text: WILSON LOOP AREA LAW 13 (cid:90) Lemma 3.1. The function 1 dw χ (z) = − γ (cid:90) 2(cid:90)πi γ z w is smooth away from γ and satisﬁes ∂¯χ = δ(0,1) in the sense of currents, i.e. γ γ − ∈ χ ( ∂¯ω) = ω, ω Ω1,0. γ C γ | − | Moreover, we have χ (z) = O(log z z ) as z approaches a point z in the image of γ. γ 0 0 (cid:90) → Proof. Let f be a smooth compactly supported function such that f dw¯ δ(0,1) as a (cid:15) (cid:15) γ current. We have 1 f (w) χ (z) = dw¯dw (cid:15)− (cid:90) (cid:15) (cid:90) 2πi C z w is a smooth function and solves ∂¯χ = f dw¯, so that (cid:15) (cid:15) − ∧ ∈ χ ( ∂¯ω) = f (z)dw¯ ω, ω Ω1,0. (cid:15) (cid:15) C C → Letting (cid:15) 0, the above equations imply lim → χ = χ and ∂¯χ = δ(0,1). The last state- (cid:3) (cid:15) 0 (cid:15) γ γ γ ment is clear. By slight abuse of notation, we will often identify γ with its image in Σ since γ is (cid:122) (cid:125)(cid:124) (cid:123) embedded. We also use the notational abbreviation n × · · · × ⊂ γn = γ γ Σn. D ⊂ (cid:122) (cid:125)(cid:124) (cid:123) Let [0, 1]n be a domain and let 0 n D × · · · × D ⊂ = ( γ γ )( ) γn (cid:90) (cid:90) 0 (cid:122) (cid:125)(cid:124) (cid:123) ∈ (cid:2) be its image under γ applied to each variable. Given a form ω (Ω1) , deﬁne n × · · · × ∗ ω = ( γ γ ) (ω) D D 0 This is just a notational shortcut so as to avoid writing pullbacks everywhere in the case where γ is not embedded. (In the end, our main results hold for γ a simple closed curve, but for the time being we can work with general curves.) (cid:2) (cid:2) Deﬁne δD to be the current of type (Ω1) n on Σn which takes an element of (Ω1) n and D (cid:2) integrates it over . Likewise, deﬁne δD(0,1) to be the current of type (Ω0,1) n that takes an (cid:2) D element of (Ω1,0) n and integrates it along . (cid:90) D ⊂ Corollary 3.2. For any domain γn, we have 1 dw · · · dw χD(z , . . . , z ) = −n − 1 (3.1) n 1 (2πi)n D z w z w n n 1 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: WILSON

text: LOOP

text: AREA

text: LAW

text: Lemma

text: The

text: function

text: dw

text: χ

text: (z)

text: γ

text: γ

text: z

text: w

text: is

text: smooth

text: away

text: from

text: γ

text: and

text: satisﬁes

text: ∂¯χ

text: in

text: the

text: sense

text: of

text: currents,

text: i.e.

text: γ

text: γ

text: χ

text: ∂¯ω)

text: ω,

text: ω

text: γ

text: C

text: γ

text: Moreover,

text: we

text: have

text: χ

text: (z)

text: O(log

text: z

text: z

text: as

text: z

text: approaches

text: a

text: point

text: z

text: in

text: the

text: image

text: of

text: γ.

text: γ

text: Proof.

text: Let

text: f

text: be

text: a

text: smooth

text: compactly

text: supported

text: function

text: such

text: that

text: f

text: dw¯

text: as

text: a

text: γ

text: current.

text: We

text: have

text: f

text: (w)

text: χ

text: (z)

text: dw¯dw

text: C

text: z

text: w

text: is

text: a

text: smooth

text: function

text: and

text: solves

text: ∂¯χ

text: f

text: dw¯,

text: so

text: that

text: χ

text: ∂¯ω)

text: f

text: (z)dw¯

text: ω,

text: ω

text: C

text: C

text: Letting

text: the

text: above

text: equations

text: imply

text: lim

text: χ

text: χ

text: and

text: ∂¯χ

text: The

text: last

text: state-

text: γ

text: γ

text: γ

text: ment

text: is

text: clear.

text: By

text: slight

text: abuse

text: of

text: notation,

text: we

text: will

text: often

text: identify

text: γ

text: with

text: its

text: image

text: in

text: Σ

text: since

text: γ

text: is

text: embedded.

text: We

text: also

text: use

text: the

text: notational

text: abbreviation

text: n

text: γn

text: γ

text: γ

text: Σn.

text: D

text: Let

text: be

text: a

text: domain

text: and

text: let

text: n

text: D

text: D

text: γ

text: γ

text: γn

text: be

text: its

text: image

text: under

text: γ

text: applied

text: to

text: each

text: variable.

text: Given

text: a

text: form

text: ω

text: deﬁne

text: n

text: ω

text: γ

text: γ

text: (ω)

text: D

text: D

text: This

text: is

text: just

text: a

text: notational

text: shortcut

text: so

text: as

text: to

text: avoid

text: writing

text: pullbacks

text: everywhere

text: in

text: the

text: case

text: where

text: γ

text: is

text: not

text: embedded.

text: (In

text: the

text: end,

text: our

text: main

text: results

text: hold

text: for

text: γ

text: a

text: simple

text: closed

text: curve,

text: but

text: for

text: the

text: time

text: being

text: we

text: can

text: work

text: with

text: general

text: curves.)

text: Deﬁne

text: δD

text: to

text: be

text: the

text: current

text: of

text: type

text: n

text: on

text: Σn

text: which

text: takes

text: an

text: element

text: of

text: n

text: and

text: D

text: integrates

text: it

text: over

text: Likewise,

text: deﬁne

text: to

text: be

text: the

text: current

text: of

text: type

text: n

text: that

text: takes

text: an

text: D

text: element

text: of

text: n

text: and

text: integrates

text: it

text: along

text: D

text: Corollary

text: For

text: any

text: domain

text: γn,

text: we

text: have

text: dw

text: dw

text: χD(z

text: z

text: −n

text: n

text: D

text: z

text: w

text: z

text: w

text: n

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 TIMOTHY NGUYEN Theorem 2. (Area Law to All Orders for Circles) Let γ be a circular contour on S2 and suppose dσ is rotationally symmetric about the center of the disk bounded by γ. Let |R||S2\R| (cid:90) ρ = | | . Then S2 2 (cid:104) (cid:105) 1 −| | W = e X 2/2ρλf(eX)dX. (1.5) f,γ gax (2πρλ)dimG/2 g R | | The same result holds on 2 by replacing ρ with R . Theorem 2 is a slight generalization of the result in [11], which dealt with the standard area form on S2. Although we state our results here for single Wilson loops for the sake of simplicity, they have straightforward analogs to products of Wilson loops. Theorem 1 extends to the case of when the loops are disjoint and Theorem 2 extends to the case when the loops are all circular and concentrically nested. See [11] for the corresponding Gaussian R matrix integral formula in the latter case. The last statement about 2 should be regarded R as a decompactiﬁcation limit as S2 increases to 2 [19]. The outline of this paper is as follows. We provide the basic deﬁnitions and setup in Section 2. Section 3 provides the main technical results concerning iterated integrals of Green’s operators that we need to evaluate Wilson loop observables. One of our key insights involves extending known results about the homotopy invariance of iterated integrals within the class of based loops to the class of free (unbased) loops. Section 4 proves the main theorems. In the appendix, we provide an important discussion of our results from the path integral point of view. Here, we provide a formal proof of invariance of the path integral under area-preserving diﬀeomorphisms and we interpret our generalized axial-gauge as a (complexiﬁed) gauge-ﬁxing procedure. The motivation of this paper stems from the author’s investigation of the conventional paradigm in physics that formal perturbation theory approximates the full quantum theory when the latter has a rigorous construction (two-dimensional Yang-Mills theory is such an example). In fact, equality to all orders in (1.4) is precisely what one would expect if perturbation theory is supposed to provide an asymptotic series for the exact Wilson loop expectation. Our work here is therefore part of a fuller discussion to be found in [19]. See also [20] for a quite diﬀerent analysis of Yang-Mills theory in (bona ﬁde) axial-gauge, in which an honest real direction is gauged away instead of a complexiﬁed one. 2. The Setup A Since we work with a trivial G-bundle over Σ, we can identify the space of connections with Ω1(Σ; g), the space of 1-forms with values in the Lie algebra g of G. The Yang-Mills G action (1.1) is invariant under the group of bundle automorphisms = Maps(Σ, G), for ∈ G which a gauge-transformation g acts on a connection via · (cid:55)→ − − g A gAg 1 + gdg 1. (2.1) If we endow Σ with a complex structure, we obtain a splitting of the complex 1-forms Ω1(Σ) into the bundle of complex-linear and complex anti-linear forms c ⊕ Ω1(Σ) = Ω1,0(Σ) Ω1,0(Σ). c ##LTLine## ##LTLine##

label: paragraph

text: TIMOTHY

text: NGUYEN

text: Theorem

text: (Area

text: Law

text: to

text: All

text: Orders

text: for

text: Circles)

text: Let

text: γ

text: be

text: a

text: circular

text: contour

text: on

text: and

text: suppose

text: dσ

text: is

text: rotationally

text: symmetric

text: about

text: the

text: center

text: of

text: the

text: disk

text: bounded

text: by

text: γ.

text: Let

label: equation

text: ρ

text: Then

text: W

text: e

text: X

text: f,γ

text: gax

text: g

text: R

text: The

text: same

text: result

text: holds

text: on

text: by

text: replacing

text: ρ

text: with

text: R

text: Theorem

text: is

text: a

text: slight

text: generalization

text: of

text: the

text: result

text: in

text: which

text: dealt

text: with

text: the

text: standard

text: area

text: form

text: on

text: Although

text: we

text: state

text: our

text: results

text: here

text: for

text: single

text: Wilson

text: loops

text: for

text: the

text: sake

text: of

text: simplicity,

text: they

text: have

text: straightforward

text: analogs

text: to

text: products

text: of

text: Wilson

text: loops.

text: Theorem

text: extends

text: to

text: the

text: case

text: of

text: when

text: the

text: loops

text: are

text: disjoint

text: and

text: Theorem

text: extends

text: to

text: the

text: case

text: when

text: the

text: loops

text: are

text: all

text: circular

text: and

text: concentrically

text: nested.

text: See

text: for

text: the

text: corresponding

text: Gaussian

text: R

text: matrix

text: integral

text: formula

text: in

text: the

text: latter

text: case.

text: The

text: last

text: statement

text: about

text: should

text: be

text: regarded

text: R

text: as

text: a

text: decompactiﬁcation

text: limit

text: as

text: increases

text: to

text: The

text: outline

text: of

text: this

text: paper

text: is

text: as

text: follows.

text: We

text: provide

text: the

text: basic

text: deﬁnitions

text: and

text: setup

text: in

text: Section

text: Section

text: provides

text: the

text: main

text: technical

text: results

text: concerning

text: iterated

text: integrals

text: of

text: Green’s

text: operators

text: that

text: we

text: need

text: to

text: evaluate

text: Wilson

text: loop

text: observables.

text: One

text: of

text: our

text: key

text: insights

text: involves

text: extending

text: known

text: results

text: about

text: the

text: homotopy

text: invariance

text: of

text: iterated

text: integrals

text: within

text: the

text: class

text: of

text: based

text: loops

text: to

text: the

text: class

text: of

text: free

text: (unbased)

text: loops.

text: Section

text: proves

text: the

text: main

text: theorems.

text: In

text: the

text: appendix,

text: we

text: provide

text: an

text: important

text: discussion

text: of

text: our

text: results

text: from

text: the

text: path

text: integral

text: point

text: of

text: view.

text: Here,

text: we

text: provide

text: a

text: formal

text: proof

text: of

text: invariance

text: of

text: the

text: path

text: integral

text: under

text: area-preserving

text: diﬀeomorphisms

text: and

text: we

text: interpret

text: our

text: generalized

text: axial-gauge

text: as

text: a

text: (complexiﬁed)

text: gauge-ﬁxing

text: procedure.

text: The

text: motivation

text: of

text: this

text: paper

text: stems

text: from

text: the

text: author’s

text: investigation

text: of

text: the

text: conventional

text: paradigm

text: in

text: physics

text: that

text: formal

text: perturbation

text: theory

text: approximates

text: the

text: full

text: quantum

text: theory

text: when

text: the

text: latter

text: has

text: a

text: rigorous

text: construction

text: (two-dimensional

text: Yang-Mills

text: theory

text: is

text: such

text: an

text: example).

text: In

text: fact,

text: equality

text: to

text: all

text: orders

text: in

text: is

text: precisely

text: what

text: one

text: would

text: expect

text: if

text: perturbation

text: theory

text: is

text: supposed

text: to

text: provide

text: an

text: asymptotic

text: series

text: for

text: the

text: exact

text: Wilson

text: loop

text: expectation.

text: Our

text: work

text: here

text: is

text: therefore

text: part

text: of

text: a

text: fuller

text: discussion

text: to

text: be

text: found

text: in

text: See

text: also

text: for

text: a

text: quite

text: diﬀerent

text: analysis

text: of

text: Yang-Mills

text: theory

text: in

text: (bona

text: ﬁde)

text: axial-gauge,

text: in

text: which

text: an

text: honest

text: real

text: direction

text: is

text: gauged

text: away

text: instead

text: of

text: a

text: complexiﬁed

text: one.

text: The

text: Setup

text: A

text: Since

text: we

text: work

text: with

text: a

text: trivial

text: G-bundle

text: over

text: Σ,

text: we

text: can

text: identify

text: the

text: space

text: of

text: connections

text: with

text: g),

text: the

text: space

text: of

text: with

text: values

text: in

text: the

text: Lie

text: algebra

text: g

text: of

text: G.

text: The

text: Yang-Mills

text: G

text: action

text: is

text: invariant

text: under

text: the

text: group

text: of

text: bundle

text: automorphisms

text: Maps(Σ,

text: G),

text: for

text: G

text: which

text: a

text: gauge-transformation

text: g

text: acts

text: on

text: a

text: connection

text: via

text: g

text: A

text: gAg

text: gdg

text: If

text: we

text: endow

text: Σ

text: with

text: a

text: complex

text: structure,

text: we

text: obtain

text: a

text: splitting

text: of

text: the

text: complex

text: into

text: the

text: bundle

text: of

text: complex-linear

text: and

text: complex

text: anti-linear

text: forms

text: c

text: c

text: ##LTLine##

text: ##LTLine##


text: 6 TIMOTHY NGUYEN R In particular, α-gauge for α = π/2 is holomorphic gauge on 2 with its standard complex structure. R ≤ ≤ Deﬁnition 2.3. We refer to the collection of α-gauges on 2, 0 α π/2, and holomorphic gauge for a general Riemann surface Σ as generalized axial-gauge. Here, we interpret the → case α = 0 as the limit α 0+, which we call Wu-Mandelstam-Liebrandt (WML) light cone gauge. The signiﬁcance of WML light cone gauge follows from the substantial physics literature which discusses perturbative quantization in light cone gauge [16, 5, 22, 6], for which a suit- able regularization scheme is needed in this highly singular gauge. The WML prescription is one of them, and it is prescribed by being continuously connected to holomorphic gauge → via Wick rotation. The signiﬁcance of α 0+ cannot be seen on the classical level, in which → setting α = 0 yields A = A dx for A in 0-gauge. Rather, letting α 0+ is signiﬁcant + + for the regularization of Feynman diagrams, see (2.9). By the remarks above, it suﬃces to consider holomorphic gauge and so we will primarily be focused on this case. The sense in which placing a connection in generalized axial-gauge is really a choice of gauge is partially addressed by the following lemma: R Lemma 2.4. On 2, any complex connection can be placed into generalized axial-gauge using a complex gauge transformation. On S2, any complex connection not holomorphically equivalent to a non-minimal Yang-Mills connection3 can be placed into holomorphic gauge using a complex gauge transformation. R C Proof. Without loss of generality, we consider holomorphic gauge. On 2 = , the C lemma is precisely the standard result that every holomorphic bundle on is equivalent to one with the trivial holomorphic structure. For S2, this follows from the work of Atiyah (cid:3) and Bott [1]. G Because Lemma 2.4 involves complex gauge transformations in and not unitary (i.e. G c ordinary) gauge transformations in , it is not clear in what sense working with α-gauge is really a choice of honest gauge. We provide some insightful remarks on how to interpret generalized axial-gauge in the sense most appropriate for the path integral approach to quantum ﬁeld theory in the appendix, one which more adequately justiﬁes the use of the word gauge. Next, we analyze the kinetic operator of Yang-Mills theory in generalized axial-gauge, which allows us to deﬁne the corresponding Green’s function (i.e. propagator). This allows us to deﬁne the perturbative expectation value of Wilson loop observables in terms of Wick contractions using these gauge-ﬁxed Green’s functions. 3Such connections form an open, dense set containing a neighborhood of the trivial connection. Hence, perturbation theory around the trivial connection does not detect non-minimal Yang-Mills connections. In the higher genus case, the presence of a continuous moduli of nontrivial holomorphic structures invalidates Lemma 2.4 for a non-negligible set of connections. ##LTLine##

text: TIMOTHY

text: NGUYEN

text: R

text: In

text: particular,

text: α-gauge

text: for

text: α

text: is

text: holomorphic

text: gauge

text: on

text: with

text: its

text: standard

text: complex

text: structure.

text: R

text: Deﬁnition

text: We

text: refer

text: to

text: the

text: collection

text: of

text: α-gauges

text: on

text: α

text: and

text: holomorphic

text: gauge

text: for

text: a

text: general

text: Riemann

text: surface

text: Σ

text: as

text: generalized

text: axial-gauge.

text: Here,

text: we

text: interpret

text: the

text: case

text: α

text: as

text: the

text: limit

text: α

text: which

text: we

text: call

text: Wu-Mandelstam-Liebrandt

text: (WML)

text: light

text: cone

text: gauge.

text: The

text: signiﬁcance

text: of

text: WML

text: light

text: cone

text: gauge

text: follows

text: from

text: the

text: substantial

text: physics

text: literature

text: which

text: discusses

text: perturbative

text: quantization

text: in

text: light

text: cone

text: gauge

text: for

text: which

text: a

text: suit-

text: able

text: regularization

text: scheme

text: is

text: needed

text: in

text: this

text: highly

text: singular

text: gauge.

text: The

text: WML

text: prescription

text: is

text: one

text: of

text: them,

text: and

text: it

text: is

text: prescribed

text: by

text: being

text: continuously

text: connected

text: to

text: holomorphic

text: gauge

text: via

text: Wick

text: rotation.

text: The

text: signiﬁcance

text: of

text: α

text: cannot

text: be

text: seen

text: on

text: the

text: classical

text: level,

text: in

text: which

text: setting

text: α

text: yields

text: A

text: A

text: dx

text: for

text: A

text: in

text: Rather,

text: letting

text: α

text: is

text: signiﬁcant

text: for

text: the

text: regularization

text: of

text: Feynman

text: diagrams,

text: see

text: By

text: the

text: remarks

text: above,

text: it

text: suﬃces

text: to

text: consider

text: holomorphic

text: gauge

text: and

text: so

text: we

text: will

text: primarily

text: be

text: focused

text: on

text: this

text: case.

text: The

text: sense

text: in

text: which

text: placing

text: a

text: connection

text: in

text: generalized

text: axial-gauge

text: is

text: really

text: a

text: choice

text: of

text: gauge

text: is

text: partially

text: addressed

text: by

text: the

text: following

text: lemma:

text: R

text: Lemma

text: On

text: any

text: complex

text: connection

text: can

text: be

text: placed

text: into

text: generalized

text: axial-gauge

text: using

text: a

text: complex

text: gauge

text: transformation.

text: On

text: any

text: complex

text: connection

text: not

text: holomorphically

text: equivalent

text: to

text: a

text: non-minimal

text: Yang-Mills

text: can

text: be

text: placed

text: into

text: holomorphic

text: gauge

text: using

text: a

text: complex

text: gauge

text: transformation.

text: R

text: C

text: Proof.

text: Without

text: loss

text: of

text: generality,

text: we

text: consider

text: holomorphic

text: gauge.

text: On

text: the

text: C

text: lemma

text: is

text: precisely

text: the

text: standard

text: result

text: that

text: every

text: holomorphic

text: bundle

text: on

text: is

text: equivalent

text: to

text: one

text: with

text: the

text: trivial

text: holomorphic

text: structure.

text: For

text: this

text: follows

text: from

text: the

text: work

text: of

text: Atiyah

text: and

text: Bott

text: G

text: Because

text: Lemma

text: involves

text: complex

text: gauge

text: transformations

text: in

text: and

text: not

text: unitary

text: (i.e.

text: G

text: c

text: ordinary)

text: gauge

text: transformations

text: in

text: it

text: is

text: not

text: clear

text: in

text: what

text: sense

text: working

text: with

text: α-gauge

text: is

text: really

text: a

text: choice

text: of

text: honest

text: gauge.

text: We

text: provide

text: some

text: insightful

text: remarks

text: on

text: how

text: to

text: interpret

text: generalized

text: axial-gauge

text: in

text: the

text: sense

text: most

text: appropriate

text: for

text: the

text: path

text: integral

text: approach

text: to

text: quantum

text: ﬁeld

text: theory

text: in

text: the

text: appendix,

text: one

text: which

text: more

text: adequately

text: justiﬁes

text: the

text: use

text: of

text: the

text: word

text: gauge.

text: Next,

text: we

text: analyze

text: the

text: kinetic

text: operator

text: of

text: Yang-Mills

text: theory

text: in

text: generalized

text: axial-gauge,

text: which

text: allows

text: us

text: to

text: deﬁne

text: the

text: corresponding

text: Green’s

text: function

text: (i.e.

text: propagator).

text: This

text: allows

text: us

text: to

text: deﬁne

text: the

text: perturbative

text: expectation

text: value

text: of

text: Wilson

text: loop

text: observables

text: in

text: terms

text: of

text: Wick

text: contractions

text: using

text: these

text: gauge-ﬁxed

text: Green’s

text: functions.

text: connections

text: form

text: an

text: open,

text: dense

text: set

text: containing

text: a

text: neighborhood

text: of

text: the

text: trivial

text: connection.

text: Hence,

text: perturbation

text: theory

text: around

text: the

text: trivial

text: connection

text: does

text: not

text: detect

text: non-minimal

text: Yang-Mills

text: connections.

text: In

text: the

text: higher

text: genus

text: case,

text: the

text: presence

text: of

text: a

text: continuous

text: moduli

text: of

text: nontrivial

text: holomorphic

text: structures

text: invalidates

text: Lemma

text: for

text: a

text: non-negligible

text: set

text: of

text: connections.

text: ##LTLine##


text: WILSON LOOP AREA LAW 9 − (cid:48) (cid:48) i (x x )2− (cid:48) P (x, x ) = lim dx − (cid:48) (cid:48) − dx . (2.9) WML (cid:15)→0+ 4π + (x x )+(x−x )− i(cid:15) + If we had set α = 0 ﬁrst in the Yang-Mills action, the kinetic operator would not be elliptic. Deﬁning a Green’s operator for the resulting operator would be ambiguous due → to the need to choose a regularization scheme; the WML prescription α 0+ provides such a scheme, yielding a well-deﬁned propagator that is continuously connected to Green’s operators of elliptic operators via Wick rotation. In the above, we worked in a coordinate-free manner as possible because then (i) it is R clear that our analysis holds for general area forms on 2; (ii) we can readily adapt the above analysis to the case of holomorphic gauge on compact surfaces. For Σ compact, the • → • operator ∂¯ : Ω (Σ) Ω (Σ) has a kernel and cokernel. For the simplest case Σ = S2, ker ∂¯ consists of just constant functions. By uniformization, we can suppose that S2 has the standard complex structure, otherwise we can apply a diﬀeomorphism (and change our area form accordingly). We then have the standard coordinate charts on the Riemann C ∪C sphere S2 = ∞, related to each other by stereographic projection interchanging z = 0 with z˜ = 1 =0∞. In the chart C , the Green’s operator for ∂¯ : Ω0 → Ω0,1 has the same z C 0 expression as for the case of : − 1 dw P (z, w) = − . ∂¯ 2πi z w The formula for the holomorphic gauge propagator on S2 involves the appropriate modiﬁ- cation of (2.5), and is given by ◦ ∗ ◦ → P = P t ˜ P : Ω0,1 Ω1,0 (2.10) hol ∂¯ ∂¯ (cid:2) with the integral kernel of Phol lying in(cid:18)Ω1,0 Ω1(cid:90),0 and w(cid:19)here ∗ → ˜ : Ω0 Ω2 (cid:55)→ ∗ − 1 f f | | fdσ S2 S2 projects Hodge star onto the top-degree forms that integrate to zero (which are thus in the → image of ∂¯ : Ω1,0 Ω2). For concreteness, we write down an explicit formula for P when hol dσ = 4d|2|z is the round are(cid:90)a form. On(cid:20)e can explicitly compute [11] (cid:21) (1+ z 2)2 · 1 1 − (cid:48) (cid:48) − 1 (cid:48) 1 P (z, w)dzdw = dzdw − dσ(u)δ(u u )d2u | |dσ(u)dσ(u ) (cid:48) − 4π2 C×C z u S2 u w − 1 1 1 z¯ w¯ = dzdw | | | | − (2.11) π 1 + z 2 1 + w 2 z w C × C ⊂ × The above integral must be interpreted in the sense of distributions on S2 S2 since, even though S2 is compact, dz and dw are not globally deﬁned coordinates. Never- theless, we obtain the correct expression for P (z, w)dzdw with respect to the holomorphic ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: WILSON

text: LOOP

text: AREA

text: LAW

text: i

text: (x

text: x

text: P

text: (x,

text: x

text: lim

text: dx

text: dx

text: WML

text: 4π

text: (x

text: x

text: )+(x−x

text: If

text: we

text: had

text: set

text: α

text: ﬁrst

text: in

text: the

text: Yang-Mills

text: action,

text: the

text: kinetic

text: operator

text: would

text: not

text: be

text: elliptic.

text: Deﬁning

text: a

text: Green’s

text: operator

text: for

text: the

text: resulting

text: operator

text: would

text: be

text: ambiguous

text: due

text: to

text: the

text: need

text: to

text: choose

text: a

text: regularization

text: scheme;

text: the

text: WML

text: prescription

text: α

text: provides

text: such

text: a

text: scheme,

text: yielding

text: a

text: well-deﬁned

text: propagator

text: that

text: is

text: continuously

text: connected

text: to

text: Green’s

text: operators

text: of

text: elliptic

text: operators

text: via

text: Wick

text: rotation.

text: In

text: the

text: above,

text: we

text: worked

text: in

text: a

text: coordinate-free

text: manner

text: as

text: possible

text: because

text: then

text: (i)

text: it

text: is

text: R

text: clear

text: that

text: our

text: analysis

text: holds

text: for

text: general

text: area

text: forms

text: on

text: (ii)

text: we

text: can

text: readily

text: adapt

text: the

text: above

text: analysis

text: to

text: the

text: case

text: of

text: holomorphic

text: gauge

text: on

text: compact

text: surfaces.

text: For

text: Σ

text: compact,

text: the

text: operator

text: Ω

text: (Σ)

text: Ω

text: (Σ)

text: has

text: a

text: kernel

text: and

text: cokernel.

text: For

text: the

text: simplest

text: case

text: Σ

text: ker

text: consists

text: of

text: just

text: constant

text: functions.

text: By

text: uniformization,

text: we

text: can

text: suppose

text: that

text: has

text: the

text: standard

text: complex

text: structure,

text: otherwise

text: we

text: can

text: apply

text: a

text: diﬀeomorphism

text: (and

text: change

text: our

text: area

text: form

text: accordingly).

text: We

text: then

text: have

text: the

text: standard

text: coordinate

text: charts

text: on

text: the

text: Riemann

text: C

text: ∪C

text: sphere

text: related

text: to

text: each

text: other

text: by

text: stereographic

text: projection

text: interchanging

text: z

text: with

text: z˜

text: In

text: the

text: chart

text: C

text: the

text: Green’s

text: operator

text: for

text: Ω0

text: has

text: the

text: same

text: z

text: C

text: expression

text: as

text: for

text: the

text: case

text: of

text: dw

text: P

text: (z,

text: w)

text: z

text: w

text: The

text: formula

text: for

text: the

text: holomorphic

text: gauge

text: propagator

text: on

text: involves

text: the

text: appropriate

text: modiﬁ-

text: cation

text: of

text: and

text: is

text: given

text: by

text: P

text: P

text: t

text: P

text: hol

text: with

text: the

text: integral

text: kernel

text: of

text: Phol

text: lying

text: and

text: Ω0

text: Ω2

text: f

text: f

text: fdσ

text: projects

text: Hodge

text: star

text: onto

text: the

text: top-degree

text: forms

text: that

text: integrate

text: to

text: zero

text: (which

text: are

text: thus

text: in

text: the

text: image

text: of

text: For

text: concreteness,

text: we

text: write

text: down

text: an

text: explicit

text: formula

text: for

text: P

text: when

text: hol

text: dσ

text: is

text: the

text: round

text: form.

text: can

text: explicitly

text: compute

text: z

text: P

text: (z,

text: w)dzdw

text: dzdw

text: dσ(u)δ(u

text: u

text: |dσ(u)dσ(u

text: C×C

text: z

text: u

text: u

text: w

text: z¯

text: w¯

text: dzdw

text: π

text: z

text: w

text: z

text: w

text: C

text: C

text: The

text: above

text: integral

text: must

text: be

text: interpreted

text: in

text: the

text: sense

text: of

text: distributions

text: on

text: since,

text: even

text: though

text: is

text: compact,

text: dz

text: and

text: dw

text: are

text: not

text: globally

text: deﬁned

text: coordinates.

text: Never-

text: theless,

text: we

text: obtain

text: the

text: correct

text: expression

text: for

text: P

text: (z,

text: w)dzdw

text: with

text: respect

text: to

text: the

text: holomorphic

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: dV − V 3 = C(V + R), V (0) = v(0) (10) dt 3 dR − 1 − = (V a + bR), R(0) = r(0) (11) dt C describe the reciprocal dependencies of the voltage compartment V across an axon membrane and a recovery compartment R summarizing outward currents. Solutions to this system of diﬀer- ential equations quickly converge for a range of starting values to periodic behavior that alternates between smooth evolution and sharp changes (bifurcations) in direction. These solutions exhibit features common to elements of biological neural networks [36]. The system has been used by a number of authors including [28, 19] to demonstrate methods of parameter estimation for data with time varying expectation given by this system of diﬀerential equations. For this example there are two random variables: the voltage across an axon membrane, Y (t), 1 and a recovery measurement of outward currents, Y (t). We assume that observed data, y (t) and 2 1 y (t) follow independent normal distributions with 2 ∼ Y (t) = V (t) + (cid:15) , (cid:15) i.i.d. N(0, σ2) (12) 1 1 1 1 ∼ Y (t) = R(t) + (cid:15) , (cid:15) i.i.d. N(0, σ2) (13) 2 2 2 2 where V (t) and R(t) satisfy the diﬀerential equations (10) and (11). The likelihood surface for data with expectation given by a nonlinear system of diﬀerential equations may be extremely complex and can contain multiple local maxima (equivalently, local minimum solutions to the nonlinear least squares regression model). For the Fitzhugh-Nagumo equations we considered two sets of parameters; C = 3, a = 0.58, b = 0.58 and C = 3, a = 0.34, b = 0.20. The parameters C = 3, a = 0.58, b = 0.58 are very near a supercritical Hopf bifurcation value of the system; from a stable steady state to an oscillating limit cycle. There is a dramatic change in the least squares surface as the parameters a and b vary about 0.58 (with C = 3 ﬁxed). Contours of the least squares surface for a set of zero noise data are shown in Figure 3 panels A, B, and C. Panel A shows the contours of the least square surface for a range of a = 0.4 to 1.4 and b = 0 to 0.2 to 2.2. The blue box contains the true solution, a = 0.58 and b = 0.58, the red box contains a local minimum. Panels B and C show close ups of the red and blue regions, respectively. Panel B suggests that there is local minimum in the least squares ﬁt surface at approximately a 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: dV

text: V

text: C(V

text: R),

text: V

text: dt

text: dR

text: (V

text: a

text: bR),

text: dt

text: C

text: describe

text: the

text: reciprocal

text: dependencies

text: of

text: the

text: voltage

text: compartment

text: V

text: across

text: an

text: axon

text: membrane

text: and

text: a

text: recovery

text: compartment

text: R

text: summarizing

text: outward

text: currents.

text: Solutions

text: to

text: this

text: system

text: of

text: diﬀer-

text: ential

text: equations

text: quickly

text: converge

text: for

text: a

text: range

text: of

text: starting

text: values

text: to

text: periodic

text: behavior

text: that

text: alternates

text: between

text: smooth

text: evolution

text: and

text: sharp

text: changes

text: (bifurcations)

text: in

text: direction.

text: These

text: solutions

text: exhibit

text: features

text: common

text: to

text: elements

text: of

text: biological

text: neural

text: networks

text: The

text: system

text: has

text: been

text: used

text: by

text: a

text: number

text: of

text: authors

text: including

text: to

text: demonstrate

text: methods

text: of

text: parameter

text: estimation

text: for

text: data

text: with

text: time

text: varying

text: expectation

text: given

text: by

text: this

text: system

text: of

text: diﬀerential

text: equations.

text: For

text: this

text: example

text: there

text: are

text: two

text: random

text: variables:

text: the

text: voltage

text: across

text: an

text: axon

text: membrane,

text: Y

text: (t),

text: and

text: a

text: recovery

text: measurement

text: of

text: outward

text: currents,

text: Y

text: (t).

text: We

text: assume

text: that

text: observed

text: data,

text: y

text: (t)

text: and

text: y

text: (t)

text: follow

text: independent

text: normal

text: distributions

text: with

text: Y

text: (t)

text: V

text: (t)

text: i.i.d.

text: Y

text: (t)

text: R(t)

text: i.i.d.

text: where

text: V

text: (t)

text: and

text: R(t)

text: satisfy

text: the

text: diﬀerential

text: equations

text: and

text: The

text: likelihood

text: surface

text: for

text: data

text: with

text: expectation

text: given

text: by

text: a

text: nonlinear

text: system

text: of

text: diﬀerential

text: equations

text: may

text: be

text: extremely

text: complex

text: and

text: can

text: contain

text: multiple

text: local

text: maxima

text: (equivalently,

text: local

text: minimum

text: solutions

text: to

text: the

text: nonlinear

text: least

text: squares

text: regression

text: model).

text: For

text: the

text: Fitzhugh-Nagumo

text: equations

text: we

text: considered

text: two

text: sets

text: of

text: parameters;

text: C

text: a

text: b

text: and

text: C

text: a

text: b

text: The

text: parameters

text: C

text: a

text: b

text: are

text: very

text: near

text: a

text: supercritical

text: Hopf

text: bifurcation

text: value

text: of

text: the

text: system;

text: from

text: a

text: stable

text: steady

text: state

text: to

text: an

text: oscillating

text: limit

text: cycle.

text: There

text: is

text: a

text: dramatic

text: change

text: in

text: the

text: least

text: squares

text: surface

text: as

text: the

text: parameters

text: a

text: and

text: b

text: vary

text: about

text: (with

text: C

text: ﬁxed).

text: Contours

text: of

text: the

text: least

text: squares

text: surface

text: for

text: a

text: set

text: of

text: zero

text: noise

text: data

text: are

text: shown

text: in

text: Figure

text: panels

text: A,

text: B,

text: and

text: C.

text: Panel

text: A

text: shows

text: the

text: contours

text: of

text: the

text: least

text: square

text: surface

text: for

text: a

text: range

text: of

text: a

text: to

text: and

text: b

text: to

text: to

text: The

text: blue

text: box

text: contains

text: the

text: true

text: solution,

text: a

text: and

text: b

text: the

text: red

text: box

text: contains

text: a

text: local

text: minimum.

text: Panels

text: B

text: and

text: C

text: show

text: close

text: ups

text: of

text: the

text: red

text: and

text: blue

text: regions,

text: respectively.

text: Panel

text: B

text: suggests

text: that

text: there

text: is

text: local

text: minimum

text: in

text: the

text: least

text: squares

text: ﬁt

text: surface

text: at

text: approximately

text: a

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: cov(Z Y , Z Y ) = cov(Z Y , (Z \ + Z )Y ) j j i i j j i (j+1) j+1 i = cov(Z Y , Z \ Y ) + cov(Z Y , Z Y ) j j i (j+1) i j j j+1 i 1 = 0 + cov(Z Y , [ h(cid:63)(Y ) + Z ]Y ) j j j j i n 1 { } = cov(Z Y , h(cid:63)(Y )Y ) + cov(Z Y , Z Y ) j j j i j j j i n 1 { − } = E[Z Y h(cid:63)(Y )Y ] M µ h(cid:63)(µ )µ + µ µ var(Z ) j j j i j j j i j i j n 1 { − } = M µ E[Y h(cid:63)(Y )] M µ h(cid:63)(µ )µ + µ µ var(Z ) j j j j j j j i j i j n 1 { − } = M µ (E[Y h(cid:63)(Y )] h(cid:63)(µ )µ ) + µ µ var(Z ) j j j j j i j i j n − = O(n 2) O(j) Similar calculations show that − − cov( Z Y , Z2β ) = O(n 2) O(j) + O(n 3) O(j2) j j i − cov( Z2β, Z Y ) = O(n 3) O(j2) j i i − − cov( Z2β, Z2β ) = O(n 3) O(j2) + O(n 4) O(j3) j i { } It follows that the contribution to var U (β) from the oﬀ-diagonal terms is: n 24 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: cov(Z

text: Y

text: Z

text: Y

text: cov(Z

text: Y

text: (Z

text: Z

text: )Y

text: j

text: j

text: i

text: i

text: j

text: j

text: i

text: i

text: cov(Z

text: Y

text: Z

text: Y

text: cov(Z

text: Y

text: Z

text: Y

text: j

text: j

text: i

text: i

text: j

text: j

text: i

text: cov(Z

text: Y

text: Z

text: ]Y

text: j

text: j

text: j

text: j

text: i

text: n

text: cov(Z

text: Y

text: )Y

text: cov(Z

text: Y

text: Z

text: Y

text: j

text: j

text: j

text: i

text: j

text: j

text: j

text: i

text: n

text: E[Z

text: Y

text: )Y

text: M

text: µ

text: )µ

text: µ

text: µ

text: var(Z

text: j

text: j

text: j

text: i

text: j

text: j

text: j

text: i

text: j

text: i

text: j

text: n

text: M

text: µ

text: E[Y

text: M

text: µ

text: )µ

text: µ

text: µ

text: var(Z

text: j

text: j

text: j

text: j

text: j

text: j

text: j

text: i

text: j

text: i

text: j

text: n

text: M

text: µ

text: (E[Y

text: )µ

text: µ

text: µ

text: var(Z

text: j

text: j

text: j

text: j

text: j

text: i

text: j

text: i

text: j

text: n

text: O(n

text: O(j)

text: Similar

text: calculations

text: show

text: that

text: cov(

text: Z

text: Y

text: O(n

text: O(j)

text: O(n

text: j

text: j

text: i

text: cov(

text: Z

text: Y

text: O(n

text: j

text: i

text: i

text: cov(

text: O(n

text: O(n

text: j

text: i

text: It

text: follows

text: that

text: the

text: contribution

text: to

text: var

text: U

text: (β)

text: from

text: the

text: oﬀ-diagonal

text: terms

text: is:

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [8] Efron, B., and Tibshirani, R. J. (1993) An Introduction to the Boostrap Chapman Hall:New York. [9] FitzHugh, R. (1961) “Impulses and Physiological States in MOndels of Nerve Membrame”, Biophysical Journal 1, 445–466. [10] Foss, S. D. (1971), “Estimates of chemical kinetic rate constants by numerical integration,” Chemical Engineering Science, 26, 485–486. [11] Freedman, H.I. (1980) Deterministic Mathematical Models in Population Ecology Marcel Dekker:New York. [12] Galecki A.T, Wolﬁnger R.D, Linares O.A, Smith M.J,Halter J.B. (2004), “Ordinary Diﬀerential Equations PK/PD Models Using the SAS Macro NLINMIX,” Journal of Biopharmaceutical Statistics 14, 483–503. [13] Gause, GF. (1934) The Struggle for Existence Willianms and Williams:Baltimore. [14] Himmelblau, D. M., Jones, C. R., and Bischoﬀ, K. B. “Determination of rate constants for complex kinetics models,” Industrial and Engineering Chemistry Fundamentals, 6, 539–543, 1967. [15] Ho, D. D., Neumann, A. U., Perelson, A. S., Chen, W., Leonard, J. M., and Markowitz M. (1995), “Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection,” Nature, 373, 123–126. [16] Hosten, L. H. (1979) “A comparative study of short cut procedures for parameter estimation in diﬀerential equations,” Computers and Chemical Engineering, 3, 117–126. [17] Hodgkin, A. L., and Huxley, A. F. (1952) “A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve,” Jorunal of Physiology 133, 444–479. [18] Jaquez JA. (1974) Compartmental Analysis in Biology and Medicine Elsevier:New York. [19] Liang, H. and Wu, H. (2008) “Parameter Estimation for diﬀerential Equation Models Using a Framework of Measurement Error in Regression Models,” Journal of the American Statistical Association, 103, 1570–1583. 27

label: reference

text: Efron,

text: B.,

text: and

text: Tibshirani,

text: R.

text: J.

text: An

text: Introduction

text: to

text: the

text: Boostrap

text: Chapman

text: Hall:New

text: York.

text: FitzHugh,

text: R.

text: “Impulses

text: and

text: Physiological

text: States

text: in

text: MOndels

text: of

text: Nerve

text: Membrame”,

text: Biophysical

text: Journal

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text: “Estimates

text: of

text: chemical

text: kinetic

text: rate

text: constants

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text: integration,”

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text: Engineering

text: Science,

text: Freedman,

text: H.I.

text: Deterministic

text: Mathematical

text: Models

text: in

text: Population

text: Ecology

text: Marcel

text: Dekker:New

text: York.

text: Galecki

text: A.T,

text: Wolﬁnger

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text: M.J,Halter

text: J.B.

text: “Ordinary

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text: Equations

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text: NLINMIX,”

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text: Biopharmaceutical

text: Statistics

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text: The

text: Struggle

text: for

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text: Williams:Baltimore.

text: Himmelblau,

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text: M.,

text: Jones,

text: C.

text: R.,

text: and

text: Bischoﬀ,

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text: “Determination

text: of

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text: constants

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text: complex

text: kinetics

text: models,”

text: Industrial

text: and

text: Engineering

text: Chemistry

text: Fundamentals,

text: Ho,

text: D.

text: D.,

text: Neumann,

text: A.

text: U.,

text: Perelson,

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text: S.,

text: Chen,

text: W.,

text: Leonard,

text: J.

text: M.,

text: and

text: Markowitz

text: M.

text: “Rapid

text: turnover

text: of

text: plasma

text: virions

text: and

text: lymphocytes

text: in

text: infection,”

text: Nature,

text: Hosten,

text: L.

text: H.

text: “A

text: comparative

text: study

text: of

text: short

text: cut

text: procedures

text: for

text: parameter

text: estimation

text: in

text: diﬀerential

text: equations,”

text: Computers

text: and

text: Chemical

text: Engineering,

text: Hodgkin,

text: A.

text: L.,

text: and

text: Huxley,

text: A.

text: F.

text: “A

text: Quantitative

text: Description

text: of

text: Membrane

text: Current

text: and

text: Its

text: Application

text: to

text: Conduction

text: and

text: Excitation

text: in

text: Nerve,”

text: Jorunal

text: of

text: Physiology

text: Jaquez

text: JA.

text: Compartmental

text: Analysis

text: in

text: Biology

text: and

text: Medicine

text: Elsevier:New

text: York.

text: Liang,

text: H.

text: and

text: Wu,

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text: “Parameter

text: Estimation

text: for

text: diﬀerential

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text: Models

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text: Framework

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text: Regression

text: Models,”

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association,


text: [32] Varah, J. M. (1982),“A spline least squares method for numerical parameter estimation in diﬀerential equations”, SIAM, Journal of Scientiﬁc and Statistical Computation, 3, 28–46. [33] Wei, X., Ghosh, S. K., Taylor, M. E., Johnson, V. A. Emini, E. A., Deutsch, P., Lifson, J. D., Bonhoeﬀer, S., Nowak, M. A., Hahn, B. H., Saag, M. S., Shaw, G. M. (1995),“Viral dynamics of HIV-1 infection,” Nature 373, 117–122. [34] Wikstrom, G. (1997) “Computation of parameters occurring linearly in systems of ordinary diﬀerential equations, part i,” Technical report, Department of Computing Science, Umea University, 1997. [35] Wikstrom, G. (1997) “Computation of parameters occurring linearly in systems of ordinary diﬀerential equations, part ii,” Technical report, Department of Computing Science, Umea University, 1997. [36] Wilson, H. (1999) Spikes, Decisions and Actions: the Dynamical Foundations of Neuroscience Oxford University Press, Oxford, England. [37] Zhang, W.B. (2005) Diﬀerential Equations, Bifurcations, and Chaos in Economics World Scientiﬁc Publishing Company. 29

text: Varah,

text: J.

text: M.

text: spline

text: least

text: squares

text: method

text: for

text: numerical

text: parameter

text: estimation

text: in

text: diﬀerential

text: equations”,

text: SIAM,

text: Journal

text: of

text: Scientiﬁc

text: and

text: Statistical

text: Computation,

text: Wei,

text: X.,

text: Ghosh,

text: S.

text: K.,

text: Taylor,

text: M.

text: E.,

text: Johnson,

text: V.

text: A.

text: Emini,

text: E.

text: A.,

text: Deutsch,

text: P.,

text: Lifson,

text: J.

text: D.,

text: Bonhoeﬀer,

text: S.,

text: Nowak,

text: M.

text: A.,

text: Hahn,

text: B.

text: H.,

text: Saag,

text: M.

text: S.,

text: Shaw,

text: G.

text: M.

text: dynamics

text: of

text: infection,”

text: Nature

text: Wikstrom,

text: G.

text: “Computation

text: of

text: parameters

text: occurring

text: linearly

text: in

text: systems

text: of

text: ordinary

text: diﬀerential

text: equations,

text: part

text: i,”

text: Technical

text: report,

text: Department

text: of

text: Computing

text: Science,

text: Umea

text: University,

text: Wikstrom,

text: G.

text: “Computation

text: of

text: parameters

text: occurring

text: linearly

text: in

text: systems

text: of

text: ordinary

text: diﬀerential

text: equations,

text: part

text: ii,”

text: Technical

text: report,

text: Department

text: of

text: Computing

text: Science,

text: Umea

text: University,

text: Wilson,

text: H.

text: Spikes,

text: Decisions

text: and

text: Actions:

text: the

text: Dynamical

text: Foundations

text: of

text: Neuroscience

text: Oxford

text: University

text: Press,

text: Oxford,

text: England.

text: Zhang,

text: W.B.

text: Diﬀerential

text: Equations,

text: Bifurcations,

text: and

text: Chaos

text: in

text: Economics

text: World

text: Scientiﬁc

text: Publishing

text: Company.


text: Table 3: Simulation Results for Parameters in FitzHugh-Nagamu Model. Least squares without bias correction (LS). Bias-corrected least squares (BCLS). Psuedo least squares with smoothing bandwidth 0.2 (PsLS bw = 0.2). Psuedo least squares with smoothing bandwidth 0.37 (PsLS bw = 0.37). Actual a = 0.34 b = 0.2 C = 3 v =-1 r =1 0 0 mean bias mean bias mean bias mean bias mean bias (s.e.) (s.e.) (s.e.) (s.e.) (s.e.) LS 0.340 -0.001 0.200 0.002 2.958 -0.014 -0.936 -0.064 1.013 0.013 ( 0.004 ) ( 0.014 ) ( 0.086 ) ( 0.281 ) ( 0.030 ) BCLS 0.340 0.000 0.200 0.001 2.967 -0.011 -0.984 -0.016 1.010 0.010 ( 0.004 ) ( 0.013 ) ( 0.084 ) ( 0.280 ) ( 0.029 ) PsLS bw = 0.2 0.340 -0.001 0.199 -0.007 2.973 -0.009 NA NA ( 0.024 ) ( 0.042 ) ( 0.045 ) PsLS bw = 0.37 0.336 -0.013 0.206 0.028 3.056 0.019 NA NA ( 0.013 ) ( 0.029 ) ( 0.046 ) Actual a = 0.58 b = 0.58 C = 3 v =-1 r =1 0 0 LS 0.581 0.002 0.578 -0.003 2.938 -0.021 -0.892 -0.108 1.020 0.020 ( 0.005 ) ( 0.014 ) ( 0.118 ) ( 0.248 ) ( 0.041 ) BCLS 0.581 0.002 0.579 -0.002 2.948 -0.017 -0.966 -0.034 1.017 0.017 ( 0.005 ) ( 0.014 ) ( 0.122 ) ( 0.242 ) ( 0.042 ) PsLS bw = 0.2 0.585 0.008 0.577 -0.005 2.959 -0.014 NA NA ( 0.025 ) ( 0.057) ( 0.060 ) PsLS bw = 0.37 0.581 0.002 0.573 -0.012 3.025 0.008 NA NA ( 0.016 ) ( 0.036) ( 0.065 ) 36 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Simulation

label: caption

text: Results

text: for

text: Parameters

text: in

text: FitzHugh-Nagamu

text: Model.

text: Least

text: squares

text: without

text: bias

text: correction

text: (LS).

text: Bias-corrected

text: least

text: squares

text: (BCLS).

text: Psuedo

text: least

text: squares

text: with

text: smoothing

text: bandwidth

text: (PsLS

text: bw

text: Psuedo

text: least

text: squares

text: with

text: smoothing

text: bandwidth

text: (PsLS

text: bw

text: Actual

label: table

text: a

text: b

text: C

text: v

text: r

text: mean

text: bias

text: mean

text: bias

text: mean

text: bias

text: mean

text: bias

text: mean

text: bias

text: (s.e.)

text: (s.e.)

text: (s.e.)

text: (s.e.)

text: (s.e.)

text: LS

text: BCLS

text: PsLS

text: bw

text: PsLS

text: bw

text: Actual

text: a

text: b

text: C

text: v

text: r

text: LS

text: BCLS

text: PsLS

text: bw

text: PsLS

text: bw

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Table 4: Simulation Results for Parameters in FitzHugh-Nagamu Model using Nonlinear Least- squares (NLS) with various starting values a=0.34 b=0.20 a =0.58 b=0.58 Starting Values Mean Mean percent Mean Mean percent a b (s.e.) (s.e.) conv. (s.e.) (s.e.) conv. 0.40 0.40 0.340 0.200 93.9% 1.154 2.018 31.7% (0.002) (0.015) (0.011) (0.032) 0.80 0.340 0.200 94.4% 0.580 0.581 94.7% (0.002) (0.015) (0.007) (0.010) 0.80 0.40 1.720 2.771 32.6 % 1.155 2.018 28.5% (0.012) (0.043) (0.010) (0.031) 0.80 1.720 2.770 33.7% 1.154 2.017 32.1% (0.013) (0.043) (0.011) (0.031) 1.2 0.40 1.720 2.770 36.2% 1.155 2.018 31.1% (0.012) (0.044) (0.011) (0.031) 0.80 1.720 2.772 36.2% 1.155 2.016 30.2% (0.013) (0.043) (0.011) (0.030) From BCLS 0.340 0.200 94.2% 0.580 0.581 95.1% (0.002) (0.015) (0.007) (0.010) 37 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Simulation

text: Results

text: for

text: Parameters

text: in

text: FitzHugh-Nagamu

text: Model

text: using

text: Nonlinear

text: Least-

text: squares

text: (NLS)

text: with

text: various

text: starting

text: values

text: a

text: Starting

text: Values

text: Mean

text: Mean

text: percent

text: Mean

text: Mean

text: percent

text: a

text: b

text: (s.e.)

text: (s.e.)

text: conv.

text: (s.e.)

text: (s.e.)

text: conv.

text: From

text: BCLS

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: matrix, although the variances associated with each state are not assumed to be the same for all states. In this work, we assume that Σ is known or will be obtained using an independent method. These variances are usually needed to construct the bias correction functions, h(cid:63). k − To estimate β, linear regression with response y(t) ( or y(t) y(t ) ) and the covariate matrix 0 z(t) is performed so that (cid:98) (cid:48) − (cid:48) β = (z(t) z(t)) 1z(t) y(t) (6) In [19], the authors refer to regression analysis of this type as pseudo-least squares, (PsLS) since the minimization that occurs with the linear regression algorithm is not minimizing a true likelihood. The same is true of bias-corrected least squares. (cid:98) To demonstrate consistency of the BCLS m(cid:88)ethod we rely on the following Theorem. Theorem 1. Assume conditions (A.1)-(A.3) (from Section 2.1 ) are satisﬁed and that β is the unique root n 1 n { − } of the estimating function, deﬁned by U (β) = zT y z β = 0 . Then: (cid:98) n n i i i (cid:80) i=1 I U (β ) is a consistent estimator of zero. n n(cid:80) { } II E 1 n ZT Z is bounded and non-zero. n i=1 i i III var{ 1 n ZT Z } → 0 as n → ∞(cid:98). n i=1 i i (cid:48) − (cid:48) Thus, the linear least squares estimate, β = (z(t) z(t)) 1z(t) y(t) is a consistent estimator of β. (cid:98) The ﬁnal conclusion, that the three conditions listed in Theorem 1 imply consistency of estimator β from standard linear regression follows from results in [4]. The key technical issue is that the derived covariates, z(t), are functions of the observed data rather than of the state means and are therefore measured with error, and that the covariates themselves are correlated with each other. However, contrary to the classical measurement error setting the variance of the covariates − is O(n 1) and therefore bias due to error in the covariates is asymptotically eliminated. It will be shown in the appendix that the BCLS method satisﬁes condition [I-III]. 3 Example: A single compartment nonlinear ODE 3.1 Data Analysis: Growth colonies of paramecium aurelium To illustrate the method using a model deﬁned by a nonlinear diﬀerential equation with a single compartment, data on the growth of four colonies of the unicellular organism, paramecium 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: matrix,

text: although

text: the

text: variances

text: associated

text: with

text: each

text: state

text: are

text: not

text: assumed

text: to

text: be

text: the

text: same

text: for

text: all

text: states.

text: In

text: this

text: work,

text: we

text: assume

text: that

text: Σ

text: is

text: known

text: or

text: will

text: be

text: obtained

text: using

text: an

text: independent

text: method.

text: These

text: variances

text: are

text: usually

text: needed

text: to

text: construct

text: the

text: bias

text: correction

text: functions,

text: k

text: To

text: estimate

text: β,

text: linear

text: regression

text: with

text: response

text: y(t)

text: or

text: y(t)

text: y(t

text: and

text: the

text: covariate

text: matrix

text: z(t)

text: is

text: performed

text: so

text: that

text: β

text: (z(t)

text: z(t))

text: y(t)

text: In

text: the

text: authors

text: refer

text: to

text: regression

text: analysis

text: of

text: this

text: type

text: as

text: pseudo-least

text: squares,

text: (PsLS)

text: since

text: the

text: minimization

text: that

text: occurs

text: with

text: the

text: linear

text: regression

text: algorithm

text: is

text: not

text: minimizing

text: a

text: true

text: likelihood.

text: The

text: same

text: is

text: true

text: of

text: bias-corrected

text: least

text: squares.

text: To

text: demonstrate

text: consistency

text: of

text: the

text: BCLS

text: we

text: rely

text: on

text: the

text: following

text: Theorem.

text: Theorem

text: Assume

text: conditions

text: (from

text: Section

text: are

text: satisﬁed

text: and

text: that

text: β

text: is

text: the

text: unique

text: root

text: n

text: n

text: of

text: the

text: estimating

text: function,

text: deﬁned

text: by

text: U

text: (β)

text: zT

text: y

text: z

text: β

text: Then:

text: n

text: n

text: i

text: i

text: i

text: I

text: U

text: (β

text: is

text: a

text: consistent

text: estimator

text: of

text: zero.

text: n

text: II

text: E

text: n

text: ZT

text: Z

text: is

text: bounded

text: and

text: non-zero.

text: n

text: i

text: i

text: III

text: var{

text: n

text: ZT

text: Z

text: as

text: n

text: n

text: i

text: i

text: Thus,

text: the

text: linear

text: least

text: squares

text: estimate,

text: β

text: (z(t)

text: z(t))

text: y(t)

text: is

text: a

text: consistent

text: estimator

text: of

text: β.

text: The

text: ﬁnal

text: conclusion,

text: that

text: the

text: three

text: conditions

text: listed

text: in

text: Theorem

text: imply

text: consistency

text: of

text: estimator

text: β

text: from

text: standard

text: linear

text: regression

text: follows

text: from

text: results

text: in

text: The

text: key

text: technical

text: issue

text: is

text: that

text: the

text: derived

text: covariates,

text: z(t),

text: are

text: functions

text: of

text: the

text: observed

text: data

text: rather

text: than

text: of

text: the

text: state

text: means

text: and

text: are

text: therefore

text: measured

text: with

text: error,

text: and

text: that

text: the

text: covariates

text: themselves

text: are

text: correlated

text: with

text: each

text: other.

text: However,

text: contrary

text: to

text: the

text: classical

text: measurement

text: error

text: setting

text: the

text: variance

text: of

text: the

text: covariates

text: is

text: O(n

text: and

text: therefore

text: bias

text: due

text: to

text: error

text: in

text: the

text: covariates

text: is

text: asymptotically

text: eliminated.

text: It

text: will

text: be

text: shown

text: in

text: the

text: appendix

text: that

text: the

text: BCLS

text: method

text: satisﬁes

text: condition

text: [I-III].

text: Example:

text: A

text: single

text: compartment

text: nonlinear

text: ODE

text: Data

text: Analysis:

text: Growth

text: colonies

text: of

text: paramecium

text: aurelium

text: To

text: illustrate

text: the

text: method

text: using

text: a

text: model

text: deﬁned

text: by

text: a

text: nonlinear

text: diﬀerential

text: equation

text: with

text: a

text: single

text: compartment,

text: data

text: on

text: the

text: growth

text: of

text: four

text: colonies

text: of

text: the

text: unicellular

text: organism,

text: paramecium

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: argued that in a stable network of relationships every triad should have an even number of negative (antagonistic) edges. In essence these networks are ones which satisfy the aphorism “the enemy of my enemy is my friend.” For example balance theory suggests that a network with three mutually antag- onistic groups is unstable, with two of the groups making common cause against the third. Harary and Cartwright [3] generalized this condition to require an even number of negative edges in every cycle, and showed that balanced graphs are exactly bi-partitite graphs where edges within a group have positive weights and edges between groups have a negative weight. The works of Heider and Cartwright and Harary are static, but the language used is strongly suggestive of a dynamical process, and there have been several attempts to introduce dynamical models of this process. One such model was introduced by Antal, Krapivsky and Redner [1]. In this discrete time model, unstable triangles transition, with some probability, to stable ones by ﬂipping the sign of an edge. Another model is the one introduced by Kulakowski, Gawr´onski and Gronek, [10] and later analyzed by Marvel, Kleinberg, Kleinberg and Strogatz [11, 12]. The Kulakowski-Gawr´onski- Gronek model takes the form of a single matrix Ricatti equation dX = X2, dt where X denotes the opinion actor i holds of actor j. This model was ij explicitly solved in the symmetric case (X = X ) by Marvel, Kleinberg, ij ji Kleinberg and Strogatz, who showed that for generic initial conditions the matrix X(t) converges in ﬁnite time to a balanced state. While these models are very interesting, they are phenomenonological: the form of the dynamics is chosen to drive the edge weights towards the bal- anced state. It would be preferable to ﬁnd a model in which the balance state was not assumed but rather emergent from the dynamics. Further in the modeling it is not necessarily desirable to assume that the underlying graph is the complete graph, where all actors know each other, but it is not clear how to extend the Kulakowski-Gawr´onski-Gronek model to a more general graph which might have few or no triangles. In this paper we propose and analyze such a model: each actor behaves in a very natural way, and balance arises naturally from the asymptotic steady states of the model. The model we consider is posed on a graph Γ, representing a network of relations. The graph has N vertices, representing a number of actors, and | | (cid:0) (cid:1) E edges, representing the relations between pairs of actors. For this paper we assume that the underlying graph is the complete graph, where all actors | | know each other, so E = N but the model extends in a straightforward 2 way to an arbitrary graph. There are two types of variables in this model: 2 ##LTLine##

text: argued

text: that

text: in

text: a

text: stable

text: network

text: of

text: relationships

text: every

text: triad

text: should

text: have

text: an

text: even

text: number

text: of

text: negative

text: (antagonistic)

text: edges.

text: In

text: essence

text: these

text: networks

text: are

text: ones

text: which

text: satisfy

text: the

text: aphorism

text: “the

text: enemy

text: of

text: my

text: enemy

text: is

text: my

text: friend.”

text: For

text: example

text: balance

text: theory

text: suggests

text: that

text: a

text: network

text: with

text: three

text: mutually

text: antag-

text: onistic

text: groups

text: is

text: unstable,

text: with

text: two

text: of

text: the

text: groups

text: making

text: common

text: cause

text: against

text: the

text: third.

text: Harary

text: and

text: Cartwright

text: generalized

text: this

text: condition

text: to

text: require

text: an

text: even

text: number

text: of

text: negative

text: edges

text: in

text: every

text: cycle,

text: and

text: showed

text: that

text: balanced

text: graphs

text: are

text: exactly

text: bi-partitite

text: graphs

text: where

text: edges

text: within

text: a

text: group

text: have

text: positive

text: weights

text: and

text: edges

text: between

text: groups

text: have

text: a

text: negative

text: weight.

text: The

text: works

text: of

text: Heider

text: and

text: Cartwright

text: and

text: Harary

text: are

text: static,

text: but

text: the

text: language

text: used

text: is

text: strongly

text: suggestive

text: of

text: a

text: dynamical

text: process,

text: and

text: there

text: have

text: been

text: several

text: attempts

text: to

text: introduce

text: dynamical

text: models

text: of

text: this

text: process.

text: One

text: such

text: model

text: was

text: introduced

text: by

text: Antal,

text: Krapivsky

text: and

text: Redner

text: In

text: this

text: discrete

text: time

text: model,

text: unstable

text: triangles

text: transition,

text: with

text: some

text: probability,

text: to

text: stable

text: ones

text: by

text: ﬂipping

text: the

text: sign

text: of

text: an

text: edge.

text: Another

text: model

text: is

text: the

text: one

text: introduced

text: by

text: Kulakowski,

text: Gawr´onski

text: and

text: Gronek,

text: and

text: later

text: analyzed

text: by

text: Marvel,

text: Kleinberg,

text: Kleinberg

text: and

text: Strogatz

text: The

text: Kulakowski-Gawr´onski-

text: Gronek

text: model

text: takes

text: the

text: form

text: of

text: a

text: single

text: matrix

text: Ricatti

text: equation

text: dX

text: dt

text: where

text: X

text: denotes

text: the

text: opinion

text: actor

text: i

text: holds

text: of

text: actor

text: j.

text: This

text: model

text: was

text: ij

text: explicitly

text: solved

text: in

text: the

text: symmetric

text: case

text: (X

text: X

text: by

text: Marvel,

text: Kleinberg,

text: ij

text: ji

text: Kleinberg

text: and

text: Strogatz,

text: who

text: showed

text: that

text: for

text: generic

text: initial

text: conditions

text: the

text: matrix

text: X(t)

text: converges

text: in

text: ﬁnite

text: time

text: to

text: a

text: balanced

text: state.

text: While

text: these

text: models

text: are

text: very

text: interesting,

text: they

text: are

text: phenomenonological:

text: the

text: form

text: of

text: the

text: dynamics

text: is

text: chosen

text: to

text: drive

text: the

text: edge

text: weights

text: towards

text: the

text: bal-

text: anced

text: state.

text: It

text: would

text: be

text: preferable

text: to

text: ﬁnd

text: a

text: model

text: in

text: which

text: the

text: balance

text: state

text: was

text: not

text: assumed

text: but

text: rather

text: emergent

text: from

text: the

text: dynamics.

text: Further

text: in

text: the

text: modeling

text: it

text: is

text: not

text: necessarily

text: desirable

text: to

text: assume

text: that

text: the

text: underlying

text: graph

text: is

text: the

text: complete

text: graph,

text: where

text: all

text: actors

text: know

text: each

text: other,

text: but

text: it

text: is

text: not

text: clear

text: how

text: to

text: extend

text: the

text: Kulakowski-Gawr´onski-Gronek

text: model

text: to

text: a

text: more

text: general

text: graph

text: which

text: might

text: have

text: few

text: or

text: no

text: triangles.

text: In

text: this

text: paper

text: we

text: propose

text: and

text: analyze

text: such

text: a

text: model:

text: each

text: actor

text: behaves

text: in

text: a

text: very

text: natural

text: way,

text: and

text: balance

text: arises

text: naturally

text: from

text: the

text: asymptotic

text: steady

text: states

text: of

text: the

text: model.

text: The

text: model

text: we

text: consider

text: is

text: posed

text: on

text: a

text: graph

text: Γ,

text: representing

text: a

text: network

text: of

text: relations.

text: The

text: graph

text: has

text: N

text: vertices,

text: representing

text: a

text: number

text: of

text: actors,

text: and

text: E

text: edges,

text: representing

text: the

text: relations

text: between

text: pairs

text: of

text: actors.

text: For

text: this

text: paper

text: we

text: assume

text: that

text: the

text: underlying

text: graph

text: is

text: the

text: complete

text: graph,

text: where

text: all

text: actors

text: know

text: each

text: other,

text: so

text: E

text: N

text: but

text: the

text: model

text: extends

text: in

text: a

text: straightforward

text: way

text: to

text: an

text: arbitrary

text: graph.

text: There

text: are

text: two

text: types

text: of

text: variables

text: in

text: this

text: model:

text: ##LTLine##


text: (cid:18) (cid:19) from which we see that v is monotone decreasing and in particular that ≤ − − v2(0) v˙ 2σ v 1 . min R A direct argument or the use of Gronwall’s inequality shows that v = 0 is exponentially attracting. Thus we have proved Theorem 4.1 (Stability of Consensus State I) Suppose (cid:15) = 0. Sup- (cid:114) pose also that L(0) is positive semi-deﬁnite with a 1 dimensional kernel. Then it holds that R lim x(t) = 1 →∞ N t N That is, the consensus state is globally asymptotically stable. The demonstration above relied mostly on the fact that we had a spectral gap. We know that 0 is always an eigenvalue of L but the assumption (cid:15) = 0 was enough to guarantee that the next eigenvalue was strictly positive if (cid:54) L(0) had all nonnegative eigenvalues. If (cid:15) = 0, we can still ﬁnd suﬃcient conditions guaranteeing the existence of a spectral gap. Theorem 4.2 (Suﬃcient Conditions for Global Stability) Suppose that − P Q2 1 < − . Q2 N 1 Then L is always positive semi-deﬁnite and the consesnsus state is a global minimizer. Proof. For the proof we will verify that, under the stated assumption, on P, Q, L is positive semi-deﬁnite with a 1-dimensional kernel. We write each opinion as a mean plus a mean-zero part, γ = Q + γ˜ ij ij where mean-zero part γ˜ now satisﬁes ij (cid:88) 1 (cid:88) γ˜ = 0 (22) ij E i<j 1 − γ˜2 = P Q2. (23) E ij i<j The corresponding graph Laplacian takes the form 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: from

text: which

text: we

text: see

text: that

text: v

text: is

text: monotone

text: decreasing

text: and

text: in

text: particular

text: that

text: v˙

text: 2σ

text: v

text: min

text: R

text: A

text: direct

text: argument

text: or

text: the

text: use

text: of

text: Gronwall’s

text: inequality

text: shows

text: that

text: v

text: is

text: exponentially

text: attracting.

text: Thus

text: we

text: have

text: proved

text: Theorem

text: (Stability

text: of

text: Consensus

text: State

text: I)

text: Suppose

text: Sup-

text: pose

text: also

text: that

text: is

text: positive

text: semi-deﬁnite

text: with

text: a

text: dimensional

text: kernel.

text: Then

text: it

text: holds

text: that

text: R

text: lim

text: x(t)

text: N

text: t

text: N

text: That

text: is,

text: the

text: consensus

text: state

text: is

text: globally

text: asymptotically

text: stable.

text: The

text: demonstration

text: above

text: relied

text: mostly

text: on

text: the

text: fact

text: that

text: we

text: had

text: a

text: spectral

text: gap.

text: We

text: know

text: that

text: is

text: always

text: an

text: eigenvalue

text: of

text: L

text: but

text: the

text: assumption

text: was

text: enough

text: to

text: guarantee

text: that

text: the

text: next

text: eigenvalue

text: was

text: strictly

text: positive

text: if

text: had

text: all

text: nonnegative

text: eigenvalues.

text: If

text: we

text: can

text: still

text: ﬁnd

text: suﬃcient

text: conditions

text: guaranteeing

text: the

text: existence

text: of

text: a

text: spectral

text: gap.

text: Theorem

text: (Suﬃcient

text: Conditions

text: for

text: Global

text: Stability)

text: Suppose

text: that

text: P

text: N

text: Then

text: L

text: is

text: always

text: positive

text: semi-deﬁnite

text: and

text: the

text: consesnsus

text: state

text: is

text: a

text: global

text: minimizer.

text: Proof.

text: For

text: the

text: proof

text: we

text: will

text: verify

text: that,

text: under

text: the

text: stated

text: assumption,

text: on

text: P,

text: Q,

text: L

text: is

text: positive

text: semi-deﬁnite

text: with

text: a

text: kernel.

text: We

text: write

text: each

text: opinion

text: as

text: a

text: mean

text: plus

text: a

text: mean-zero

text: part,

text: γ

text: Q

text: γ˜

text: ij

text: ij

text: where

text: mean-zero

text: part

text: γ˜

text: now

text: satisﬁes

text: ij

text: γ˜

text: ij

text: E

text: i<j

text: γ˜2

text: P

text: E

text: ij

text: i<j

text: The

text: corresponding

text: graph

text: Laplacian

text: takes

text: the

text: form

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: √ R| | constraint g = P deﬁnes a sphere of radius P in E . If P = Q2, 2 ∗ ∗ then they intersect tangentially at the single point γ = Q, i.e. γ = ∗ ij Q(1, 1, . . . , 1)t. Note that L(γ ) = L and that the spectrum of L consists ij 0 − 0 of 0 which is simple and NQ of multiplicity N 1. This is clearly positive semi-deﬁnite with a one dimensional kernel as long as Q > 0. What we have − ∼ really shown is that for P Q2 small then L still satisﬁes this property as well. Theorem 4.3 Let (cid:15) > 0 be arbitrary but ﬁxed. Suppose that σ (M(0)) > min 0 i.e., L(0) is positive semi-deﬁnite with a 1 dimensional kernel. Then there is a positive δ = δ((cid:15), σ (M(0)), R, N, P, Q) such that θ = 0 is attracting min in the neighborhood N (0). In other words, if L(0) satisﬁes the stated hy- δ pothesis then the consensus state is locally stable. 5 Stability of The Bipartite State In this section we shall address the issue of stability of the explicitly con- structed bipartite states. The approach is to linearize the ﬂow about the bipartite equilibria and to count the number of negative eigenvalues, i.e. the index, of the resulting linear map. Recall that the critical points of the ﬂow are precisely the constrained ex- trema of the Dirichlet energy D i.e. the critical points of D. The method of Lagrange multipliers and a standard Lyapunov function argument gives the following result ∈ Lemma 5.1 Let ω Ω be a critical point of D i.e. a local extrema of D 0 subject to the constraints (1),(2), (3). Let the associa(cid:12)ted Hessian (cid:12) (cid:12)  (cid:12)  (cid:12)  ∂2D ∂2D (cid:12)  (cid:12)  ∂x ∂x ∂γ ∂x (cid:12) − i k lk i (cid:12) H(ω ) = (cid:12) . (26) 0 ∂2D ∂2D (cid:15) (cid:15) ∂x ∂γ ∂γ ∂γ k ij lk ij ω0 | Let T Ω be the tangent space to Ω at ω . If H(ω ) is negative deﬁnite, ω0 0 0 Tω0Ω then the gradient ﬂow is stable near ω . 0 Thus we need only determine the number of negative eigenvalues of the Hessian, H, restricted to T Ω. To facilitate the computation we need the following R Lemma 5.2 Let A be a symmetric invertible m(cid:12)atrix in n. Given any ⊂ R (cid:12) linear subspace S n then | − n±(A) = n±(A ) + n±(A 1 ⊥). S S 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: R|

text: constraint

text: g

text: P

text: deﬁnes

text: a

text: sphere

text: of

text: radius

text: P

text: in

text: E

text: If

text: P

text: then

text: they

text: intersect

text: tangentially

text: at

text: the

text: single

text: point

text: γ

text: Q,

text: i.e.

text: γ

text: ij

text: Note

text: that

text: L(γ

text: L

text: and

text: that

text: the

text: spectrum

text: of

text: L

text: consists

text: ij

text: of

text: which

text: is

text: simple

text: and

text: NQ

text: of

text: multiplicity

text: N

text: This

text: is

text: clearly

text: positive

text: semi-deﬁnite

text: with

text: a

text: one

text: dimensional

text: kernel

text: as

text: long

text: as

text: Q

text: What

text: we

text: have

text: really

text: shown

text: is

text: that

text: for

text: P

text: small

text: then

text: L

text: still

text: satisﬁes

text: this

text: property

text: as

text: well.

text: Theorem

text: Let

text: be

text: arbitrary

text: but

text: ﬁxed.

text: Suppose

text: that

text: σ

text: min

text: i.e.,

text: is

text: positive

text: semi-deﬁnite

text: with

text: a

text: dimensional

text: kernel.

text: Then

text: there

text: is

text: a

text: positive

text: δ

text: σ

text: R,

text: N,

text: P,

text: Q)

text: such

text: that

text: θ

text: is

text: attracting

text: min

text: in

text: the

text: neighborhood

text: N

text: In

text: other

text: words,

text: if

text: satisﬁes

text: the

text: stated

text: hy-

text: δ

text: pothesis

text: then

text: the

text: consensus

text: state

text: is

text: locally

text: stable.

text: Stability

text: of

text: The

text: Bipartite

text: State

text: In

text: this

text: section

text: we

text: shall

text: address

text: the

text: issue

text: of

text: stability

text: of

text: the

text: explicitly

text: con-

text: structed

text: bipartite

text: states.

text: The

text: approach

text: is

text: to

text: linearize

text: the

text: ﬂow

text: about

text: the

text: bipartite

text: equilibria

text: and

text: to

text: count

text: the

text: number

text: of

text: negative

text: eigenvalues,

text: i.e.

text: the

text: index,

text: of

text: the

text: resulting

text: linear

text: map.

text: Recall

text: that

text: the

text: critical

text: points

text: of

text: the

text: ﬂow

text: are

text: precisely

text: the

text: constrained

text: ex-

text: trema

text: of

text: the

text: Dirichlet

text: energy

text: D

text: i.e.

text: the

text: critical

text: points

text: of

text: D.

text: The

text: method

text: of

text: Lagrange

text: multipliers

text: and

text: a

text: standard

text: Lyapunov

text: function

text: argument

text: gives

text: the

text: following

text: result

text: Lemma

text: Let

text: ω

text: Ω

text: be

text: a

text: critical

text: point

text: of

text: D

text: i.e.

text: a

text: local

text: extrema

text: of

text: D

text: subject

text: to

text: the

text: constraints

text: Let

text: the

text: Hessian

text: ∂x

text: ∂x

text: ∂γ

text: ∂x

text: i

text: k

text: lk

text: i

text: H(ω

text: ∂x

text: ∂γ

text: ∂γ

text: ∂γ

text: k

text: ij

text: lk

text: ij

text: ω0

text: Let

text: T

text: Ω

text: be

text: the

text: tangent

text: space

text: to

text: Ω

text: at

text: ω

text: If

text: H(ω

text: is

text: negative

text: deﬁnite,

text: ω0

text: Tω0Ω

text: then

text: the

text: gradient

text: ﬂow

text: is

text: stable

text: near

text: ω

text: Thus

text: we

text: need

text: only

text: determine

text: the

text: number

text: of

text: negative

text: eigenvalues

text: of

text: the

text: Hessian,

text: H,

text: restricted

text: to

text: T

text: Ω.

text: To

text: facilitate

text: the

text: computation

text: we

text: need

text: the

text: following

text: R

text: Lemma

text: Let

text: A

text: be

text: a

text: symmetric

text: invertible

text: in

text: n.

text: Given

text: any

text: R

text: linear

text: subspace

text: S

text: n

text: then

text: n±(A)

text: n±(A

text: n±(A

text: S

text: S

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: • Positions x (t) are associated with the vertices, and represent the posi- i tion of actor i on some issue which can be represented as a continuum: conservative vs. liberal, tastes great vs. less ﬁlling, etc. • Opinions γ are associated with the edges in the graph, and represent ij the degree of friendliness or respect between actor i and actor j, with γ > 0 representing friendliness and γ < 0 antagonism. ij ij We will not initially assume that the opinions γ are symmetric: but we will ij show that this emerges naturally from the dynamics: the steady states of the model all have the property that the opinions are symmetric: γ = γ . ij ji (cid:88) We associate to the quantities x , γ a Dirichlet energy i ij D − (x, γ) = γ (x x )2, ij i j i>j which represents the total amount of disharmony in the system. Note that γ above can be of either sign. If all γ > 0 (friendly relations) then the ij ij energy is minimized when the actors take the same position, x = x : friends i j like to agree. If, on the other hand, γ < 0, then the energy is minimized − ij when (x x )2 is large: antagonists prefer to disagree. i j The basic dynamics of the model is as follows: we assume that all actors D act continuously in time so as to minimize ((cid:126)x,(cid:126)γ) subject to the following constraints. (cid:88) 1 (cid:88) γ = Q > 0 (1) ij 2E (cid:54) i=j 1 γ2 = P (2) 2E (cid:88)(cid:54) ij i=j N x2 = R (3) i i=1 The ﬁrst constraint requires that the sum of the opinions must be a positive constant. This can be interpreted as a societal pressure towards civil dis- course: while actors may hold negative opinions of each other the average opinion must be positive. The second constraint guarantees that no actor can hold an opinion that is too extreme. Note that the Cauchy-Schwartz inequality implies that − ≥ P Q2 0. − The quantity P Q2 represents some socially acceptable range of opinions, and thus is analogous to an entropy. The Lagrange multiplier that enforces 3 ##LTLine## ##LTLine##

label: list

text: Positions

text: x

text: (t)

text: are

text: associated

text: with

text: the

text: vertices,

text: and

text: represent

text: the

text: posi-

text: i

text: tion

text: of

text: actor

text: i

text: on

text: some

text: issue

text: which

text: can

text: be

text: represented

text: as

text: a

text: continuum:

text: conservative

text: vs.

text: liberal,

text: tastes

text: great

text: vs.

text: less

text: ﬁlling,

text: etc.

text: Opinions

text: γ

text: are

text: associated

text: with

text: the

text: edges

text: in

text: the

text: graph,

text: and

text: represent

text: ij

text: the

text: degree

text: of

text: friendliness

text: or

text: respect

text: between

text: actor

text: i

text: and

text: actor

text: j,

text: with

text: γ

text: representing

text: friendliness

text: and

text: γ

text: antagonism.

text: ij

text: ij

text: We

text: will

text: not

text: initially

text: assume

text: that

text: the

text: opinions

text: γ

text: are

text: symmetric:

text: but

text: we

text: will

text: ij

text: show

text: that

text: this

text: emerges

text: naturally

text: from

text: the

text: dynamics:

text: the

text: steady

text: states

text: of

text: the

text: model

text: all

text: have

text: the

text: property

text: that

text: the

text: opinions

text: are

text: symmetric:

text: γ

text: γ

text: ij

text: ji

text: We

text: associate

text: to

text: the

text: quantities

text: x

text: γ

text: a

text: Dirichlet

text: energy

text: i

text: ij

text: D

text: (x,

text: γ)

text: γ

text: (x

text: x

text: ij

text: i

text: j

text: i>j

text: which

text: represents

text: the

text: total

text: amount

text: of

text: disharmony

text: in

text: the

text: system.

text: Note

text: that

text: γ

text: above

text: can

text: be

text: of

text: either

text: sign.

text: If

text: all

text: γ

text: (friendly

text: relations)

text: then

text: the

text: ij

text: ij

text: energy

text: is

text: minimized

text: when

text: the

text: actors

text: take

text: the

text: same

text: position,

text: x

text: x

text: friends

text: i

text: j

text: like

text: to

text: agree.

text: If,

text: on

text: the

text: other

text: hand,

text: γ

text: then

text: the

text: energy

text: is

text: minimized

text: ij

text: when

text: (x

text: x

text: is

text: large:

text: antagonists

text: prefer

text: to

text: disagree.

text: i

text: j

text: The

text: basic

text: dynamics

text: of

text: the

text: model

text: is

text: as

text: follows:

text: we

text: assume

text: that

text: all

text: actors

text: D

text: act

text: continuously

text: in

text: time

text: so

text: as

text: to

text: minimize

text: subject

text: to

text: the

text: following

text: constraints.

text: γ

text: Q

text: ij

text: i=j

text: γ2

text: P

text: ij

text: i=j

text: N

text: R

text: i

text: The

text: ﬁrst

text: constraint

text: requires

text: that

text: the

text: sum

text: of

text: the

text: opinions

text: must

text: be

text: a

text: positive

text: constant.

text: This

text: can

text: be

text: interpreted

text: as

text: a

text: societal

text: pressure

text: towards

text: civil

text: dis-

text: course:

text: while

text: actors

text: may

text: hold

text: negative

text: opinions

text: of

text: each

text: other

text: the

text: average

text: opinion

text: must

text: be

text: positive.

text: The

text: second

text: constraint

text: guarantees

text: that

text: no

text: actor

text: can

text: hold

text: an

text: opinion

text: that

text: is

text: too

text: extreme.

text: Note

text: that

text: the

text: Cauchy-Schwartz

text: inequality

text: implies

text: that

text: P

text: The

text: quantity

text: P

text: represents

text: some

text: socially

text: acceptable

text: range

text: of

text: opinions,

text: and

text: thus

text: is

text: analogous

text: to

text: an

text: entropy.

text: The

text: Lagrange

text: multiplier

text: that

text: enforces

text: ##LTLine##

text: ##LTLine##


text: [13] M Mobilia. Does a single zealot aﬀect an inﬁnite group of voters. Phys. Rev. Lett., 91, 2003. [14] F. Shi, P.J. Mucha, and R. Durrett. Multiopinion coevolving voter model with inﬁnitely many phase transitions. Phys. Rev. E, 88, 2013. [15] W. Zhang, C. Lim, S. Screenivasan, J. Xie, B.K. Szymanski, and G. Ko- rniss. Social inﬂuencing and associated random walk models: Asymp- totic consensus times on the complete graph. Chaos, 21, 2011. 8 Appendix In this section we show the existence of a locally attracting neighborhood of the consensus state. The argument is similar in spirit to that for global stability. The main complication is that we have to explicitly handle the evolution equation for the spectrum of L. For convenience, we deﬁne 1ˆ = √1 1 to be the normalized vector of all N N { ∈ R | · N} ones. Let W = y N y 1ˆ = 0 be the subspace of vectors orthogonal N(cid:0) (cid:1) to the “consensus state”. Following a similar argument as before, we will write x = c cos(θ)1ˆ + sin(θ)y N ∈ (cid:107) (cid:107) → with y W and y = 1. The goal is to show that θ 0 which will imply the stability of the consensus state. We will ﬁrst obtain equations governing the dynamics of the relevant variables. · From the constraint x x˙ = 0, it follows that (cid:104) (cid:105) λ = sin2(θ) Ly, y . − − T(cid:16)he original equation x˙ = 2(L(cid:126)x(cid:17) λx) a(cid:0)nd the fact th(cid:0)at L1ˆN = 0 then (cid:1)(cid:1) implies − − − sin(θ)θ˙1ˆ + cos(θ)θ˙y + sin(θ)y˙ = 2 sin(θ)Ly λ cos(θ)1ˆ + sin(θ)y . N N Taking the inner product of the above equation with 1ˆ and simplifying N gives − θ˙ = sin(2θ)(Ly, y). (35) Using the equation for θ, we can also determine that − − (cid:104) (cid:105) y˙ = 2(L(cid:126)y Ly, y y). (36) 24 ##LTLine## ##LTLine##

text: M

text: Mobilia.

text: Does

text: a

text: single

text: zealot

text: aﬀect

text: an

text: inﬁnite

text: group

text: of

text: voters.

text: Phys.

text: Rev.

text: Lett.,

text: F.

text: Shi,

text: P.J.

text: Mucha,

text: and

text: R.

text: Durrett.

text: Multiopinion

text: coevolving

text: voter

text: model

text: with

text: inﬁnitely

text: many

text: phase

text: transitions.

text: Phys.

text: Rev.

text: E,

text: W.

text: Zhang,

text: C.

text: Lim,

text: S.

text: Screenivasan,

text: J.

text: Xie,

text: B.K.

text: Szymanski,

text: and

text: G.

text: Ko-

text: rniss.

text: Social

text: inﬂuencing

text: and

text: associated

text: random

text: walk

text: models:

text: Asymp-

text: totic

text: consensus

text: times

text: on

text: the

text: complete

text: graph.

text: Chaos,

label: section

text: Appendix

text: In

text: this

text: section

text: we

text: show

text: the

text: existence

text: of

text: a

text: locally

text: attracting

text: neighborhood

text: of

text: the

text: consensus

text: state.

text: The

text: argument

text: is

text: similar

text: in

text: spirit

text: to

text: that

text: for

text: global

text: stability.

text: The

text: main

text: complication

text: is

text: that

text: we

text: have

text: to

text: explicitly

text: handle

text: the

text: evolution

text: equation

text: for

text: the

text: spectrum

text: of

text: L.

text: For

text: convenience,

text: we

text: deﬁne

text: 1ˆ

text: √1

text: to

text: be

text: the

text: normalized

text: vector

text: of

text: all

text: N

text: N

text: R

text: N}

text: ones.

text: Let

text: W

text: y

text: N

text: y

text: 1ˆ

text: be

text: the

text: subspace

text: of

text: vectors

text: orthogonal

text: to

text: the

text: “consensus

text: state”.

text: Following

text: a

text: similar

text: argument

text: as

text: before,

text: we

text: will

text: write

text: x

text: c

text: sin(θ)y

text: N

text: with

text: y

text: W

text: and

text: y

text: The

text: goal

text: is

text: to

text: show

text: that

text: θ

text: which

text: will

text: imply

text: the

text: stability

text: of

text: the

text: consensus

text: state.

text: We

text: will

text: ﬁrst

text: obtain

text: equations

text: governing

text: the

text: dynamics

text: of

text: the

text: relevant

text: variables.

text: From

text: the

text: constraint

text: x

text: x˙

text: it

text: follows

text: that

text: λ

text: Ly,

text: y

text: original

text: equation

text: x˙

text: λx)

text: the

text: fact

text: then

text: implies

text: cos(θ)θ˙y

text: sin(θ)y˙

text: sin(θ)Ly

text: λ

text: sin(θ)y

text: N

text: N

text: Taking

text: the

text: inner

text: product

text: of

text: the

text: above

text: equation

text: with

text: 1ˆ

text: and

text: simplifying

text: N

text: gives

text: θ˙

text: y).

text: Using

text: the

text: equation

text: for

text: θ,

text: we

text: can

text: also

text: determine

text: that

text: y˙

text: Ly,

text: y

text: y).

text: ##LTLine##

text: ##LTLine##


text: These are simultaneous polynomial equations, so in general it is diﬃcult to ﬁnd all solutions, but we have been able to ﬁnd a number of exact solutions corresponding to all of the observed behaviors in the system. We begin by noting that the vector x = (1, 1, 1 . . . , 1)t is always in the null- space of L and thus is always a ﬁxed point of the model regardless of the opinions γ . This makes sense: if all actors hold the same position there is ij nothing to drive the conﬂict. Thus we refer to this as the consensus state. Technically speaking the consensus state is not a critical point since, as we have observed, it exists for all values of γ: (cid:112) C Deﬁnition 3.1 The consensus state is the (critical) manifold deﬁned by C { ∈ } := (x, γ) Ω : x = R/N(1, 1, 1 . . . , 1)t . To describe the other set of ﬁxed points we introduce some notation. Let ∩ ∅ ∪ I, J be subsets of the vertex set V (Γ) such that I J = and I J = V (Γ).  Consider the matrix      − (cid:54) α; i = j in same subset − (cid:54)   β; i = j in diﬀerent subset M = | | − | | ∈ (12) ij ( I 1)α + J β; i = j I | | − | | ∈ ( J 1)α + I β; i = j J. Note that M, deﬁned above, is symmetric and also has row and column sum to zero. We then have the following characterization if its spectrum:  Lemma 3.2 The spectrum of M, deﬁned by (12), is given by      0, of multiplicity = 1    Nβ, of multiplicity = 1 σ(M) = | | | | | | − (13) I α + J β, of multiplicity = I 1 | | | | | | − (cid:18) (cid:19) I β + J α, of multiplicity = J 1 a1| | Proof. The proof amounts to a computation. Let (cid:126)u = 1 and (cid:126)v = I , N b1| | | | J where 1| | is a vector of all ones of leng(cid:18)th J . Then(cid:19)M(cid:126)u = 0 since the matrix J M has row sum 0. Another calculation shows that | | − β J 1| | L(cid:126)v = (a b) − | | I β I 1| | J 8 ##LTLine##

text: These

text: are

text: simultaneous

text: polynomial

text: equations,

text: so

text: in

text: general

text: it

text: is

text: diﬃcult

text: to

text: ﬁnd

text: all

text: solutions,

text: but

text: we

text: have

text: been

text: able

text: to

text: ﬁnd

text: a

text: number

text: of

text: exact

text: solutions

text: corresponding

text: to

text: all

text: of

text: the

text: observed

text: behaviors

text: in

text: the

text: system.

text: We

text: begin

text: by

text: noting

text: that

text: the

text: vector

text: x

text: is

text: always

text: in

text: the

text: null-

text: space

text: of

text: L

text: and

text: thus

text: is

text: always

text: a

text: ﬁxed

text: point

text: of

text: the

text: model

text: regardless

text: of

text: the

text: opinions

text: γ

text: This

text: makes

text: sense:

text: if

text: all

text: actors

text: hold

text: the

text: same

text: position

text: there

text: is

text: ij

text: nothing

text: to

text: drive

text: the

text: conﬂict.

text: Thus

text: we

text: refer

text: to

text: this

text: as

text: the

text: consensus

text: state.

text: Technically

text: speaking

text: the

text: consensus

text: state

text: is

text: not

text: a

text: critical

text: point

text: since,

text: as

text: we

text: have

text: observed,

text: it

text: exists

text: for

text: all

text: values

text: of

text: γ:

text: C

text: Deﬁnition

text: The

text: consensus

text: state

text: is

text: the

text: (critical)

text: manifold

text: deﬁned

text: by

text: C

text: (x,

text: γ)

text: Ω

text: x

text: To

text: describe

text: the

text: other

text: set

text: of

text: ﬁxed

text: points

text: we

text: introduce

text: some

text: notation.

text: Let

text: I,

text: J

text: be

text: subsets

text: of

text: the

text: vertex

text: set

text: V

text: (Γ)

text: such

text: that

text: I

text: J

text: and

text: I

text: J

text: V

text: (Γ).

text: Consider

text: the

text: matrix

text: α;

text: i

text: j

text: in

text: same

text: subset

text: β;

text: i

text: j

text: in

text: diﬀerent

text: subset

text: M

text: ij

text: I

text: J

text: β;

text: i

text: j

text: I

text: J

text: I

text: β;

text: i

text: j

text: J.

text: Note

text: that

text: M,

text: deﬁned

text: above,

text: is

text: symmetric

text: and

text: also

text: has

text: row

text: and

text: column

text: sum

text: to

text: zero.

text: We

text: then

text: have

text: the

text: following

text: characterization

text: if

text: its

text: spectrum:

text: Lemma

text: The

text: spectrum

text: of

text: M,

text: deﬁned

text: by

text: is

text: given

text: by

text: of

text: multiplicity

text: Nβ,

text: of

text: multiplicity

text: σ(M)

text: I

text: α

text: J

text: β,

text: of

text: multiplicity

text: I

text: I

text: β

text: J

text: α,

text: of

text: multiplicity

text: J

text: Proof.

text: The

text: proof

text: amounts

text: to

text: a

text: computation.

text: Let

text: and

text: I

text: N

text: J

text: where

text: is

text: a

text: vector

text: of

text: all

text: ones

text: of

text: J

text: since

text: the

text: matrix

text: J

text: M

text: has

text: row

text: sum

text: Another

text: calculation

text: shows

text: that

text: β

text: J

text: (a

text: b)

text: I

text: β

text: I

text: J

text: ##LTLine##