Mate´rn covariance function possibly with a nugget, { } { } { − } C(h; α, β, ν, τ2) = α (2ν)1/2h/β νK (2ν)1/2h/β / Γ(ν)2ν 1 + τ21(h = 0), (5) ν · · where K ( ) is the modiﬁed Bessel function of the second kind of order ν, Γ( ) is the gamma ν · ≥ function, 1( ) is the indicator function, h 0 is the distance between two locations, α > 0 is the sill parameter, β > 0 is the range parameter, ν > 0 is the smoothness parameter, and τ2 is the nugget effect. For n irregularly spaced locations, the description of the ﬁve methods considered is shown in Table 1. § In –, we present the Kullback–Leibler divergence calculated from different settings for the § ﬁve methods with α ﬁxed at 1 and n = 900. In , we discuss the effect of sample size n and the rank r. 3.2 Dependence level In the Mate´rn model in equation (5), the range parameter β controls the dependence of the 11

label: section

text: Numerical

text: study

text: Design

text: setup

text: In

text: the

text: numerical

text: study

text: in

text: this

text: section

text: and

text: the

text: following

text: simulation

text: study

text: in

text: we

text: focus

text: on

text: irregularly

text: spaced

text: data

text: with

text: an

text: unstructured

text: covariance

text: matrix

text: (Sun

text: and

text: Stein

text: The

text: obser-

text: vations

text: are

text: generated

text: at

text: the

text: locations

text: n

text: X

text: Y

text: for

text: r,

text: where

text: n

text: is

text: the

text: number

text: of

text: locations,

text: and

text: X

text: s

text: and

text: Y

text: s

text: are

text: independent

text: and

text: identically

text: distributed,

text: uniform

text: on

text: The

text: advantage

text: of

text: this

text: design

text: is

text: that

text: it

text: is

text: irregular,

text: and

text: we

text: can

text: guarantee

text: that

text: no

text: two

text: locations

text: are

text: too

text: close.

text: Here,

text: we

text: study

text: the

text: performances

text: of

text: different

text: approximation

text: methods

text: proposed

text: in

text: and

text: in

text: different

text: settings.

text: We

text: consider

text: a

text: zero-mean

text: Gaussian

text: process

text: model

text: with

text: Mate´rn

text: covariance

text: function

text: possibly

text: with

text: a

text: nugget,

text: C(h;

text: α,

text: β,

text: ν,

text: α

text: νK

text: Γ(ν)2ν

text: ν

text: where

text: K

text: is

text: the

text: modiﬁed

text: Bessel

text: function

text: of

text: the

text: second

text: kind

text: of

text: order

text: ν,

text: Γ(

text: is

text: the

text: gamma

text: ν

text: function,

text: is

text: the

text: indicator

text: function,

text: h

text: is

text: the

text: distance

text: between

text: two

text: locations,

text: α

text: is

text: the

text: sill

text: parameter,

text: β

text: is

text: the

text: range

text: parameter,

text: ν

text: is

text: the

text: smoothness

text: parameter,

text: and

text: τ2

text: is

text: the

text: nugget

text: effect.

text: For

text: n

text: irregularly

text: spaced

text: locations,

text: the

text: description

text: of

text: the

text: ﬁve

text: methods

text: considered

text: is

text: shown

text: in

text: Table

text: In

text: we

text: present

text: the

text: Kullback–Leibler

text: divergence

text: calculated

text: from

text: different

text: settings

text: for

text: the

text: ﬁve

text: methods

text: with

text: α

text: ﬁxed

text: at

text: and

text: n

text: In

text: we

text: discuss

text: the

text: effect

text: of

text: sample

text: size

text: n

text: and

text: the

text: rank

text: r.

text: Dependence

text: level

text: In

text: the

text: Mate´rn

text: model

text: in

text: equation

text: the

text: range

text: parameter

text: β

text: controls

text: the

text: dependence

text: of

text: the


text: Table 1: Description of the ﬁve methods used in the numerical study. IND, independent blocks method; NN, nearest neighbors method; SUM, nearest neighboring sets method; NNSUM, near- est neighbors and nearest neighboring sets method; HLR, the hierarchical low rank approximation method. Method Description (cid:100) (cid:101) IND Divide the locations into n/r blocks, each of which contains r (cid:100) (cid:101) points. n/r means the largest integer that is no larger than n/r. NN A number of r nearest neighbors are selected to construct A . j,r SUM A number of r nearest neighboring sets are selected and each set has 2 locations. Then a total number of 2r nearest neighbors are used to construct A . j,r (cid:100) (cid:101) NNSUM A number of r/2 nearest neighbors are ﬁrst selected, then the fol- − (cid:100) (cid:101) − (cid:100) (cid:101) lowing 2(r r/2 ) nearest neighbors are divided into r r/2 sets of size 2. HLR A number of 2r nearest neighbors are considered, where L is a j × r r diagonal matrix with elements corresponding to the r leading eigenvalues. P consists of the r corresponding eigenvectors. process. In this section, we consider different β. Given ν = 0.5, which corresponds to an exponential covariance function and τ2 = 0.15, the ﬁrst row of Fig. 2 shows the Kullback–Leibler divergence for β = 0.1, which means a weaker dependence, and β = 0.5, which indicates a stronger dependence, as the rank r increases from 2 to 8. We can see that the HLR approximation is always the best with the smallest Kullback–Leibler divergence, and SUM and NNSUM win against NN only when r = 2 for β = 0.1, while for β = 0.5, the improvement of SUM and NNSUM exists up to r = 6. It implies that when a strong correlation is present, a small number 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Description

text: of

text: the

text: ﬁve

text: methods

text: used

text: in

text: the

text: numerical

text: study.

text: IND,

text: independent

text: blocks

text: method;

text: NN,

text: nearest

text: neighbors

text: method;

text: SUM,

text: nearest

text: neighboring

text: sets

text: method;

text: NNSUM,

text: near-

text: est

text: neighbors

text: and

text: nearest

text: neighboring

text: sets

text: method;

text: HLR,

text: the

text: hierarchical

text: low

text: rank

text: approximation

text: method.

text: Method

label: table

text: Description

text: IND

text: Divide

text: the

text: locations

text: into

text: n/r

text: blocks,

text: each

text: of

text: which

text: contains

text: r

text: points.

text: n/r

text: means

text: the

text: largest

text: integer

text: that

text: is

text: no

text: larger

text: than

text: n/r.

text: NN

text: A

text: number

text: of

text: r

text: nearest

text: neighbors

text: are

text: selected

text: to

text: construct

text: A

text: j,r

text: SUM

text: A

text: number

text: of

text: r

text: nearest

text: neighboring

text: sets

text: are

text: selected

text: and

text: each

text: set

text: has

text: locations.

text: Then

text: a

text: total

text: number

text: of

text: nearest

text: neighbors

text: are

text: used

text: to

text: construct

text: A

text: j,r

text: NNSUM

text: A

text: number

text: of

text: nearest

text: neighbors

text: are

text: ﬁrst

text: selected,

text: then

text: the

text: fol-

text: lowing

text: nearest

text: neighbors

text: are

text: divided

text: into

text: r

text: sets

text: of

text: size

text: HLR

text: A

text: number

text: of

text: nearest

text: neighbors

text: are

text: considered,

text: where

text: L

text: is

text: a

text: j

text: r

text: r

text: diagonal

text: matrix

text: with

text: elements

text: corresponding

text: to

text: the

text: r

text: leading

text: eigenvalues.

text: P

text: consists

text: of

text: the

text: r

text: corresponding

text: eigenvectors.

text: process.

text: In

text: this

text: section,

text: we

text: consider

text: different

text: β.

text: Given

text: ν

text: which

text: corresponds

text: to

text: an

text: exponential

text: covariance

text: function

text: and

text: τ2

text: the

text: ﬁrst

text: row

text: of

text: Fig.

text: shows

text: the

text: Kullback–Leibler

text: divergence

text: for

text: β

text: which

text: means

text: a

text: weaker

text: dependence,

text: and

text: β

text: which

text: indicates

text: a

text: stronger

text: dependence,

text: as

text: the

text: rank

text: r

text: increases

text: from

text: to

text: We

text: can

text: see

text: that

text: the

text: HLR

text: approximation

text: is

text: always

text: the

text: best

text: with

text: the

text: smallest

text: Kullback–Leibler

text: divergence,

text: and

text: SUM

text: and

text: NNSUM

text: win

text: against

text: NN

text: only

text: when

text: r

text: for

text: β

text: while

text: for

text: β

text: the

text: improvement

text: of

text: SUM

text: and

text: NNSUM

text: exists

text: up

text: to

text: r

text: It

text: implies

text: that

text: when

text: a

text: strong

text: correlation

text: is

text: present,

text: a

text: small

text: number

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: τ2 = 0.15. We run the optimization for α, β, τ2 while ﬁxing ν at the true value and obtain the estimates of α, β, τ2 by maximizing the approximated likelihoods with r = 2. We repeat the estimates procedure 500 times and the boxplots of α and β are shown in Fig. 4. We see that the NN NNSUM SUM HLR MLE NN NNSUM SUM HLR MLE Figure 4: Two panels showing the boxplot of parameter estimates and mean squared error times 10 (left) or 1000 (right). The solid line is a reference for the true parameter value, and the dash line is the corresponding mean squared error of the 500 number of estimates in each method. Left: illustration for estimated α; Right: illustration for estimated β. estimates obtained by the hierarchical low rank approximation method have the smallest mean squared error among the 3 approximation methods and are close to the exact maximum likelihood estimation. 5 Application 5.1 Dataset description In this section, we apply our method to modeling soil moisture, a key factor in evaluating 16 α detamitse 8.1 6.1 4.1 2.1 0.1 8.0 9 8 7 6 5 4 01 × ESM β detamitse 04.0 53.0 03.0 52.0 02.0 51.0 01.0 50.0 0.4 5.3 0.3 5.2 0.2 5.1 0.1 0001 × ESM ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: τ2

text: We

text: run

text: the

text: optimization

text: for

text: α,

text: β,

text: τ2

text: while

text: ﬁxing

text: ν

text: at

text: the

text: true

text: value

text: and

text: obtain

text: the

text: estimates

text: of

text: α,

text: β,

text: τ2

text: by

text: maximizing

text: the

text: approximated

text: likelihoods

text: with

text: r

text: We

text: repeat

text: the

text: estimates

text: procedure

text: times

text: and

text: the

text: boxplots

text: of

text: α

text: and

text: β

text: are

text: shown

text: in

text: Fig.

text: We

text: see

text: that

text: the

text: NN

text: NNSUM

text: SUM

text: HLR

text: MLE

text: NN

text: NNSUM

text: SUM

text: HLR

text: MLE

text: Figure

text: Two

text: panels

text: showing

text: the

text: boxplot

text: of

text: parameter

text: estimates

text: and

text: mean

text: squared

text: error

text: times

text: (left)

text: or

text: (right).

text: The

text: solid

text: line

text: is

text: a

text: reference

text: for

text: the

text: true

text: parameter

text: value,

text: and

text: the

text: dash

text: line

text: is

text: the

text: corresponding

text: mean

text: squared

text: error

text: of

text: the

text: number

text: of

text: estimates

text: in

text: each

text: method.

text: Left:

text: illustration

text: for

text: estimated

text: α;

text: Right:

text: illustration

text: for

text: estimated

text: β.

text: estimates

text: obtained

text: by

text: the

text: hierarchical

text: low

text: rank

text: approximation

text: method

text: have

text: the

text: smallest

text: mean

text: squared

text: error

text: among

text: the

text: approximation

text: methods

text: and

text: are

text: close

text: to

text: the

text: exact

text: maximum

text: likelihood

text: estimation.

text: Application

text: Dataset

text: description

text: In

text: this

text: section,

text: we

text: apply

text: our

text: method

text: to

text: modeling

text: soil

text: moisture,

text: a

text: key

text: factor

text: in

text: evaluating

text: α

text: detamitse

text: ESM

text: β

text: detamitse

text: ESM

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: approximation methods can evaluate the approximated likelihood at each iteration in the opti- mization procedure within 5 and 14 minutes, respectively. The fast computation makes it highly practical for applying the proposed methods to a large real-world spatial dataset problem. The experiment is performed with the Intel Xeon E5-2680 v3@2.50GHz processor. Next, we use the ﬁtted Mate´rn model by the hierarchical low rank approximation method to predict soil moisture at the 1000 left out locations by kriging, which is known to provide the best linear unbiased prediction as well as the prediction standard errors (Cressie 1993). However, the problem here is of size n = 2, 000, 000, hence kriging cannot be employed directly, because it involves a linear solve of size n (Furrer et al. 2006). In fact, the proposed methods in this paper can be adopted for approximating kriging equations as well. But for the purpose of validating the ﬁtted model, we explore the exact computation method by treating the irregularly spaced data as observations on a ﬁner regular grid with missing values. The resulting covariance matrix has a block Toeplitz Toeplitz block structure, which can be embedded in a block circulant circulant block matrix (Kozintsev 1999). Then kriging can be done by fast Fourier transformation. More details can be found in Chan and Ng (1996). The mean squared prediction errors over the 1000 × − validation locations is 4.53 10 5, which is notably small. 6 Discussion The implementation in this paper was done with a single-thread program, however as afore- § mentioned in , computation in each hierarchy can be paralleled, which would reduce the com- putation time dramatically and make applications even more practical. The proposed method can be also extended to more complicated settings. For example, although the rank was ﬁxed to the same in each hierarchy, it can be chosen ﬂexibly in accordance with the number of “past” observations that are involved in the hierarchy, which, we believe, would give a better approxi- 20

text: approximation

text: methods

text: can

text: evaluate

text: the

text: approximated

text: likelihood

text: at

text: each

text: iteration

text: in

text: the

text: opti-

text: mization

text: procedure

text: within

text: and

text: minutes,

text: respectively.

text: The

text: fast

text: computation

text: makes

text: it

text: highly

text: practical

text: for

text: applying

text: the

text: proposed

text: methods

text: to

text: a

text: large

text: real-world

text: spatial

text: dataset

text: problem.

text: The

text: experiment

text: is

text: performed

text: with

text: the

text: Intel

text: Xeon

text: processor.

text: Next,

text: we

text: use

text: the

text: ﬁtted

text: Mate´rn

text: model

text: by

text: the

text: hierarchical

text: low

text: rank

text: approximation

text: method

text: to

text: predict

text: soil

text: moisture

text: at

text: the

text: left

text: out

text: locations

text: by

text: kriging,

text: which

text: is

text: known

text: to

text: provide

text: the

text: best

text: linear

text: unbiased

text: prediction

text: as

text: well

text: as

text: the

text: prediction

text: standard

text: errors

text: (Cressie

text: However,

text: the

text: problem

text: here

text: is

text: of

text: size

text: n

text: hence

text: kriging

text: cannot

text: be

text: employed

text: directly,

text: because

text: it

text: involves

text: a

text: linear

text: solve

text: of

text: size

text: n

text: (Furrer

text: et

text: al.

text: In

text: fact,

text: the

text: proposed

text: methods

text: in

text: this

text: paper

text: can

text: be

text: adopted

text: for

text: approximating

text: kriging

text: equations

text: as

text: well.

text: But

text: for

text: the

text: purpose

text: of

text: validating

text: the

text: ﬁtted

text: model,

text: we

text: explore

text: the

text: exact

text: computation

text: method

text: by

text: treating

text: the

text: irregularly

text: spaced

text: data

text: as

text: observations

text: on

text: a

text: ﬁner

text: regular

text: grid

text: with

text: missing

text: values.

text: The

text: resulting

text: covariance

text: matrix

text: has

text: a

text: block

text: Toeplitz

text: Toeplitz

text: block

text: structure,

text: which

text: can

text: be

text: embedded

text: in

text: a

text: block

text: circulant

text: circulant

text: block

text: matrix

text: (Kozintsev

text: Then

text: kriging

text: can

text: be

text: done

text: by

text: fast

text: Fourier

text: transformation.

text: More

text: details

text: can

text: be

text: found

text: in

text: Chan

text: and

text: Ng

text: The

text: mean

text: squared

text: prediction

text: errors

text: over

text: the

text: validation

text: locations

text: is

text: which

text: is

text: notably

text: small.

text: Discussion

text: The

text: implementation

text: in

text: this

text: paper

text: was

text: done

text: with

text: a

text: single-thread

text: program,

text: however

text: as

text: afore-

text: mentioned

text: in

text: computation

text: in

text: each

text: hierarchy

text: can

text: be

text: paralleled,

text: which

text: would

text: reduce

text: the

text: com-

text: putation

text: time

text: dramatically

text: and

text: make

text: applications

text: even

text: more

text: practical.

text: The

text: proposed

text: method

text: can

text: be

text: also

text: extended

text: to

text: more

text: complicated

text: settings.

text: For

text: example,

text: although

text: the

text: rank

text: was

text: ﬁxed

text: to

text: the

text: same

text: in

text: each

text: hierarchy,

text: it

text: can

text: be

text: chosen

text: ﬂexibly

text: in

text: accordance

text: with

text: the

text: number

text: of

text: “past”

text: observations

text: that

text: are

text: involved

text: in

text: the

text: hierarchy,

text: which,

text: we

text: believe,

text: would

text: give

text: a

text: better

text: approxi-


text: Morrison–Woodbury formula, − − − − − − − (P L P T + (cid:15)2I ) 1 = (cid:15) 2I (cid:15) 4P (L 1 + (cid:15) 2P TP ) 1P T, (4) j j j j mr j mr j j j j j j j − × then (AT Σ A ) 1 in equation (2) can be approximated by inverting only an r r matrix j,mr jj j,mr L . This approach shares the same spirits with the predictive process (Banerjee et al. 2008) and j ﬁxed rank kriging (Cressie and Johannesson 2008). However both methods approximate the covariance function by a low rank representation while the low rank approximation is done for each j > r hierarchy in our method, and the resulting approximated covariance is no longer low rank. The detailed choice for P were discussed by Cressie and Johannesson (2008). In this j paper, we use the eigenfunctions. § To help comprehend, Fig. 1 illustrates the methods described in and for n = 5 observations Z = (z , . . . , z )T. Let r = 2, then IND considers 3 independent blocks, and f(Z) is approxi- 1 5 z 5 z 4 z 3 z 2 z 1 Figure 1: A random ﬁeld where n = 5 locations have observations . 8 ##LTFigure##

text: Morrison–Woodbury

text: formula,

text: (P

text: L

text: P

text: T

text: (L

text: TP

text: T,

text: j

text: j

text: j

text: j

text: mr

text: j

text: mr

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: then

text: (AT

text: Σ

text: A

text: in

text: equation

text: can

text: be

text: approximated

text: by

text: inverting

text: only

text: an

text: r

text: r

text: matrix

text: j,mr

text: jj

text: j,mr

text: L

text: This

text: approach

text: shares

text: the

text: same

text: spirits

text: with

text: the

text: predictive

text: process

text: (Banerjee

text: et

text: al.

text: and

text: j

text: ﬁxed

text: rank

text: kriging

text: (Cressie

text: and

text: Johannesson

text: However

text: both

text: methods

text: approximate

text: the

text: covariance

text: function

text: by

text: a

text: low

text: rank

text: representation

text: while

text: the

text: low

text: rank

text: approximation

text: is

text: done

text: for

text: each

text: j

text: r

text: hierarchy

text: in

text: our

text: method,

text: and

text: the

text: resulting

text: approximated

text: covariance

text: is

text: no

text: longer

text: low

text: rank.

text: The

text: detailed

text: choice

text: for

text: P

text: were

text: discussed

text: by

text: Cressie

text: and

text: Johannesson

text: In

text: this

text: j

text: paper,

text: we

text: use

text: the

text: eigenfunctions.

text: To

text: help

text: comprehend,

text: Fig.

text: illustrates

text: the

text: methods

text: described

text: in

text: and

text: for

text: n

text: observations

text: Z

text: (z

text: z

text: )T.

text: Let

text: r

text: then

text: IND

text: considers

text: independent

text: blocks,

text: and

text: f(Z)

text: is

text: approxi-

text: z

text: z

text: z

text: z

text: z

text: Figure

text: A

text: random

text: ﬁeld

text: where

text: n

text: locations

text: have

text: observations

text: ##LTFigure##


text: mated by f(z )f(z , z )f(z , z ). For the other four methods, the conditional density is required 5 4 3 2 1 to calculate in each hierarchy. For instance, in hierarchy j = 5, NN approximates the condi- | | | tional density f(z z , . . . , z ) by f(z z , z ); SUM by f(z z + z , z + z ); NNSUM by 5 4 1 5 4 3 5 4 3 2 1 | | f(z z , z +z ); and HLR by f(z a z +a z +a z +a z , a z +a z +a z +a z ), 5 4 3 2 5 14 4 13 3 12 2 11 1 24 4 23 3 22 2 21 1 where a s are determined by the low rank approximation. ij 2.4 Assessing model quality There are various ways to measure the performance of approximation methods, including the Kullback–Leibler divergence, the Godambe information matrix, and the Frobenius norm. The Kullback–Leibler divergence computes the divergence of the approximated from the ex- act distributions. For the zero-mean Gaussian process, the Kullback–Leibler divergence has the (cid:8) (cid:9) closed form, (cid:107) 1 − | | − | | − D (N N ) = tr(Σ 1Σ ) + log( Σ ) log( Σ ) n , K-L e a 2 a e a e where N and N stand for the exact and the approximated distributions, respectively, Σ and Σ e a e a are the corresponding covariance matrices, and n is the dimension of the distribution. The Godambe information matrix gives the asymptotic variances and covariances for the estimated parameters in the Gaussian process, as used by Kaufman et al. (2008) and Sun and Stein (2014). The Frobenius norm is another way to think about this problem. However, it is a matrix norm and does not penalize the positive deﬁniteness of a covariance matrix (Stein 2014). § For our numerical and simulation studies in 3, we choose the Kullback–Leibler divergence and the Godambe Information matrix to assess the quality of the approximation. Because the results in terms of showing the different performances are similar, we only present the results of Kullback–Leibler divergence. It will be shown numerically that the Kullback–Leibler divergence of the hierarchical low rank approximation method is always the smallest. This is due to the fact that for each j > r, the hierarchical low rank approximation method provides a better approxima- 9 ##LTLine##

text: mated

text: by

text: f(z

text: )f(z

text: z

text: )f(z

text: z

text: For

text: the

text: other

text: four

text: methods,

text: the

text: conditional

text: density

text: is

text: required

text: to

text: calculate

text: in

text: each

text: hierarchy.

text: For

text: instance,

text: in

text: hierarchy

text: j

text: NN

text: approximates

text: the

text: condi-

text: tional

text: density

text: f(z

text: z

text: z

text: by

text: f(z

text: z

text: z

text: SUM

text: by

text: f(z

text: z

text: z

text: z

text: z

text: NNSUM

text: by

text: f(z

text: z

text: z

text: +z

text: and

text: HLR

text: by

text: f(z

text: a

text: z

text: +a

text: z

text: +a

text: z

text: +a

text: z

text: a

text: z

text: +a

text: z

text: +a

text: z

text: +a

text: z

text: where

text: a

text: s

text: are

text: determined

text: by

text: the

text: low

text: rank

text: approximation.

text: ij

text: Assessing

text: model

text: quality

text: There

text: are

text: various

text: ways

text: to

text: measure

text: the

text: performance

text: of

text: approximation

text: methods,

text: including

text: the

text: Kullback–Leibler

text: divergence,

text: the

text: Godambe

text: information

text: matrix,

text: and

text: the

text: Frobenius

text: norm.

text: The

text: Kullback–Leibler

text: divergence

text: computes

text: the

text: divergence

text: of

text: the

text: approximated

text: from

text: the

text: ex-

text: act

text: distributions.

text: For

text: the

text: zero-mean

text: Gaussian

text: process,

text: the

text: Kullback–Leibler

text: divergence

text: has

text: the

text: closed

text: form,

text: D

text: (N

text: N

text: tr(Σ

text: 1Σ

text: log(

text: Σ

text: log(

text: Σ

text: n

text: K-L

text: e

text: a

text: a

text: e

text: a

text: e

text: where

text: N

text: and

text: N

text: stand

text: for

text: the

text: exact

text: and

text: the

text: approximated

text: distributions,

text: respectively,

text: Σ

text: and

text: Σ

text: e

text: a

text: e

text: a

text: are

text: the

text: corresponding

text: covariance

text: matrices,

text: and

text: n

text: is

text: the

text: dimension

text: of

text: the

text: distribution.

text: The

text: Godambe

text: information

text: matrix

text: gives

text: the

text: asymptotic

text: variances

text: and

text: covariances

text: for

text: the

text: estimated

text: parameters

text: in

text: the

text: Gaussian

text: process,

text: as

text: used

text: by

text: Kaufman

text: et

text: al.

text: and

text: Sun

text: and

text: Stein

text: The

text: Frobenius

text: norm

text: is

text: another

text: way

text: to

text: think

text: about

text: this

text: problem.

text: However,

text: it

text: is

text: a

text: matrix

text: norm

text: and

text: does

text: not

text: penalize

text: the

text: positive

text: deﬁniteness

text: of

text: a

text: covariance

text: matrix

text: (Stein

text: For

text: our

text: numerical

text: and

text: simulation

text: studies

text: in

text: we

text: choose

text: the

text: Kullback–Leibler

text: divergence

text: and

text: the

text: Godambe

text: Information

text: matrix

text: to

text: assess

text: the

text: quality

text: of

text: the

text: approximation.

text: Because

text: the

text: results

text: in

text: terms

text: of

text: showing

text: the

text: different

text: performances

text: are

text: similar,

text: we

text: only

text: present

text: the

text: results

text: of

text: Kullback–Leibler

text: divergence.

text: It

text: will

text: be

text: shown

text: numerically

text: that

text: the

text: Kullback–Leibler

text: divergence

text: of

text: the

text: hierarchical

text: low

text: rank

text: approximation

text: method

text: is

text: always

text: the

text: smallest.

text: This

text: is

text: due

text: to

text: the

text: fact

text: that

text: for

text: each

text: j

text: r,

text: the

text: hierarchical

text: low

text: rank

text: approximation

text: method

text: provides

text: a

text: better

text: approxima-

text: ##LTLine##


text: 3   transaction between a pair of agents. The entries of the matrix (cid:88)   F range from zero (no transaction possible) to unity (larger amount of goods and money exchanged). We assume that dτ − 1 − 1 ki agent i prefers to buy if the value of the goods in its possession dti = t Ri(t) (cid:2) si(t) k (cid:3) sj(t) , is smaller than the amount of money it has, and favours sell- 0 i j=1 ing otherwise, in order to maintain a balanced stock of money − 1 − (6) = R (t) s (t) sk(t) , and goods, and the same for agent j. To reﬂect this, one could t i i i 0 deﬁne the matrix elements of F as follows, ds 1 − i = [R τ (t) gs (t)] , dt t i i i 0 − F (t) = R Θ(sˆ(t)y (t) x (t)). (4) ij j i i where τ is a variable that enables agent i to decide how to i modify its prices according to trust and the information avail- where sˆ(t) is the average price over the whole network of able to it. The parameter t is the time scale for price changes 0 the goods at time t, which could be very different from the and g is a constant that represents reluctance for the agent to price the individual i is pricing its goods. Here it is assumed lower the prices, which for simplicity is assumed to be very that agent i is buying from agent j, and therefore the reputa- small and the same for all the agents. tion of agent j is the factor that matters. Trust on sellers when We can eliminate the variable τ from this equations and buying is important because one buys from a reliable agents obtain a second order equation for s, which we identify as a set only. On the contrary, when selling one is happy to do it to of damped harmonic oscillators, which admits an analytical whomever is able to pay, independently of the reputation of solution, namely, (cid:20) (cid:21) the buyer. Note that the transpose F , should be used on the ji transactions in which agent i sells goods, in which case the − − γ reputation of agent i matters. s (t) = s (0)e γt cos(ω t)(cid:112) sin(ω t) + sk(t), (7) Taking this new feature into account, the dynamical equa- i i i ωi i i tions of our model read as follows, (cid:88) − where γ = g/2 and ω = (4R2 g2)/2. Notice that i i this solution is approximate and valid only when the prices do dx − − i = [ x β F s y α F not change appreciably within a time step dt, which means dt i ij ij j j ji ij that t is large and g is small. In any case one could ﬁnd j 0 the solution of this non-linear system of coupled oscillators (cid:20) +x β F + s y α F , ] , (cid:88) j ji ji i i ij ji numerically. The solutions of Eqs. (6) produce an apparently (5) chaotic behaviour for s(t) showing qualitatively similar to the dyi = xiβijFij + y α F (cid:21) price variations of real markets (there are even some models dt s j ji ij of the stock market that use damped oscillators to analyse the j j price variation [10]). −x β F − III. RESULTS. j ji ji y α F . s i ij ji i The numerical calculations were made in a random network Notice that these model equations reduce to those of the of 500 agents with the average degree of 12, using a simple ∈ basic model (Eqs. (1)) if∀one eliminates the effect of trust by Euler method with random initial conditions for x, and y setting Fij = Fji, = 1, i, j. In this sense, the effect of trust [0, 1]. The appropriate step size for convergence was found to is to regulate the amount of goods and money exchanged in be dt = 0.05, reached within 200000 iterations. At time t = 0 each transaction, but also in deciding if one sells or buys. we set s(0) randomly using a ﬂat distribution with the mean C. Evolution of prices s(t) s¯ = 1.2 and the width ds = 0.4, and set a time scale t = 400 0 and g = dt/t . For this t the system reaches dynamical 0 0 The main quantity that regulates the dynamics of goods and equilibrium within the running time. The diagonal elements money in our trading model is the time dependence of the of the trading matrices A(t) and B(t) were set to zero. price of the goods. Therefore, one needs to suggest a mech- In Fig. 1 we depict the structure of the ﬁnal network by anism by which an agent modiﬁes the price of its goods. We showing only the active links, where the height of the 3D assume that in general people would like to sell at the high- plot is stratiﬁed according to the wealth of the agent i, i.e. est possible price, but we need to consider to what extend it w = x + y . Here it is seen that there are poor agents and i i i is limited by the agent’s reputation. If the agent is reputed to only a few very rich ones. In addition we observed in numer- be a successful trader, it could increase its prices by a large ous calculations that if the width of the distribution of prices amount, provided that the price the agent i is using is smaller is increased, the rich become richer and the number of poor than the average price the agents connected to it are offering, agents increases. otherwise lowering the value of s is a way of attracting more In Fig. 2 we show the time histories of the variables for i transactions. In a mathematical language, one can represent the typical calculation of the previous ﬁgure. Note that this situation by the following coupled ﬁrst order equations, most agents have converged to certain amounts of money and ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: transaction

text: between

text: a

text: pair

text: of

text: agents.

text: The

text: entries

text: of

text: the

text: matrix

text: F

text: range

text: from

text: zero

text: (no

text: transaction

text: possible)

text: to

text: unity

text: (larger

text: amount

text: of

text: goods

text: and

text: money

text: exchanged).

text: We

text: assume

text: that

text: dτ

text: ki

text: agent

text: i

text: prefers

text: to

text: buy

text: if

text: the

text: value

text: of

text: the

text: goods

text: in

text: its

text: possession

text: dti

text: t

text: Ri(t)

text: si(t)

text: k

text: sj(t)

text: is

text: smaller

text: than

text: the

text: amount

text: of

text: money

text: it

text: has,

text: and

text: favours

text: sell-

text: i

text: ing

text: otherwise,

text: in

text: order

text: to

text: maintain

text: a

text: balanced

text: stock

text: of

text: money

text: R

text: (t)

text: s

text: (t)

text: sk(t)

text: and

text: goods,

text: and

text: the

text: same

text: for

text: agent

text: j.

text: To

text: reﬂect

text: this,

text: one

text: could

text: t

text: i

text: i

text: i

text: deﬁne

text: the

text: matrix

text: elements

text: of

text: F

text: as

text: follows,

text: ds

text: i

text: [R

text: τ

text: (t)

text: gs

text: (t)]

text: dt

text: t

text: i

text: i

text: i

text: F

text: (t)

text: R

text: Θ(sˆ(t)y

text: (t)

text: x

text: (t)).

text: ij

text: j

text: i

text: i

text: where

text: τ

text: is

text: a

text: variable

text: that

text: enables

text: agent

text: i

text: to

text: decide

text: how

text: to

text: i

text: modify

text: its

text: prices

text: according

text: to

text: trust

text: and

text: the

text: information

text: avail-

text: where

text: sˆ(t)

text: is

text: the

text: average

text: price

text: over

text: the

text: whole

text: network

text: of

text: able

text: to

text: it.

text: The

text: parameter

text: t

text: is

text: the

text: time

text: scale

text: for

text: price

text: changes

text: the

text: goods

text: at

text: time

text: t,

text: which

text: could

text: be

text: very

text: different

text: from

text: the

text: and

text: g

text: is

text: a

text: constant

text: that

text: represents

text: reluctance

text: for

text: the

text: agent

text: to

text: price

text: the

text: individual

text: i

text: is

text: pricing

text: its

text: goods.

text: Here

text: it

text: is

text: assumed

text: lower

text: the

text: prices,

text: which

text: for

text: simplicity

text: is

text: assumed

text: to

text: be

text: very

text: that

text: agent

text: i

text: is

text: buying

text: from

text: agent

text: j,

text: and

text: therefore

text: the

text: reputa-

text: small

text: and

text: the

text: same

text: for

text: all

text: the

text: agents.

text: tion

text: of

text: agent

text: j

text: is

text: the

text: factor

text: that

text: matters.

text: Trust

text: on

text: sellers

text: when

text: We

text: can

text: eliminate

text: the

text: variable

text: τ

text: from

text: this

text: equations

text: and

text: buying

text: is

text: important

text: because

text: one

text: buys

text: from

text: a

text: reliable

text: agents

text: obtain

text: a

text: second

text: order

text: equation

text: for

text: s,

text: which

text: we

text: identify

text: as

text: a

text: set

text: only.

text: On

text: the

text: contrary,

text: when

text: selling

text: one

text: is

text: happy

text: to

text: do

text: it

text: to

text: of

text: damped

text: harmonic

text: oscillators,

text: which

text: admits

text: an

text: analytical

text: whomever

text: is

text: able

text: to

text: pay,

text: independently

text: of

text: the

text: reputation

text: of

text: solution,

text: namely,

text: the

text: buyer.

text: Note

text: that

text: the

text: transpose

text: F

text: should

text: be

text: used

text: on

text: the

text: ji

text: transactions

text: in

text: which

text: agent

text: i

text: sells

text: goods,

text: in

text: which

text: case

text: the

text: γ

text: reputation

text: of

text: agent

text: i

text: matters.

text: s

text: (t)

text: s

text: γt

text: cos(ω

text: sin(ω

text: t)

text: sk(t),

text: Taking

text: this

text: new

text: feature

text: into

text: account,

text: the

text: dynamical

text: equa-

text: i

text: i

text: i

text: ωi

text: i

text: i

text: tions

text: of

text: our

text: model

text: read

text: as

text: follows,

text: where

text: γ

text: and

text: ω

text: Notice

text: that

text: i

text: i

text: this

text: solution

text: is

text: approximate

text: and

text: valid

text: only

text: when

text: the

text: prices

text: do

text: dx

text: i

text: x

text: β

text: F

text: s

text: y

text: α

text: F

text: not

text: change

text: appreciably

text: within

text: a

text: time

text: step

text: dt,

text: which

text: means

text: dt

text: i

text: ij

text: ij

text: j

text: j

text: ji

text: ij

text: that

text: t

text: is

text: large

text: and

text: g

text: is

text: small.

text: In

text: any

text: case

text: one

text: could

text: ﬁnd

text: j

text: the

text: solution

text: of

text: this

text: non-linear

text: system

text: of

text: coupled

text: oscillators

text: +x

text: β

text: F

text: s

text: y

text: α

text: F

text: j

text: ji

text: ji

text: i

text: i

text: ij

text: ji

text: numerically.

text: The

text: solutions

text: of

text: Eqs.

text: produce

text: an

text: apparently

text: chaotic

text: behaviour

text: for

text: s(t)

text: showing

text: qualitatively

text: similar

text: to

text: the

text: dyi

text: xiβijFij

text: y

text: α

text: F

text: price

text: variations

text: of

text: real

text: markets

text: (there

text: are

text: even

text: some

text: models

text: dt

text: s

text: j

text: ji

text: ij

text: of

text: the

text: stock

text: market

text: that

text: use

text: damped

text: oscillators

text: to

text: analyse

text: the

text: j

text: j

text: price

text: variation

text: −x

text: β

text: F

text: III.

text: RESULTS.

text: j

text: ji

text: ji

text: y

text: α

text: F

text: s

text: i

text: ij

text: ji

text: i

text: The

text: numerical

text: calculations

text: were

text: made

text: in

text: a

text: random

text: network

text: Notice

text: that

text: these

text: model

text: equations

text: reduce

text: to

text: those

text: of

text: the

text: of

text: agents

text: with

text: the

text: average

text: degree

text: of

text: using

text: a

text: simple

text: basic

text: model

text: (Eqs.

text: if∀one

text: eliminates

text: the

text: effect

text: of

text: trust

text: by

text: Euler

text: method

text: with

text: random

text: initial

text: conditions

text: for

text: x,

text: and

text: y

text: setting

text: Fij

text: Fji,

text: i,

text: j.

text: In

text: this

text: sense,

text: the

text: effect

text: of

text: trust

text: The

text: appropriate

text: step

text: size

text: for

text: convergence

text: was

text: found

text: to

text: is

text: to

text: regulate

text: the

text: amount

text: of

text: goods

text: and

text: money

text: exchanged

text: in

text: be

text: dt

text: reached

text: within

text: iterations.

text: At

text: time

text: t

text: each

text: transaction,

text: but

text: also

text: in

text: deciding

text: if

text: one

text: sells

text: or

text: buys.

text: we

text: set

text: randomly

text: using

text: a

text: ﬂat

text: distribution

text: with

text: the

text: mean

text: C.

text: Evolution

text: of

text: prices

text: s(t)

text: s¯

text: and

text: the

text: width

text: ds

text: and

text: set

text: a

text: time

text: scale

text: t

text: and

text: g

text: dt/t

text: For

text: this

text: t

text: the

text: system

text: reaches

text: dynamical

text: The

text: main

text: quantity

text: that

text: regulates

text: the

text: dynamics

text: of

text: goods

text: and

text: equilibrium

text: within

text: the

text: running

text: time.

text: The

text: diagonal

text: elements

text: money

text: in

text: our

text: trading

text: model

text: is

text: the

text: time

text: dependence

text: of

text: the

text: of

text: the

text: trading

text: matrices

text: A(t)

text: and

text: B(t)

text: were

text: set

text: to

text: zero.

text: price

text: of

text: the

text: goods.

text: Therefore,

text: one

text: needs

text: to

text: suggest

text: a

text: mech-

text: In

text: Fig.

text: we

text: depict

text: the

text: structure

text: of

text: the

text: ﬁnal

text: network

text: by

text: anism

text: by

text: which

text: an

text: agent

text: modiﬁes

text: the

text: price

text: of

text: its

text: goods.

text: We

text: showing

text: only

text: the

text: active

text: links,

text: where

text: the

text: height

text: of

text: the

text: assume

text: that

text: in

text: general

text: people

text: would

text: like

text: to

text: sell

text: at

text: the

text: high-

text: plot

text: is

text: stratiﬁed

text: according

text: to

text: the

text: wealth

text: of

text: the

text: agent

text: i,

text: i.e.

text: est

text: possible

text: price,

text: but

text: we

text: need

text: to

text: consider

text: to

text: what

text: extend

text: it

text: w

text: x

text: y

text: Here

text: it

text: is

text: seen

text: that

text: there

text: are

text: poor

text: agents

text: and

text: i

text: i

text: i

text: is

text: limited

text: by

text: the

text: agent’s

text: reputation.

text: If

text: the

text: agent

text: is

text: reputed

text: to

text: only

text: a

text: few

text: very

text: rich

text: ones.

text: In

text: addition

text: we

text: observed

text: in

text: numer-

text: be

text: a

text: successful

text: trader,

text: it

text: could

text: increase

text: its

text: prices

text: by

text: a

text: large

text: ous

text: calculations

text: that

text: if

text: the

text: width

text: of

text: the

text: distribution

text: of

text: prices

text: amount,

text: provided

text: that

text: the

text: price

text: the

text: agent

text: i

text: is

text: using

text: is

text: smaller

text: is

text: increased,

text: the

text: rich

text: become

text: richer

text: and

text: the

text: number

text: of

text: poor

text: than

text: the

text: average

text: price

text: the

text: agents

text: connected

text: to

text: it

text: are

text: offering,

text: agents

text: increases.

text: otherwise

text: lowering

text: the

text: value

text: of

text: s

text: is

text: a

text: way

text: of

text: attracting

text: more

text: In

text: Fig.

text: we

text: show

text: the

text: time

text: histories

text: of

text: the

text: variables

text: for

text: i

text: transactions.

text: In

text: a

text: mathematical

text: language,

text: one

text: can

text: represent

text: the

text: typical

text: calculation

text: of

text: the

text: previous

text: ﬁgure.

text: Note

text: that

text: this

text: situation

text: by

text: the

text: following

text: coupled

text: ﬁrst

text: order

text: equations,

text: most

text: agents

text: have

text: converged

text: to

text: certain

text: amounts

text: of

text: money

text: and

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##