text: (cid:88) ∞ (cid:39) (cid:80) We look for an eigenvalue Eh hje j j=0(cid:39) ∞ and an associated eigenfunction f hjf , so (cid:88) h j=0 j ∞ − − (cid:39) eφ/h(P h(x, hD) EhI)(e φ/hf ) hjκ h j j=0 with − κ (x) = (W (x) e I)f (x) 0 0 −0(cid:80) 0 − κ (x) = (W (x, D) e I)f (x) + (W (x) e I)f (x) 1 1 − 1 0 0 −0 1 − κ (x) = (W (x, D) e I)f (x) + (W (x, D) e I)f (x) + (W (x) e I)f (x) 2 − 2 − 2− 0 1 1− 1 0 0 2 κ (x) = e f (x) j 3 e − f (x) + (W (x, D) e I)f − (x) j j 0 − (cid:96)=1 j (cid:96) (cid:96) 2 − 2 j 2 + (W (x, D) e I)f − (x) + (W (x) e I)f (x), (j > 2). 1 1 j 1 0 0 j (cid:26) − ∀ We recall that f = (f+, f ) and we want that κ (x) = 0, j. j j j j 1) Term of order 0 −|∇ | − | | − ⇔ ( φ(x) 2 + v(x) w(x) e )f+(x) = 0 As κ (x) = 0 −|∇ | | | − 0 0− 0 ( φ(x) 2 + v(x) + w(x) e )f (x) = 0 , 0 0 choosing φ satisfying (5.3), then −|∇ | | | − | | e = E and φ(x) 2 + v(x) + w(x) e = 2(cid:18)w(x) > 0 imp(cid:19)lies that 0 0 0 − − 0 0 f (x) = 0 , e = E and W (x) e I = | | . (5.8) 0 0 0 0 0 0 2 w(x) 2) Term of order 1. − The components of κ (x) = (κ+(x), κ (x)) become 1 1 1 − − ∇ ∇ ∇ κ+(x) = (∆φ(x) e 2ia (x). φ(x))f+(x) + 2 φ(x). f+(x) 1− | | − 1 − .,11 ∇ 0 0 κ (x) = 2 w(x) f (x) 2i(a (x). φ(x))f+(x) , 1 1 .,21 0 |∇ | As φ(x) has a simple zero at x = 0, the equation κ+(x) = 0 can be solved 0 1 only when e = ∆φ(0). In this case there exists a unique function f+(x) such 1 0 that f+(0) = 1 and κ+(x) = 0. We can conclude that the study of the term of  0 1 order 1 leads to   e = ∆φ(0) , 1∇ ∇ − ∇ 2 φ(x). f+(x) = [e ∆φ(x) + 2ia (x). φ(x)]f+(x) , (5.9) − 0 ∇ 1 .,11 0 f (x) = | i |( φ(x).a (x))f+(x) . 1 w(x) .,21 0 3) Term of order 2. − The components of κ (x) = (κ+(x), κ (x)) become 2 2 2 14 ##LTLine##

text: We

text: look

text: for

text: an

text: eigenvalue

text: Eh

text: hje

text: j

text: and

text: an

text: associated

text: eigenfunction

text: f

text: hjf

text: so

text: h

text: j

text: eφ/h(P

text: h(x,

text: hD)

text: EhI)(e

text: φ/hf

text: hjκ

text: h

text: j

text: with

text: κ

text: (x)

text: (W

text: (x)

text: e

text: I)f

text: (x)

text: κ

text: (x)

text: (W

text: (x,

text: D)

text: e

text: I)f

text: (x)

text: (W

text: (x)

text: e

text: I)f

text: (x)

text: −0

text: κ

text: (x)

text: (W

text: (x,

text: D)

text: e

text: I)f

text: (x)

text: (W

text: (x,

text: D)

text: e

text: I)f

text: (x)

text: (W

text: (x)

text: e

text: I)f

text: (x)

text: 2−

text: 1−

text: κ

text: (x)

text: e

text: f

text: (x)

text: j

text: e

text: f

text: (x)

text: (W

text: (x,

text: D)

text: e

text: I)f

text: (x)

text: j

text: j

text: j

text: j

text: (W

text: (x,

text: D)

text: e

text: I)f

text: (x)

text: (W

text: (x)

text: e

text: I)f

text: (x),

text: (j

text: j

text: j

text: We

text: recall

text: that

text: f

text: (f+,

text: f

text: and

text: we

text: want

text: that

text: κ

text: (x)

text: j.

text: j

text: j

text: j

text: j

text: Term

text: of

text: order

text: φ(x)

text: v(x)

text: w(x)

text: e

text: )f+(x)

text: As

text: κ

text: (x)

text: 0−

text: φ(x)

text: v(x)

text: w(x)

text: e

text: )f

text: (x)

text: choosing

text: φ

text: satisfying

text: then

text: e

text: E

text: and

text: φ(x)

text: v(x)

text: w(x)

text: e

text: that

text: f

text: (x)

text: e

text: E

text: and

text: W

text: (x)

text: e

text: I

text: w(x)

text: Term

text: of

text: order

text: The

text: components

text: of

text: κ

text: (x)

text: (κ+(x),

text: κ

text: (x))

text: become

text: κ+(x)

text: (∆φ(x)

text: e

text: (x).

text: φ(x))f+(x)

text: φ(x).

text: f+(x)

text: 1−

text: κ

text: (x)

text: w(x)

text: f

text: (x)

text: (x).

text: φ(x))f+(x)

text: As

text: φ(x)

text: has

text: a

text: simple

text: zero

text: at

text: x

text: the

text: equation

text: κ+(x)

text: can

text: be

text: solved

text: only

text: when

text: e

text: In

text: this

text: case

text: there

text: exists

text: a

text: unique

text: function

text: f+(x)

text: such

text: that

text: and

text: κ+(x)

text: We

text: can

text: conclude

text: that

text: the

text: study

text: of

text: the

text: term

text: of

text: order

text: leads

text: to

text: e

text: 1∇

text: φ(x).

text: f+(x)

text: [e

text: ∆φ(x)

text: (x).

text: φ(x)]f+(x)

text: f

text: (x)

text: i

text: φ(x).a

text: (x))f+(x)

text: w(x)

text: Term

text: of

text: order

text: The

text: components

text: of

text: κ

text: (x)

text: (κ+(x),

text: κ

text: (x))

text: become

text: ##LTLine##


text: − ∇ ∇ − − ∇ − ∇ (cid:88) [ 2 φ(x). f − (x) + (e 2ia (x). φ(x))f − (x) + 2ia (x). φ(x)f+− (x) j 1 1 .,11 j 1 .,21 j 1 − j 3 − ∇− − − − ∇ − +(e (i a )2)f − (x) idiv(a (x))f+− (x) 2ia (x). f+− (x)+ e − f (x)] 2 .,11 j 2 .,21 j 2 .,21 j 2 j (cid:96) (cid:96) (cid:96)=0 5) End of the proof. Let χ(x) be a cut-oﬀ function equal to 1 in a neighbourhood of 0 and supported in Ω. Then taking χ(x)f (x) instead of f (x), we get a function ϕh,Ω satisfying j j Dirichlet boundary condition and Theorem 5.1. The self-adjointness of the related (cid:3) operator ensures that the computed sequence (e ) is real j 6 Sharp asymptotic for the width of the ﬁrst band Returning to the proof of Theorem 4.2, we have to study carefully the τh,θ deﬁned in (4.16), using t(cid:88)he m(cid:16)ethod of [He-Sj-1] performed in(cid:17)[He] and [Ou]. Using (4.14)–(4.16), we have − − − τh,θ = e iθω(ρ+ + ρ ) + eiθω(ρ+ + ρ ) , (6.1) ω ω ω ω ∈ ω Γ+ 0 (cid:90) { } with Γ+ = β , . . . , β , 0 1 n 1 − ρ+ = (cid:107) (cid:107) (cid:90) ([P h(x, hD); χ]ϕh,Ω)(x ω).(χϕh,Ω)(x)dx ω ϕh,θ 2L2(Wn;C2) Wn − 1 and ρ = (cid:107) (cid:107) ([P h(x, hD); χ]ϕh,Ω)(x).(χϕh,Ω)(x + ω)dx . ω ϕh,θ 2L2(Wn;C2) Wn We get from the formula o(cid:90)f [P h(x, hD); χ] and from the estimate (4.13) that − h2 ∇ − ∇ − ρ+ = (cid:107) (cid:107) ( χ(x ω) ϕh,Ω)(x ω)).(χϕh,Ω)(x)dx + (6.2) ω ϕh,θ 2L2(Wn;C2) Wn − + O(he S0/h) (cid:90) and − − h2 ∇ ∇ − ρ = (cid:107) (cid:107) ( χ(x) ϕh,Ω)(x)).(χϕh,Ω)(x + ω)dx + O(he S0/h) . ω ϕh,θ 2L2(Wn;C2) Wn ∈ {± ± } Theorem 6.1 Under the assumption of Theorem 4.2, if for any ω β , . . . , β 1 n R such that the Agmon distance in n between 0 and ω is the least one, (i.e. 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: φ(x).

text: f

text: (x)

text: (e

text: (x).

text: φ(x))f

text: (x)

text: (x).

text: φ(x)f+−

text: (x)

text: j

text: j

text: j

text: j

text: +(e

text: (i

text: a

text: (x)

text: idiv(a

text: (x))f+−

text: (x)

text: (x).

text: f+−

text: (x)+

text: e

text: f

text: (x)]

text: j

text: j

text: j

text: j

text: End

text: of

text: the

text: proof.

text: Let

text: χ(x)

text: be

text: a

text: cut-oﬀ

text: function

text: equal

text: to

text: in

text: a

text: neighbourhood

text: of

text: and

text: supported

text: in

text: Ω.

text: Then

text: taking

text: χ(x)f

text: (x)

text: instead

text: of

text: f

text: (x),

text: we

text: get

text: a

text: function

text: ϕh,Ω

text: satisfying

text: j

text: j

text: Dirichlet

text: boundary

text: condition

text: and

text: Theorem

text: The

text: self-adjointness

text: of

text: the

text: related

text: operator

text: ensures

text: that

text: the

text: computed

text: sequence

text: (e

text: is

text: real

text: j

text: Sharp

text: asymptotic

text: for

text: the

text: width

text: of

text: the

text: ﬁrst

text: band

text: Returning

text: to

text: the

text: proof

text: of

text: Theorem

text: we

text: have

text: to

text: study

text: carefully

text: the

text: τh,θ

text: deﬁned

text: in

text: using

text: of

text: performed

text: and

text: [Ou].

text: Using

text: we

text: have

text: τh,θ

text: e

text: iθω(ρ+

text: ρ

text: eiθω(ρ+

text: ρ

text: ω

text: ω

text: ω

text: ω

text: ω

text: Γ+

text: with

text: Γ+

text: β

text: β

text: n

text: ρ+

text: ([P

text: h(x,

text: hD);

text: χ]ϕh,Ω)(x

text: ω).(χϕh,Ω)(x)dx

text: ω

text: ϕh,θ

text: Wn

text: and

text: ρ

text: ([P

text: h(x,

text: hD);

text: χ]ϕh,Ω)(x).(χϕh,Ω)(x

text: ω)dx

text: ω

text: ϕh,θ

text: Wn

text: We

text: get

text: from

text: the

text: formula

text: [P

text: h(x,

text: hD);

text: χ]

text: and

text: from

text: the

text: estimate

text: that

text: ρ+

text: χ(x

text: ω)

text: ϕh,Ω)(x

text: ω)).(χϕh,Ω)(x)dx

text: ω

text: ϕh,θ

text: Wn

text: O(he

text: and

text: ρ

text: χ(x)

text: ϕh,Ω)(x)).(χϕh,Ω)(x

text: ω)dx

text: O(he

text: ω

text: ϕh,θ

text: Wn

text: Theorem

text: Under

text: the

text: assumption

text: of

text: Theorem

text: if

text: for

text: any

text: ω

text: β

text: β

text: n

text: R

text: such

text: that

text: the

text: Agmon

text: distance

text: in

text: n

text: between

text: and

text: ω

text: is

text: the

text: least

text: one,

text: (i.e.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: References [Ag] S. Agmon : Lectures in exponential decay of solutions of second-order elliptic equations. Princeton University Press, Math. Notes 29. [Ca-Yu] M. Cardona, P. Y. Yur : Fundamentals of Semiconductors Physics and Materials Properties. Springer, Berlin 2010. [Co] N. R. Cooper : Optical Flux Lattices for Ultracold Atomic Gases. Phys. Rev. Lett. 106, (2011), 175301. [Co-Da] N. R. Cooper, J. Dalibard : Optical ﬂux lattices for two-photon dressed states. Europhysics Letters, 95 (6), (2011), 66004. [Da-al] J. Dalibard, F. Gerbier, G. Juzeliunas, P. Ohberg : Colloquium: Artiﬁcial gauge potentials for neutral atoms. Rev. Mod. Phys. 83 (4), (2011), p. 1523- 1543. [Di-Sj] M. Dimassi, J. Sj¨ostrand : Spectral asymptotics in the semi-classical limit. Cambridge University Press, 1999. [Ea] M. S. P. Eastham : The spectral theory of periodic diﬀerential equations. Scottish Academic, London 1974. [He] B. Helﬀer : Semi-classical analysis for the Schr¨odinger operator and appli- cations. Lecture Notes in Math. 1336, Springer , 1988. [He-Sj-1] B. Helﬀer, J. Sj¨ostrand : Multiple wells in the semi-classical limit. Comm. in P.D.E., 8 (4), (1984), p.337-408. [He-Sj-2] B. Helﬀer, J. Sjo¨strand : Eﬀet tunnel pour l’´equation de Schro¨dinger avec champ magn´etique. Ann. Scuola Norm. Sup. Pisa, 14 (4), (1987), p. 625-657. [Hi-Si] P. D. Hislop, I. M. Sigal : Introduction to spectral theory. With ap- plications to Schr¨odinger operators. Applied Mathematical Sciences, 113. Springer-Verlag, 1996. [Ou] A. Outassourt : Comportement semi-classique pour l’op´erateur de Schro¨dinger a` potentiel p´eriodique. J. of Functional Analysis 72, (1987), p. 65-93. [Re-Si] M. Reed, B. Simon : Methods of modern mathematical physics, Vol 4. Academic Press, New York 1972. [Ro] D. Robert : Autour de l’approximation semi-classique. Birkhauser, 1986. 18

text: References

text: [Ag]

label: reference

text: S.

text: Agmon

text: Lectures

text: in

text: exponential

text: decay

text: of

text: solutions

text: of

text: second-order

text: elliptic

text: equations.

text: Princeton

text: University

text: Press,

text: Math.

text: Notes

text: [Ca-Yu]

text: M.

text: Cardona,

text: P.

text: Y.

text: Yur

text: Fundamentals

text: of

text: Semiconductors

text: Physics

text: and

text: Materials

text: Properties.

text: Springer,

text: Berlin

text: [Co]

text: N.

text: R.

text: Cooper

text: Optical

text: Flux

text: Lattices

text: for

text: Ultracold

text: Atomic

text: Gases.

text: Phys.

text: Rev.

text: Lett.

text: [Co-Da]

text: N.

text: R.

text: Cooper,

text: J.

text: Dalibard

text: Optical

text: ﬂux

text: lattices

text: for

text: two-photon

text: dressed

text: states.

text: Europhysics

text: Letters,

text: [Da-al]

text: J.

text: Dalibard,

text: F.

text: Gerbier,

text: G.

text: Juzeliunas,

text: P.

text: Ohberg

text: Colloquium:

text: Artiﬁcial

text: gauge

text: potentials

text: for

text: neutral

text: atoms.

text: Rev.

text: Mod.

text: Phys.

text: p.

text: [Di-Sj]

text: M.

text: Dimassi,

text: J.

text: Sj¨ostrand

text: Spectral

text: asymptotics

text: in

text: the

text: semi-classical

text: limit.

text: Cambridge

text: University

text: Press,

text: [Ea]

text: M.

text: S.

text: P.

text: Eastham

text: The

text: spectral

text: theory

text: of

text: periodic

text: diﬀerential

text: equations.

text: Scottish

text: Academic,

text: London

text: [He]

text: B.

text: Helﬀer

text: Semi-classical

text: analysis

text: for

text: the

text: Schr¨odinger

text: operator

text: and

text: appli-

text: cations.

text: Lecture

text: Notes

text: in

text: Math.

text: Springer

text: B.

text: Helﬀer,

text: J.

text: Sj¨ostrand

text: Multiple

text: wells

text: in

text: the

text: semi-classical

text: limit.

text: Comm.

text: in

text: P.D.E.,

text: B.

text: Helﬀer,

text: J.

text: Sjo¨strand

text: Eﬀet

text: tunnel

text: pour

text: l’´equation

text: de

text: Schro¨dinger

text: avec

text: champ

text: magn´etique.

text: Ann.

text: Scuola

text: Norm.

text: Sup.

text: Pisa,

text: p.

text: [Hi-Si]

text: P.

text: D.

text: Hislop,

text: I.

text: M.

text: Sigal

text: Introduction

text: to

text: spectral

text: theory.

text: With

text: ap-

text: plications

text: to

text: Schr¨odinger

text: operators.

text: Applied

text: Mathematical

text: Sciences,

text: Springer-Verlag,

text: [Ou]

text: A.

text: Outassourt

text: Comportement

text: semi-classique

text: pour

text: l’op´erateur

text: de

text: Schro¨dinger

text: a`

text: potentiel

text: p´eriodique.

text: J.

text: of

text: Functional

text: Analysis

text: p.

text: [Re-Si]

text: M.

text: Reed,

text: B.

text: Simon

text: Methods

text: of

text: modern

text: mathematical

text: physics,

text: Vol

text: Academic

text: Press,

text: New

text: York

text: [Ro]

text: D.

text: Robert

text: Autour

text: de

text: l’approximation

text: semi-classique.

text: Birkhauser,


text: 3 Proof of Theorem 2.1 Proof. According to the above discussion, (cid:18)we can assume that (cid:19) P h (x, hD) P h (x, hD) Hh = P h(x, hD) , with P h(x, hD) = 11 12 , (3.1) P h (x, hD) P h (x, hD) 21 22 with       − − | |   P h (x, hD) = h2(D a (x))2 + v(x) w(x) + h2r (x) 11 .,11 | | 11 P h (x, hD) = h2(D + a (x))2 + v(x) + w(x) + h2r (x)  22 − .,11 − 22−  P h (x, hD) = h2a (x).(D + a (x)) h2a (x).(D a (x))  12 .,12 .,11 .,12 .,11  (3.2) + ih2div(a (x)) + h2r (x) − .,12 − 12 − P h (x, hD) = h2a (x)(D a (x)) h2a (x).(D + a (x)) 21 .,21 .,11 .,21 .,11 + ih2div(a (x)) + h2r (x) . .,21 21 (D = (D , D , . . . , D ) and a (x) = (a (x), a (x), . . . , a (x)) .) x1 x2 − xn .,ij 1,ij 2,ij n,ij (We used that a = a ). .,22 .,11 Let us denote by Hh,θ and Hh,θ the operators associated with − 11 2−2 T C P h (x, h(D θ)) and P h (x, h(D θ)) on L2( n; ). 11 | 2|2 (cid:107) (cid:107) Then, if c = min w(x) and c = max R(x) , 0 1 ⊂ − ∞ ⊂ − ∞ sp(Hh,θ) [E h2c , + [ and sp(Hh,θ) [E h2c + 2c , + [. 11 0 1 22 0 1 0 To prove the theorem it is then enough to prove the proposition below. Proposition 3.1 Let us consider a constant c, 0 < c < c . Then there exists ∈ − ∞ 0 (cid:27) C > 0 such that, for any Eh ] , E + 2c[, we have 0 0 ∈ ⇒ ≤ Eh sp(Hh,θ) distance(Eh, sp(Hh,θ) C h2 ∈ ⇒ 11 ≤ 0 (3.3) Eh sp(Hh,θ) distance(Eh, sp(Hh,θ)) C h2 . 11 0 − − Proof. For such Eh, (Hh,θ Eh) 1 exists and, thanks to semiclassical pseudod- 22 iﬀerential calculus of [Ro] (see also [Di-Sj] ), for h > 0 small, if 0 < h < h then (cid:107) − − (cid:107) (cid:107) − − − (cid:107)0 0 (Hh,θ Eh) 1 T + h(D θ)(Hh,θ Eh) 1 T (cid:107) 22 − − L2( n)− (cid:107) (cid:107) 22 − L−2( n) − − (cid:107) ≤ + (Hh,θ Eh) 1h(D θ) T + h(D θ)(Hh,θ Eh) 1h(D θ) T C, 22 L2( n) 22 L2( n) and then (cid:107) − − − − (cid:107) ≤ P h (x, h(D θ))(Hh,θ Eh) 1P h (x, h(D θ)) T h2C. 12 22 21 L2( n) ∈ (cid:54) So if Eh sp(Hh,θ), then uh = (uh, uh) = (0, 0) is an eigenfunction of Hh,θ 1 2 associated with Eh iﬀ 6

label: section

text: Proof

text: of

text: Theorem

text: Proof.

text: According

text: to

text: the

text: above

text: discussion,

label: equation

text: can

text: assume

text: that

text: P

text: h

text: (x,

text: hD)

text: P

text: h

text: (x,

text: hD)

text: Hh

text: P

text: h(x,

text: hD)

text: with

text: P

text: h(x,

text: hD)

text: P

text: h

text: (x,

text: hD)

text: P

text: h

text: (x,

text: hD)

text: with

text: P

text: h

text: (x,

text: hD)

text: a

text: v(x)

text: w(x)

text: (x)

text: P

text: h

text: (x,

text: hD)

text: a

text: v(x)

text: w(x)

text: (x)

text: P

text: h

text: (x,

text: hD)

text: (x).(D

text: a

text: (x))

text: (x).(D

text: a

text: (x))

text: (x))

text: (x)

text: P

text: h

text: (x,

text: hD)

text: (x)(D

text: a

text: (x))

text: (x).(D

text: a

text: (x))

text: (x))

text: (x)

text: (D

text: (D

text: D

text: D

text: and

text: a

text: (x)

text: (a

text: (x),

text: a

text: (x),

text: a

text: (x))

text: xn

text: .,ij

text: n,ij

text: (We

text: used

text: that

text: a

text: a

text: Let

text: us

text: denote

text: by

text: Hh,θ

text: and

text: Hh,θ

text: the

text: operators

text: associated

text: with

text: T

text: C

text: P

text: h

text: (x,

text: h(D

text: θ))

text: and

text: P

text: h

text: (x,

text: h(D

text: θ))

text: on

text: n;

text: Then,

text: if

text: c

text: min

text: w(x)

text: and

text: c

text: max

text: R(x)

text: sp(Hh,θ)

text: [E

text: and

text: sp(Hh,θ)

text: [E

text: To

text: prove

text: the

text: theorem

text: it

text: is

text: then

text: enough

text: to

text: prove

text: the

text: proposition

text: below.

text: Proposition

text: Let

text: us

text: consider

text: a

text: constant

text: c,

text: c

text: c

text: Then

text: there

text: exists

text: C

text: such

text: that,

text: for

text: any

text: Eh

text: E

text: we

text: have

text: Eh

text: sp(Hh,θ)

text: distance(Eh,

text: sp(Hh,θ)

text: C

text: Eh

text: sp(Hh,θ)

text: distance(Eh,

text: sp(Hh,θ))

text: C

text: Proof.

text: For

text: such

text: Eh,

text: (Hh,θ

text: Eh)

text: exists

text: and,

text: thanks

text: to

text: semiclassical

text: pseudod-

text: iﬀerential

text: calculus

text: of

text: [Ro]

text: (see

text: also

text: [Di-Sj]

text: for

text: h

text: small,

text: if

text: h

text: h

text: then

text: (Hh,θ

text: Eh)

text: T

text: h(D

text: θ)(Hh,θ

text: Eh)

text: T

text: n)−

text: n)

text: (Hh,θ

text: Eh)

text: θ)

text: T

text: h(D

text: θ)(Hh,θ

text: Eh)

text: θ)

text: T

text: C,

text: n)

text: n)

text: and

text: then

text: P

text: h

text: (x,

text: h(D

text: θ))(Hh,θ

text: Eh)

text: h

text: (x,

text: h(D

text: θ))

text: T

text: n)

text: So

text: if

text: Eh

text: sp(Hh,θ),

text: then

text: uh

text: (uh,

text: uh)

text: is

text: an

text: eigenfunction

text: of

text: Hh,θ

text: associated

text: with

text: Eh

text: iﬀ


text: (cid:27) − Hh,θuh + P h (x, h(D θ))uh = Ehuh 11 −1 12 − − 2 −1 (3.4) uh = (Hh,θ EhI) 1P h (x, h(D θ))uh . 2 22 21 1 ∈ − ∞ In fact Eh ] , E + c[ will be an eigenvalue of Hh,θ iﬀ there exists 0 T C (cid:107) (cid:107) (cid:54) uh in the Sobolev space W 2( n; ), uh T = 0, such that 1 1 L2( n) − − − − − Hh,θuh P h (x, h(D θ))(Hh,θ EhI) 1P h (x, h(D θ))uh = Ehuh , 11 1 12 22 21 1 1 then we get the ﬁrst part of Proposition 3.1. If Eh is an eigenvalue of Hh,θ satisfying the assumption of Proposition 3.1, 11 and uh an associated eigenfunction, then with 1 − − − − uh = (uh, (Hh,θ Eh) 1P h (x, h(D θ))uh), one has 1 22 21 1 (cid:107) − (cid:107) (Hh,θ EhI)uh T C (cid:107) − L2( n; 2) − − − (cid:107) = P h (x, h(D θ))(Hh,θ Eh) 1P h (x, h(D θ))uh T C ≤ 12(cid:107) (cid:107) 22 21 1 L2( n; ) h2C uh T C , L2( n; 2) we get the second part of Proposition 3.1. Theorem 2.1 follows from Proposition 3.1 and [Si-1], [Si-2], [He-Sj-1] and [He-Sj-2] results, (see also [He]), which guarantee that the sequence of eigenvalues ∀ ∃ ∀ of Hh,θ, (λ (Hh,θ)) ∈N satisﬁes N > 1, h > 0, C > 0 s.t. h, 0 < h < h ∀11 ≤ j 11| j (cid:63) − 0 | ≤ 0 (cid:3) 0 0 and j N , λ (Hh,θ) (hµh + E ) C h2 0 j 11 j 0 0 4 Asymptotic of the ﬁrst band ∈ T C For any real Lip(cid:0)schitz Γ periodic function φ, and for a(cid:1)ny u W 2( n; 2), we have the identity   (cid:88)  − | Re < P h(x, h(D θ))u e2φ/hu > T C = L2( n; 2)   n (cid:107) − − (cid:107)  h2 (((cid:101)Dxk θk)I Ak)eφ/hu 2L2(Tn;C2) (4.1) k=1 − |∇ | | + < (V φ 2I + h2R)u e2φ/hu > T C . L2( n; 2) This identity enables us to apply the method used in [He-Sj-1], (see also [He] (cid:90) (cid:112) R and [Ou]). We deﬁne the Agmon [Ag] distance on n 1 − | | − | | d(y, x) = inf v(γ(t)) w(γ(t)) E γ˙ (t) dt , (4.2) 0 γ 0 the inf is taken among paths such that γ(0) = y and γ(1) = x. For common properties of the Agmon distance, one can see for example [Hi-Si]. ∈ R We will use that, for any ﬁxed y n, the function d(y, x) is a Lipschitz R |∇ | ≤ − | | − R function on n and d(y, x) 2 v(x) w(x) E almost everywhere on n. x 0 7 ##LTLine##

text: Hh,θuh

text: P

text: h

text: (x,

text: h(D

text: θ))uh

text: Ehuh

text: −1

text: −1

text: uh

text: (Hh,θ

text: EhI)

text: h

text: (x,

text: h(D

text: θ))uh

text: In

text: fact

text: Eh

text: E

text: c[

text: will

text: be

text: an

text: eigenvalue

text: of

text: Hh,θ

text: iﬀ

text: there

text: exists

text: T

text: C

font: default

text: uh

text: in

text: the

text: Sobolev

text: space

text: W

text: n;

text: uh

text: T

text: such

text: that

text: n)

text: Hh,θuh

text: P

text: h

text: (x,

text: h(D

text: θ))(Hh,θ

text: EhI)

text: h

text: (x,

text: h(D

text: θ))uh

text: Ehuh

text: then

text: we

text: get

text: the

text: ﬁrst

text: part

text: of

text: Proposition

text: If

text: Eh

text: is

text: an

text: eigenvalue

text: of

text: Hh,θ

text: satisfying

text: the

text: assumption

text: of

text: Proposition

text: and

text: uh

text: an

text: associated

text: eigenfunction,

text: then

text: with

text: uh

text: (uh,

text: (Hh,θ

text: Eh)

text: h

text: (x,

text: h(D

text: θ))uh),

text: one

text: has

text: (Hh,θ

text: EhI)uh

text: T

text: C

text: n;

text: P

text: h

text: (x,

text: h(D

text: θ))(Hh,θ

text: Eh)

text: h

text: (x,

text: h(D

text: θ))uh

text: T

text: C

text: n;

text: uh

text: T

text: C

text: n;

text: we

text: get

text: the

text: second

text: part

text: of

text: Proposition

text: Theorem

text: follows

text: from

text: Proposition

text: and

text: and

text: results,

text: (see

text: also

text: [He]),

text: which

text: guarantee

text: that

text: the

text: sequence

text: of

text: eigenvalues

text: of

text: Hh,θ,

text: (λ

text: (Hh,θ))

text: ∈N

text: satisﬁes

text: N

text: h

text: C

text: s.t.

text: h,

text: h

text: h

text: j

text: j

text: and

text: j

text: N

text: λ

text: (Hh,θ)

text: (hµh

text: E

text: C

text: j

text: j

text: Asymptotic

text: of

text: the

text: ﬁrst

text: band

text: T

text: C

text: For

text: any

text: real

text: Γ

text: periodic

text: function

text: φ,

text: and

text: for

text: u

text: W

text: n;

text: we

text: have

text: the

text: identity

text: Re

text: P

text: h(x,

text: h(D

text: θ))u

text: T

text: C

text: n;

text: n

text: θk)I

text: Ak)eφ/hu

text: (V

text: φ

text: T

text: C

text: n;

text: This

text: identity

text: enables

text: us

text: to

text: apply

text: the

text: method

text: used

text: in

text: (see

text: also

text: [He]

text: R

text: and

text: [Ou]).

text: We

text: deﬁne

text: the

text: Agmon

text: [Ag]

text: distance

text: on

text: n

text: d(y,

text: x)

text: inf

text: v(γ(t))

text: w(γ(t))

text: E

text: γ˙

text: (t)

text: dt

text: γ

text: the

text: inf

text: is

text: taken

text: among

text: paths

text: such

text: that

text: y

text: and

text: x.

text: For

text: common

text: properties

text: of

text: the

text: Agmon

text: distance,

text: one

text: can

text: see

text: for

text: example

text: [Hi-Si].

text: R

text: We

text: will

text: use

text: that,

text: for

text: any

text: ﬁxed

text: y

text: n,

text: the

text: function

text: d(y,

text: x)

text: is

text: a

text: Lipschitz

text: R

text: R

text: function

text: on

text: n

text: and

text: d(y,

text: x)

text: v(x)

text: w(x)

text: E

text: almost

text: everywhere

text: on

text: n.

text: x

text: ##LTLine##


text: **Volume Title** ASP Conference Series, Vol. **Volume Number** **Author** (cid:13)c **Copyright Year** Astronomical Society of the Paciﬁc An ADER-WENO Finite Volume AMR code for Astrophysics O. Zanotti1, M. Dumbser1, A. Hidalgo2 and D. Balsara3 1Laboratory of Applied Mathematics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano 77, I-38123 Trento, Italy 2Departamento de Matema´tica Aplicada y Me´todos Informa´ticos, Universidad Polite´cnica de Madrid, Calle R´ıos Rosas 21, E-28003 Madrid, Spain 3Physics Department, University of Notre Dame du Lac, 225 Nieuwland Science Hall, Notre Dame, IN 46556, USA Abstract. A high order one-step ADER-WENO ﬁnite volume scheme with Adap- tive Mesh Reﬁnement (AMR) in multiple space dimensions is presented. A high order one-step time discretization is achieved using a local space-time discontinuous Galerkin predictor method, while a high order spatial accuracy is obtained through a WENO re- construction. Thanks to the one-step nature of the underlying scheme, the resulting al- gorithm can be eﬃciently imported within an AMR framework on space-time adaptive meshes. We provide convincing evidence that the presented high order AMR scheme behaves better than traditional second order AMR methods. Tests are shown of the new scheme for nonlinear systems of hyperbolic conservation laws, including the clas- sical Euler equations and the equations of ideal magnetohydrodynamics. The proposed scheme is likely to become a useful tool in several astrophysical scenarios. 1. Introduction ADER schemes were introduced more than ten years ago by Toro et al. (2001)1. Though presenting several attractive features, like the possibility of reaching an arbi- trary high order of accuracy and the fact that the time update is performed through a single step with no need for Runge-Kutta, in their original form they have never been applied to astrophysics. The reason for this is that original ADER schemes required re- sorting to the so-called Cauchy-Kovalewski procedure, that is based on a repeated use of the governing conservation laws in diﬀerential form. In spite of being conceptually clear, such a procedure becomes rather cumbersome in multiple space dimensions for the classical Euler equations. It has been implemented by Toro & Titarev (2005) and Dumbser et al. (2007), but a wider use has not been reached. A fundamental breakthrough was achieved by Dumbser et al. (2008), who pro- posed a variant of the ADER approach in which the local time evolution of the re- constructed polynomials is obtained through an element-local space-time Galerkin pre- dictor that is based on the weak integral form of the PDEs. This new strategy, which 1See also the subsequent works by Titarev & Toro (2005); Toro & Titarev (2005). 1 ##LTLine##

text: **Volume

text: Title**

text: ASP

text: Conference

text: Series,

text: Vol.

text: **Volume

text: Number**

text: **Author**

text: **Copyright

text: Year**

text: Astronomical

text: Society

text: of

text: the

text: Paciﬁc

text: An

label: title

text: ADER-WENO

text: Finite

text: Volume

text: AMR

text: code

text: for

text: Astrophysics

text: O.

label: author

text: M.

text: A.

text: and

text: D.

text: of

text: Applied

text: Mathematics,

text: Department

text: of

text: Civil,

text: Environmental

text: and

text: Mechanical

text: Engineering,

text: University

text: of

text: Trento,

text: Via

text: Mesiano

text: Trento,

text: Italy

text: de

text: Matema´tica

text: Aplicada

text: y

text: Me´todos

text: Informa´ticos,

text: Universidad

text: Polite´cnica

text: de

text: Madrid,

text: Calle

text: R´ıos

text: Rosas

text: Madrid,

text: Spain

text: Department,

text: University

text: of

text: Notre

text: Dame

text: du

text: Lac,

text: Nieuwland

text: Science

text: Hall,

text: Notre

text: Dame,

text: IN

text: USA

text: Abstract.

text: A

label: abstract

text: high

text: order

text: one-step

text: ADER-WENO

text: ﬁnite

text: volume

text: scheme

text: with

text: Adap-

text: tive

text: Mesh

text: Reﬁnement

text: (AMR)

text: in

text: multiple

text: space

text: dimensions

text: is

text: presented.

text: A

text: high

text: order

text: one-step

text: time

text: discretization

text: is

text: achieved

text: using

text: a

text: local

text: space-time

text: discontinuous

text: Galerkin

text: predictor

text: method,

text: while

text: a

text: high

text: order

text: spatial

text: accuracy

text: is

text: obtained

text: through

text: a

text: WENO

text: re-

text: construction.

text: Thanks

text: to

text: the

text: one-step

text: nature

text: of

text: the

text: underlying

text: scheme,

text: the

text: resulting

text: al-

text: gorithm

text: can

text: be

text: eﬃciently

text: imported

text: within

text: an

text: AMR

text: framework

text: on

text: space-time

text: adaptive

text: meshes.

text: We

text: provide

text: convincing

text: evidence

text: that

text: the

text: presented

text: high

text: order

text: AMR

text: scheme

text: behaves

text: better

text: than

text: traditional

text: second

text: order

text: AMR

text: methods.

text: Tests

text: are

text: shown

text: of

text: the

text: new

text: scheme

text: for

text: nonlinear

text: systems

text: of

text: hyperbolic

text: conservation

text: laws,

text: including

text: the

text: clas-

text: sical

text: Euler

text: equations

text: and

text: the

text: equations

text: of

text: ideal

text: magnetohydrodynamics.

text: The

text: proposed

text: scheme

text: is

text: likely

text: to

text: become

text: a

text: useful

text: tool

text: in

text: several

text: astrophysical

text: scenarios.

text: Introduction

text: ADER

text: schemes

text: were

text: introduced

text: more

text: than

text: ten

text: years

text: ago

text: by

text: Toro

text: et

text: al.

text: Though

text: presenting

text: several

text: attractive

text: features,

text: like

text: the

text: possibility

text: of

text: reaching

text: an

text: arbi-

text: trary

text: high

text: order

text: of

text: accuracy

text: and

text: the

text: fact

text: that

text: the

text: time

text: update

text: is

text: performed

text: through

text: a

text: single

text: step

text: with

text: no

text: need

text: for

text: Runge-Kutta,

text: in

text: their

text: original

text: form

text: they

text: have

text: never

text: been

text: applied

text: to

text: astrophysics.

text: The

text: reason

text: for

text: this

text: is

text: that

text: original

text: ADER

text: schemes

text: required

text: re-

text: sorting

text: to

text: the

text: so-called

text: Cauchy-Kovalewski

text: procedure,

text: that

text: is

text: based

text: on

text: a

text: repeated

text: use

text: of

text: the

text: governing

text: conservation

text: laws

text: in

text: diﬀerential

text: form.

text: In

text: spite

text: of

text: being

text: conceptually

text: clear,

text: such

text: a

text: procedure

text: becomes

text: rather

text: cumbersome

text: in

text: multiple

text: space

text: dimensions

text: for

text: the

text: classical

text: Euler

text: equations.

text: It

text: has

text: been

text: implemented

text: by

text: Toro

text: Titarev

text: and

text: Dumbser

text: et

text: al.

text: but

text: a

text: wider

text: use

text: has

text: not

text: been

text: reached.

text: A

text: fundamental

text: breakthrough

text: was

text: achieved

text: by

text: Dumbser

text: et

text: al.

text: who

text: pro-

text: posed

text: a

text: variant

text: of

text: the

text: ADER

text: approach

text: in

text: which

text: the

text: local

text: time

text: evolution

text: of

text: the

text: re-

text: constructed

text: polynomials

text: is

text: obtained

text: through

text: an

text: element-local

text: space-time

text: Galerkin

text: pre-

text: dictor

text: that

text: is

text: based

text: on

text: the

text: weak

text: integral

text: form

text: of

text: the

text: PDEs.

text: This

text: new

text: strategy,

text: which

text: also

text: the

text: subsequent

text: works

text: by

text: Titarev

text: Toro

text: Toro

text: Titarev

text: ##LTLine##


text: about possible sexual contacts between the latter (which could potentially lead to con- nect some of the disjoint components aforementioned) is available. There are 1,627 isolated individuals, most of which are men, especially heterosexual men. The isolated couples are mostly composed of man-woman couples with an over- representation of women compared to the rest of the population. The main reasons why these individuals are isolated are that they have few positive contacts and there is a high fraction (22.06%) who did not provide their contacts (8.96% and 2.80% in the whole population and in the giant graph resp.). The network density is globally low and very heterogeneous. But although the connectivity of the network seems fragile at ﬁrst glance, density may be locally very high. The harmonic average of the geodesic path lengths equals 10.24 and 12.2 for the oriented graph. Most of the graph connectivity is concentrated in the largest component (3,168 edges out of 4,073). The largest component has a diameter of 26 (36 when taking into account the direction of the infections) and the harmonic average of the geodesic path lengths are the same inside the largest component. These values are slightly higher than those of other real networks mentioned in [31] but remain well below the number of vertices and compatible with the logarithmic scaling related to the so-termed small world eﬀect. Figure 4 shows the layout obtained by means of the force directed placement algorithm from [19, 43]. The general picture seems quite clear, with what appears to be two parts in the graph: the lower part of the graph (on the ﬁgure) seems to be dominated by MSM while the upper part gathers almost all persons from the giant component that have only heterosexual contacts. However, the upper part is quite diﬃcult to read as it seems denser than the lower part. The layout shows what might be interpreted as cycles and also a lot of small trees connected to denser parts. The actual connection patterns between the upper part and the lower part are also very unclear. Because of these crowding eﬀects, structural properties of the network from Figure 4 appears quite diﬃcult and probably misleading. We rely therefore on the simpliﬁcation technique outlined in Section leveraging a clustering of the giant component to get an insight into its general organization. A clustering of high modularity. A graphical representation of the partition obtained by the method from [41] is displayed in Figure 5. The clustering thus pro- duced exhibits a modularity of 0.8522 and is made up of 37 clusters. This modularity is very high compared to the random level and strongly supports the hypothesis of a speciﬁc (“non random”) underlying community structure. For comparison purpose, the average maximal modularity attained by random graphs built from a conﬁgura- tion model with the same size and degree distribution as those of the giant component observed over a collection of 100 simulated replications (using the same partitioning method) is of the order 0.74, with a maximum of 0.7435. Considering that the modules are meaningful, the visual representation provided by Figure 5 is more faithful to the underlying graphical structure than the ﬁner displays of Figure 4. That said, the two graphs tend to agree as the pie charts of Figure 5 clearly show two parts in the network: the lower left part seems to gather most of the women and heterosexual men (as the upper part of Figure 4), while the upper right part contains clusters made almost entirely of MSM, as the lower part of Figure 4. While the display of Figure 5 might seem cluttered, it is in fact very readable if one considers that only 328 edges of the giant component connect persons from diﬀerent clusters while 2,840 connections happen inside clusters. Then most of the edges on Figure 5 could be disregarded as they corresponds to only one pair of connected persons (this is the case of 94 of such edges out of 142 and the former are represented as dashed 12

text: about

text: possible

text: sexual

text: contacts

text: between

text: the

text: latter

text: (which

text: could

text: potentially

text: lead

text: to

text: con-

text: nect

text: some

text: of

text: the

text: disjoint

text: components

text: aforementioned)

text: is

text: available.

text: There

text: are

text: isolated

text: individuals,

text: most

text: of

text: which

text: are

text: men,

text: especially

text: heterosexual

text: men.

text: The

text: isolated

text: couples

text: are

text: mostly

text: composed

text: of

text: man-woman

text: couples

text: with

text: an

text: over-

text: representation

text: of

text: women

text: compared

text: to

text: the

text: rest

text: of

text: the

text: population.

text: The

text: main

text: reasons

text: why

text: these

text: individuals

text: are

text: isolated

text: are

text: that

text: they

text: have

text: few

text: positive

text: contacts

text: and

text: there

text: is

text: a

text: high

text: fraction

text: who

text: did

text: not

text: provide

text: their

text: contacts

text: and

text: in

text: the

text: whole

text: population

text: and

text: in

text: the

text: giant

text: graph

text: resp.).

text: The

text: network

text: density

text: is

text: globally

text: low

text: and

text: very

text: heterogeneous.

text: But

text: although

text: the

text: connectivity

text: of

text: the

text: network

text: seems

text: fragile

text: at

text: ﬁrst

text: glance,

text: density

text: may

text: be

text: locally

text: very

text: high.

text: The

text: harmonic

text: average

text: of

text: the

text: geodesic

text: path

text: lengths

text: equals

text: and

text: for

text: the

text: oriented

text: graph.

text: Most

text: of

text: the

text: graph

text: connectivity

text: is

text: concentrated

text: in

text: the

text: largest

text: component

text: edges

text: out

text: of

text: The

text: largest

text: component

text: has

text: a

text: diameter

text: of

text: when

text: taking

text: into

text: account

text: the

text: direction

text: of

text: the

text: infections)

text: and

text: the

text: harmonic

text: average

text: of

text: the

text: geodesic

text: path

text: lengths

text: are

text: the

text: same

text: inside

text: the

text: largest

text: component.

text: These

text: values

text: are

text: slightly

text: higher

text: than

text: those

text: of

text: other

text: real

text: networks

text: mentioned

text: in

text: but

text: remain

text: well

text: below

text: the

text: number

text: of

text: vertices

text: and

text: compatible

text: with

text: the

text: logarithmic

text: scaling

text: related

text: to

text: the

text: so-termed

text: small

text: world

text: eﬀect.

text: Figure

text: shows

text: the

text: layout

text: obtained

text: by

text: means

text: of

text: the

text: force

text: directed

text: placement

text: algorithm

text: from

text: The

text: general

text: picture

text: seems

text: quite

text: clear,

text: with

text: what

text: appears

text: to

text: be

text: two

text: parts

text: in

text: the

text: graph:

text: the

text: lower

text: part

text: of

text: the

text: graph

text: (on

text: the

text: ﬁgure)

text: seems

text: to

text: be

text: dominated

text: by

text: MSM

text: while

text: the

text: upper

text: part

text: gathers

text: almost

text: all

text: persons

text: from

text: the

text: giant

text: component

text: that

text: have

text: only

text: heterosexual

text: contacts.

text: However,

text: the

text: upper

text: part

text: is

text: quite

text: diﬃcult

text: to

text: read

text: as

text: it

text: seems

text: denser

text: than

text: the

text: lower

text: part.

text: The

text: layout

text: shows

text: what

text: might

text: be

text: interpreted

text: as

text: cycles

text: and

text: also

text: a

text: lot

text: of

text: small

text: trees

text: connected

text: to

text: denser

text: parts.

text: The

text: actual

text: connection

text: patterns

text: between

text: the

text: upper

text: part

text: and

text: the

text: lower

text: part

text: are

text: also

text: very

text: unclear.

text: Because

text: of

text: these

text: crowding

text: eﬀects,

text: structural

text: properties

text: of

text: the

text: network

text: from

text: Figure

text: appears

text: quite

text: diﬃcult

text: and

text: probably

text: misleading.

text: We

text: rely

text: therefore

text: on

text: the

text: simpliﬁcation

text: technique

text: outlined

text: in

text: Section

text: leveraging

text: a

text: clustering

text: of

text: the

text: giant

text: component

text: to

text: get

text: an

text: insight

text: into

text: its

text: general

text: organization.

text: A

text: clustering

text: of

text: high

text: modularity.

text: A

text: graphical

text: representation

text: of

text: the

text: partition

text: obtained

text: by

text: the

text: method

text: from

text: is

text: displayed

text: in

text: Figure

text: The

text: clustering

text: thus

text: pro-

text: duced

text: exhibits

text: a

text: modularity

text: of

text: and

text: is

text: made

text: up

text: of

text: clusters.

text: This

text: modularity

text: is

text: very

text: high

text: compared

text: to

text: the

text: random

text: level

text: and

text: strongly

text: supports

text: the

text: hypothesis

text: of

text: a

text: speciﬁc

text: (“non

text: random”)

text: underlying

text: community

text: structure.

text: For

text: comparison

text: purpose,

text: the

text: average

text: maximal

text: modularity

text: attained

text: by

text: random

text: graphs

text: built

text: from

text: a

text: conﬁgura-

text: tion

text: model

text: with

text: the

text: same

text: size

text: and

text: degree

text: distribution

text: as

text: those

text: of

text: the

text: giant

text: component

text: observed

text: over

text: a

text: collection

text: of

text: simulated

text: replications

text: (using

text: the

text: same

text: partitioning

text: method)

text: is

text: of

text: the

text: order

text: with

text: a

text: maximum

text: of

text: Considering

text: that

text: the

text: modules

text: are

text: meaningful,

text: the

text: visual

text: representation

text: provided

text: by

text: Figure

text: is

text: more

text: faithful

text: to

text: the

text: underlying

text: graphical

text: structure

text: than

text: the

text: ﬁner

text: displays

text: of

text: Figure

text: That

text: said,

text: the

text: two

text: graphs

text: tend

text: to

text: agree

text: as

text: the

text: pie

text: charts

text: of

text: Figure

text: clearly

text: show

text: two

text: parts

text: in

text: the

text: network:

text: the

text: lower

text: left

text: part

text: seems

text: to

text: gather

text: most

text: of

text: the

text: women

text: and

text: heterosexual

text: men

text: (as

text: the

text: upper

text: part

text: of

text: Figure

text: while

text: the

text: upper

text: right

text: part

text: contains

text: clusters

text: made

text: almost

text: entirely

text: of

text: MSM,

text: as

text: the

text: lower

text: part

text: of

text: Figure

text: While

text: the

text: display

text: of

text: Figure

text: might

text: seem

text: cluttered,

text: it

text: is

text: in

text: fact

text: very

text: readable

text: if

text: one

text: considers

text: that

text: only

text: edges

text: of

text: the

text: giant

text: component

text: connect

text: persons

text: from

text: diﬀerent

text: clusters

text: while

text: connections

text: happen

text: inside

text: clusters.

text: Then

text: most

text: of

text: the

text: edges

text: on

text: Figure

text: could

text: be

text: disregarded

text: as

text: they

text: corresponds

text: to

text: only

text: one

text: pair

text: of

text: connected

text: persons

text: (this

text: is

text: the

text: case

text: of

text: of

text: such

text: edges

text: out

text: of

text: and

text: the

text: former

text: are

text: represented

text: as

text: dashed


text: [7] T. Britton, M. Deijfen, A. Lager˚as, and M. Lindholm. Epidemics on random graphs with tunable clustering. Journal of Applied Probability, 45:743--756, 2008. [8] S.K. Chan, L.R. Thornton, K.J. Chronister, J. Meyer, M. Wolverton, C.K. Johnson, R.R. Arafat, P. Joyce, W.M. Switzer, W. Heneine, A. Shankar, T. Granade, S. Michele Owen, P. Sprinkle, and V. Sullivan. Likely Female-to-Female Sexual Transmission of HIV | Texas. Morbidity and Mortality Weekly Report, Centers for Disease Control and Prevention, 63(10):209--212, 2014. [9] A. Clauset, C. Shalizi, and M. Newman. Power-law distributions in empirical data. SIAM Review, 51:661--703, 2009. [10] S. Cl´emen¸con, H. De Arazoza, F. Rossi, and V. Tran. Hierarchical clustering for graph visualization. In Proceedings of XVIIIth European Symposium on Artificial Neural Networks (ESANN 2011), pages 227--232, Bruges, Belgique, April 2011. [11] S. Cl´emen¸con, H. De Arazoza, F. Rossi, and V. Tran. Visual mining of epidemic networks. In J. Cabestany, I. Rojas, and G. Joya, editors, Advances in Computational Intelligence, Proceedings of 11th International Work-Conference on Artificial Neural Networks (IWANN 2011), number 6692 in Lecture Notes in Computer Science, pages 276--283. Springer Berlin / Heidelberg, Malaga, Spain, 2011. [12] S. Cl´emen¸con, H. De Arazoza, F. Rossi, and V. Tran. Supplementary materias for ‘‘a network analysis of the HIV-AIDS epidemics in Cuba". 2013. [13] S. Cl´emen¸con, V. Tran, and H. D. Arazoza. A stochastic SIR model with contact-tracing: large population limits and statistical inference. Journal of Biological Dynamics, 2(4):391--414, 2008. [14] H. de Arazoza and R. Lounes. A nonlinear model for sexually transmitted disease with contact-tracing. J. Math. Appl. Med. Biol., 19:221--234, 2002. [15] L. Decreusefond, J.-S. Dhersin, P. Moyal, and V. Tran. Large graph limit for a sir process in random network with heterogeneous connectivity. Annals of Applied Probability, 22(2):541--575, 2011. [16] G. Di Battista, P. Eades, R. Tamassia, and I. G. Tollis. Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, 1999. [17] R. Durrett. Random graph dynamics. Cambridge University Press, 2006. [18] S. Fortunato and M. Barth´elemy. Resolution limit in community detection. Proceedings of the National Academy of Sciences, 104(1):36--41, 2007. [19] T. Fruchterman and B. Reingold. Graph drawing by force-directed placement. Software-Practice and Experience, 21:1129--1164, 1991. [20] M. Graham and T. House. Dynamics of stochastic epidemics on heterogeneous networks. Journal of Mathematical Biology, 2013. published online. 21

text: T.

text: Britton,

text: M.

text: Deijfen,

text: A.

text: Lager˚as,

text: and

text: M.

text: Lindholm.

text: Epidemics

text: on

text: random

text: graphs

text: with

text: tunable

text: clustering.

text: Journal

text: of

text: Applied

text: Probability,

text: S.K.

text: Chan,

text: L.R.

text: Thornton,

text: K.J.

text: Chronister,

text: J.

text: Meyer,

text: M.

text: Wolverton,

text: C.K.

text: Johnson,

text: R.R.

text: Arafat,

text: P.

text: Joyce,

text: W.M.

text: Switzer,

text: W.

text: Heneine,

text: A.

text: Shankar,

text: T.

text: Granade,

text: S.

text: Michele

text: Owen,

text: P.

text: Sprinkle,

text: and

text: V.

text: Sullivan.

text: Likely

text: Female-to-Female

text: Sexual

text: Transmission

text: of

text: HIV

text: Texas.

text: Morbidity

text: and

text: Mortality

text: Weekly

text: Report,

text: Centers

text: for

text: Disease

text: Control

text: and

text: Prevention,

text: A.

text: Clauset,

text: C.

text: Shalizi,

text: and

text: M.

text: Newman.

text: Power-law

text: distributions

text: in

text: empirical

text: data.

text: SIAM

text: Review,

text: S.

text: Cl´emen¸con,

text: H.

text: De

text: Arazoza,

text: F.

text: Rossi,

text: and

text: V.

text: Tran.

text: Hierarchical

text: clustering

text: for

text: graph

text: visualization.

text: In

text: Proceedings

text: of

text: XVIIIth

text: European

text: Symposium

text: on

text: Artificial

text: Neural

text: Networks

text: (ESANN

text: pages

text: Bruges,

text: Belgique,

text: April

text: S.

text: Cl´emen¸con,

text: H.

text: De

text: Arazoza,

text: F.

text: Rossi,

text: and

text: V.

text: Tran.

text: Visual

text: mining

text: of

text: epidemic

text: networks.

text: In

text: J.

text: Cabestany,

text: I.

text: Rojas,

text: and

text: G.

text: Joya,

text: editors,

text: Advances

text: in

text: Computational

text: Intelligence,

text: Proceedings

text: of

text: International

text: Work-Conference

text: on

text: Artificial

text: Neural

text: Networks

text: (IWANN

text: number

text: in

text: Lecture

text: Notes

text: in

text: Computer

text: Science,

text: pages

text: Springer

text: Berlin

text: Heidelberg,

text: Malaga,

text: Spain,

text: S.

text: Cl´emen¸con,

text: H.

text: De

text: Arazoza,

text: F.

text: Rossi,

text: and

text: V.

text: Tran.

text: Supplementary

text: materias

text: for

text: ‘‘a

text: network

text: analysis

text: of

text: the

text: HIV-AIDS

text: epidemics

text: in

text: Cuba".

text: S.

text: Cl´emen¸con,

text: V.

text: Tran,

text: and

text: H.

text: D.

text: Arazoza.

text: A

text: stochastic

text: SIR

text: model

text: with

text: contact-tracing:

text: large

text: population

text: limits

text: and

text: statistical

text: inference.

text: Journal

text: of

text: Biological

text: Dynamics,

text: H.

text: de

text: Arazoza

text: and

text: R.

text: Lounes.

text: A

text: nonlinear

text: model

text: for

text: sexually

text: transmitted

text: disease

text: with

text: contact-tracing.

text: J.

text: Math.

text: Appl.

text: Med.

text: Biol.,

text: L.

text: Decreusefond,

text: J.-S.

text: Dhersin,

text: P.

text: Moyal,

text: and

text: V.

text: Tran.

text: Large

text: graph

text: limit

text: for

text: a

text: sir

text: process

text: in

text: random

text: network

text: with

text: heterogeneous

text: connectivity.

text: Annals

text: of

text: Applied

text: Probability,

text: G.

text: Di

text: Battista,

text: P.

text: Eades,

text: R.

text: Tamassia,

text: and

text: I.

text: G.

text: Tollis.

text: Graph

text: Drawing:

text: Algorithms

text: for

text: the

text: Visualization

text: of

text: Graphs.

text: Prentice

text: Hall,

text: R.

text: Durrett.

text: Random

text: graph

text: dynamics.

text: Cambridge

text: University

text: Press,

text: S.

text: Fortunato

text: and

text: M.

text: Barth´elemy.

text: Resolution

text: limit

text: in

text: community

text: detection.

text: Proceedings

text: of

text: the

text: National

text: Academy

text: of

text: Sciences,

text: T.

text: Fruchterman

text: and

text: B.

text: Reingold.

text: Graph

text: drawing

text: by

text: force-directed

text: placement.

text: Software-Practice

text: and

text: Experience,

text: M.

text: Graham

text: and

text: T.

text: House.

text: Dynamics

text: of

text: stochastic

text: epidemics

text: on

text: heterogeneous

text: networks.

text: Journal

text: of

text: Mathematical

text: Biology,

text: published

text: online.


text: distribution as the epidemics graph. Using the above algorithm, we compute a maxi- mal modularity clustering on each of those graphs. The modularities of the clustering provide an estimate of the distribution of the maximal modularity in random graphs with our degree distribution. If a partition of this graph exhibits a higher modularity, we conclude that it must be the result of some actual modular structure rather than a random outcome. The maximal modularity clustering is visualised using the force directed placement algorithm described above. In addition to giving a general idea of the global structure of the graph, the obtained visual representation can be used to display distributions of covariates at the cluster level. Homogeneity tests are performed in order to assess possible signiﬁcant diﬀerences between these statistical subpopulations. However, as demonstrated in [18], ﬁnding the maximal modularity clustering can lead to ignoring small modular structures that fall below the resolution limit of the modularity measure. It is then recommended in [18] to recursively apply maximal modularity clustering to the original clusters in order to investigate potential smaller scale modules. We follow this strategy coupled with the MCMC approach described above: each cluster is tested for substructure by applying the maximal modularity clustering technique from [41] and by assessing the actual signiﬁcance of a potential sub-structure via comparison with similar random graphs. 3 Results and Discussion 3.1 Overview of the Cuban HIV epidemics with contact- tracing We start oﬀ with a brief review of some macroscopic statistics (on the basis of Sections 2 and 3 of the ESM, to which we refer for further ﬁgures and details). Most HIV infections in Cuba occurred among MSM individuals (68.92%) and among people living in Ciudad de la Habana (53.48%). However, the evolution of the number of HIV detections within groups of diﬀerent sexual orientations, Fig. 1, reveals that, at the very beginning of the epidemic, infections occurred predominantly in the heterosexual men subpopulation. In the following years, women and then MSMs were then contaminated. One also observes a signiﬁcant increase of the reported HIV diagnoses, starting from 1995. Further analysis is carried out in Section 4 of the ESM by studying the infection tree, and a ﬁlm showing the emergence of the giant com- ponent is also given in the supplementary materials. Age distributions show a slight ageing of newly infected individuals. Over the observation period 1986-2006, the mean age at detection is 29.43 years for the whole population, while it is lower for women. There are several ways of HIV detection: random screening, contact-tracing or captation (i.e. following a medical examination by the family doctor). The proportion of new HIV cases diagnosed thanks to contact-tracing detection continually increased over the observation period. Among the individuals detected as HIV positive be- tween 1986 and 2006, 1,515 (respectively 821 and 3,053) persons have been detected by contact-tracing (respectively by captation and random-screening methods), which amounts to 28% (respectively 15% and 57%) of the total number of HIV detections. The “nonrandom detection methods” (i.e. detections by contact-tracing or captation) account for 65.5%, 33.5% and 38.2% of detections for women, heterosexual men and MSM respectively. A χ2-test reveals that the variables ’gender/sexual orientation’ and ’way of detection’ are not independent, even when restricted to a geographical region. 7

text: distribution

text: as

text: the

text: epidemics

text: graph.

text: Using

text: the

text: above

text: algorithm,

text: we

text: compute

text: a

text: maxi-

text: mal

text: modularity

text: clustering

text: on

text: each

text: of

text: those

text: graphs.

text: The

text: modularities

text: of

text: the

text: clustering

text: provide

text: an

text: estimate

text: of

text: the

text: distribution

text: of

text: the

text: maximal

text: modularity

text: in

text: random

text: graphs

text: with

text: our

text: degree

text: distribution.

text: If

text: a

text: partition

text: of

text: this

text: graph

text: exhibits

text: a

text: higher

text: modularity,

text: we

text: conclude

text: that

text: it

text: must

text: be

text: the

text: result

text: of

text: some

text: actual

text: modular

text: structure

text: rather

text: than

text: a

text: random

text: outcome.

text: The

text: maximal

text: modularity

text: clustering

text: is

text: visualised

text: using

text: the

text: force

text: directed

text: placement

text: algorithm

text: described

text: above.

text: In

text: addition

text: to

text: giving

text: a

text: general

text: idea

text: of

text: the

text: global

text: structure

text: of

text: the

text: graph,

text: the

text: obtained

text: visual

text: representation

text: can

text: be

text: used

text: to

text: display

text: distributions

text: of

text: covariates

text: at

text: the

text: cluster

text: level.

text: Homogeneity

text: tests

text: are

text: performed

text: in

text: order

text: to

text: assess

text: possible

text: signiﬁcant

text: diﬀerences

text: between

text: these

text: statistical

text: subpopulations.

text: However,

text: as

text: demonstrated

text: in

text: ﬁnding

text: the

text: maximal

text: modularity

text: clustering

text: can

text: lead

text: to

text: ignoring

text: small

text: modular

text: structures

text: that

text: fall

text: below

text: the

text: resolution

text: limit

text: of

text: the

text: modularity

text: measure.

text: It

text: is

text: then

text: recommended

text: in

text: to

text: recursively

text: apply

text: maximal

text: modularity

text: clustering

text: to

text: the

text: original

text: clusters

text: in

text: order

text: to

text: investigate

text: potential

text: smaller

text: scale

text: modules.

text: We

text: follow

text: this

text: strategy

text: coupled

text: with

text: the

text: MCMC

text: approach

text: described

text: above:

text: each

text: cluster

text: is

text: tested

text: for

text: substructure

text: by

text: applying

text: the

text: maximal

text: modularity

text: clustering

text: technique

text: from

text: and

text: by

text: assessing

text: the

text: actual

text: signiﬁcance

text: of

text: a

text: potential

text: sub-structure

text: via

text: comparison

text: with

text: similar

text: random

text: graphs.

text: Results

text: and

text: Discussion

text: Overview

text: of

text: the

text: Cuban

text: HIV

text: epidemics

text: with

text: contact-

text: tracing

text: We

text: start

text: oﬀ

text: with

text: a

text: brief

text: review

text: of

text: some

text: macroscopic

text: statistics

text: (on

text: the

text: basis

text: of

text: Sections

text: and

text: of

text: the

text: ESM,

text: to

text: which

text: we

text: refer

text: for

text: further

text: ﬁgures

text: and

text: details).

text: Most

text: HIV

text: infections

text: in

text: Cuba

text: occurred

text: among

text: MSM

text: individuals

text: and

text: among

text: people

text: living

text: in

text: Ciudad

text: de

text: la

text: Habana

text: However,

text: the

text: evolution

text: of

text: the

text: number

text: of

text: HIV

text: detections

text: within

text: groups

text: of

text: diﬀerent

text: sexual

text: orientations,

text: Fig.

text: reveals

text: that,

text: at

text: the

text: very

text: beginning

text: of

text: the

text: epidemic,

text: infections

text: occurred

text: predominantly

text: in

text: the

text: heterosexual

text: men

text: subpopulation.

text: In

text: the

text: following

text: years,

text: women

text: and

text: then

text: MSMs

text: were

text: then

text: contaminated.

text: One

text: also

text: observes

text: a

text: signiﬁcant

text: increase

text: of

text: the

text: reported

text: HIV

text: diagnoses,

text: starting

text: from

text: Further

text: analysis

text: is

text: carried

text: out

text: in

text: Section

text: of

text: the

text: ESM

text: by

text: studying

text: the

text: infection

text: tree,

text: and

text: a

text: ﬁlm

text: showing

text: the

text: emergence

text: of

text: the

text: giant

text: com-

text: ponent

text: is

text: also

text: given

text: in

text: the

text: supplementary

text: materials.

text: Age

text: distributions

text: show

text: a

text: slight

text: ageing

text: of

text: newly

text: infected

text: individuals.

text: Over

text: the

text: observation

text: period

text: the

text: mean

text: age

text: at

text: detection

text: is

text: years

text: for

text: the

text: whole

text: population,

text: while

text: it

text: is

text: lower

text: for

text: women.

text: There

text: are

text: several

text: ways

text: of

text: HIV

text: detection:

text: random

text: screening,

text: contact-tracing

text: or

text: captation

text: (i.e.

text: following

text: a

text: medical

text: examination

text: by

text: the

text: family

text: doctor).

text: The

text: proportion

text: of

text: new

text: HIV

text: cases

text: diagnosed

text: thanks

text: to

text: contact-tracing

text: detection

text: continually

text: increased

text: over

text: the

text: observation

text: period.

text: Among

text: the

text: individuals

text: detected

text: as

text: HIV

text: positive

text: be-

text: tween

text: and

text: (respectively

text: and

text: persons

text: have

text: been

text: detected

text: by

text: contact-tracing

text: (respectively

text: by

text: captation

text: and

text: random-screening

text: methods),

text: which

text: amounts

text: to

text: (respectively

text: and

text: of

text: the

text: total

text: number

text: of

text: HIV

text: detections.

text: The

text: “nonrandom

text: detection

text: methods”

text: (i.e.

text: detections

text: by

text: contact-tracing

text: or

text: captation)

text: account

text: for

text: and

text: of

text: detections

text: for

text: women,

text: heterosexual

text: men

text: and

text: MSM

text: respectively.

text: A

text: reveals

text: that

text: the

text: variables

text: ’gender/sexual

text: orientation’

text: and

text: ’way

text: of

text: detection’

text: are

text: not

text: independent,

text: even

text: when

text: restricted

text: to

text: a

text: geographical

text: region.


text: the giant component with probability one. Joint degree distribution of sexual partners. The independence assump- tion between the degrees of adjacent vertices does not hold here, see Fig. 3, in con- trast to what is assumed for the vast majority of graph-based SIR models of epidemic disease, e.g. [17, 31]. Indeed, the linear correlation coeﬃcient between the degree distribution of alters and egos is equal to 0.68 and independence is rejected by a χ2- − test with a p-value of 6.85 10 6. In particular, highly connected vertices tend to be connected to vertices with a high number of connections too. From the perspective of mathematical modeling, this suggests to consider graph models with a dependence structure between the degrees of adjacent nodes, in opposition to most percolation processes on (conﬁguration model) networks used to describe the spread of epidemics [28, 3, 44, 15, 20]. However, it is worth noticing that, if we restrict our analysis to some speciﬁc, more homogeneous, subgroups, the independence assumption may be grounded in evidence. So if assumptions such that the network is generated by a con- ﬁguration model do not hold globally, they may be valid for smaller clusters, which is another motivation for clustering. Preferential attachment. The network clearly exhibits a phenomenon of “assor- tative mixing” with respect to the area of residence, age, detection mode: individuals sharing these characteristics are connected or have short connecting paths. The vast majority of connections unsurprisingly occur between individuals in the same geo- graphical region, and among the latter, 1,678 (41%) link two individuals living both in the Ciudad de la Habana, resulting in a very high assortative mixing coeﬃcient, 0.7753. Those who are between 20 and 40 at the time of detection make 61.16% of the edges of the whole graph and 62.88% of the edges in the infection tree. Among these edges, 28.32% (resp. 25.80% in the infection tree) link individuals of the same group [20, 25), [25, 30), [30, 35) or [35, 40). Other relations and average geodesic distances in the giant component with respect to the various attributes are computed in the ESM. As edges correspond to sexual contacts in the present graph, the gender/sexual orientation of adjacent vertices cannot be arbitrary of course. When considering the sexual orientation of two neighbours, we see that more than a half of the edges (56.47%) link two MSM. Links between MSM and women make 1,208 edges (29.66%) and there are 439 edges (10.78%) between women and heterosexual men. Looking at the infection tree provided similar proportions: 1,202 edges (52.56%), 667 edges (29.16%) and 375 edges (16.40%) respectively. Figures reveal an asymmetry in HIV infection: among (oriented) infection edges involving women, the latter are more often alters than egos (66.13% of the edges shared with heterosexual men and 74.21% of the edges shared with MSM). The declarative degree shows a smaller mean degree for heterosexual men and comparable degree distributions between women and MSM. MSM are expected to contribute most to the connectivity of the graph, especially bisexual men who act as contact points between women and MSM who declare only contacts with men. 3.3 Analysis of the ”giant component” Several connected components form the sexual contact network of the Cuban HIV database. One is much larger than the others with 2,386 HIV-infected individuals (44.28%), while the other components count less than 17 vertices (see ESM). Inciden- tally, we underline that, although the number of non HIV-infected sexual partners of the individuals infected by the HIV between 1986 and 2006 is known, no information 10

text: the

text: giant

text: component

text: with

text: probability

text: one.

text: Joint

text: degree

text: distribution

text: of

text: sexual

text: partners.

text: The

text: independence

text: assump-

text: tion

text: between

text: the

text: degrees

text: of

text: adjacent

text: vertices

text: does

text: not

text: hold

text: here,

text: see

text: Fig.

text: in

text: con-

text: trast

text: to

text: what

text: is

text: assumed

text: for

text: the

text: vast

text: majority

text: of

text: graph-based

text: SIR

text: models

text: of

text: epidemic

text: disease,

text: e.g.

text: Indeed,

text: the

text: linear

text: correlation

text: coeﬃcient

text: between

text: the

text: degree

text: distribution

text: of

text: alters

text: and

text: egos

text: is

text: equal

text: to

text: and

text: independence

text: is

text: rejected

text: by

text: a

text: test

text: with

text: a

text: p-value

text: of

text: In

text: particular,

text: highly

text: connected

text: vertices

text: tend

text: to

text: be

text: connected

text: to

text: vertices

text: with

text: a

text: high

text: number

text: of

text: connections

text: too.

text: From

text: the

text: perspective

text: of

text: mathematical

text: modeling,

text: this

text: suggests

text: to

text: consider

text: graph

text: models

text: with

text: a

text: dependence

text: structure

text: between

text: the

text: degrees

text: of

text: adjacent

text: nodes,

text: in

text: opposition

text: to

text: most

text: percolation

text: processes

text: on

text: (conﬁguration

text: model)

text: networks

text: used

text: to

text: describe

text: the

text: spread

text: of

text: epidemics

text: However,

text: it

text: is

text: worth

text: noticing

text: that,

text: if

text: we

text: restrict

text: our

text: analysis

text: to

text: some

text: speciﬁc,

text: more

text: homogeneous,

text: subgroups,

text: the

text: independence

text: assumption

text: may

text: be

text: grounded

text: in

text: evidence.

text: So

text: if

text: assumptions

text: such

text: that

text: the

text: network

text: is

text: generated

text: by

text: a

text: con-

text: ﬁguration

text: model

text: do

text: not

text: hold

text: globally,

text: they

text: may

text: be

text: valid

text: for

text: smaller

text: clusters,

text: which

text: is

text: another

text: motivation

text: for

text: clustering.

text: Preferential

text: attachment.

text: The

text: network

text: clearly

text: exhibits

text: a

text: phenomenon

text: of

text: “assor-

text: tative

text: mixing”

text: with

text: respect

text: to

text: the

text: area

text: of

text: residence,

text: age,

text: detection

text: mode:

text: individuals

text: sharing

text: these

text: characteristics

text: are

text: connected

text: or

text: have

text: short

text: connecting

text: paths.

text: The

text: vast

text: majority

text: of

text: connections

text: unsurprisingly

text: occur

text: between

text: individuals

text: in

text: the

text: same

text: geo-

text: graphical

text: region,

text: and

text: among

text: the

text: latter,

text: link

text: two

text: individuals

text: living

text: both

text: in

text: the

text: Ciudad

text: de

text: la

text: Habana,

text: resulting

text: in

text: a

text: very

text: high

text: assortative

text: mixing

text: coeﬃcient,

text: Those

text: who

text: are

text: between

text: and

text: at

text: the

text: time

text: of

text: detection

text: make

text: of

text: the

text: edges

text: of

text: the

text: whole

text: graph

text: and

text: of

text: the

text: edges

text: in

text: the

text: infection

text: tree.

text: Among

text: these

text: edges,

text: (resp.

text: in

text: the

text: infection

text: tree)

text: link

text: individuals

text: of

text: the

text: same

text: group

text: or

text: Other

text: relations

text: and

text: average

text: geodesic

text: distances

text: in

text: the

text: giant

text: component

text: with

text: respect

text: to

text: the

text: various

text: attributes

text: are

text: computed

text: in

text: the

text: ESM.

text: As

text: edges

text: correspond

text: to

text: sexual

text: contacts

text: in

text: the

text: present

text: graph,

text: the

text: gender/sexual

text: orientation

text: of

text: adjacent

text: vertices

text: cannot

text: be

text: arbitrary

text: of

text: course.

text: When

text: considering

text: the

text: sexual

text: orientation

text: of

text: two

text: neighbours,

text: we

text: see

text: that

text: more

text: than

text: a

text: half

text: of

text: the

text: edges

text: link

text: two

text: MSM.

text: Links

text: between

text: MSM

text: and

text: women

text: make

text: edges

text: and

text: there

text: are

text: edges

text: between

text: women

text: and

text: heterosexual

text: men.

text: Looking

text: at

text: the

text: infection

text: tree

text: provided

text: similar

text: proportions:

text: edges

text: edges

text: and

text: edges

text: respectively.

text: Figures

text: reveal

text: an

text: asymmetry

text: in

text: HIV

text: infection:

text: among

text: (oriented)

text: infection

text: edges

text: involving

text: women,

text: the

text: latter

text: are

text: more

text: often

text: alters

text: than

text: egos

text: of

text: the

text: edges

text: shared

text: with

text: heterosexual

text: men

text: and

text: of

text: the

text: edges

text: shared

text: with

text: MSM).

text: The

text: declarative

text: degree

text: shows

text: a

text: smaller

text: mean

text: degree

text: for

text: heterosexual

text: men

text: and

text: comparable

text: degree

text: distributions

text: between

text: women

text: and

text: MSM.

text: MSM

text: are

text: expected

text: to

text: contribute

text: most

text: to

text: the

text: connectivity

text: of

text: the

text: graph,

text: especially

text: bisexual

text: men

text: who

text: act

text: as

text: contact

text: points

text: between

text: women

text: and

text: MSM

text: who

text: declare

text: only

text: contacts

text: with

text: men.

text: Analysis

text: of

text: the

text: ”giant

text: component”

text: Several

text: connected

text: components

text: form

text: the

text: sexual

text: contact

text: network

text: of

text: the

text: Cuban

text: HIV

text: database.

text: One

text: is

text: much

text: larger

text: than

text: the

text: others

text: with

text: HIV-infected

text: individuals

text: while

text: the

text: other

text: components

text: count

text: less

text: than

text: vertices

text: (see

text: ESM).

text: Inciden-

text: tally,

text: we

text: underline

text: that,

text: although

text: the

text: number

text: of

text: non

text: HIV-infected

text: sexual

text: partners

text: of

text: the

text: individuals

text: infected

text: by

text: the

text: HIV

text: between

text: and

text: is

text: known,

text: no

text: information


text: 2 dark matter numerical simulations [58]. However, it is possible that the dark matter halo proﬁle remains diver- (cid:68)1(cid:45) ge∝nt c−lose to the centre such that proﬁles may behave as s ρ r γ with γ > 1 (γ = 1 in the NFW proﬁle). As an 2(cid:45) example, the Via Lactea II simulation favours a proﬁle m c 10(cid:45)7 with γ = 1.24 [59]. Given that the γ-ray emission traces V the morphology of the proﬁle, the consequence of a more e strongly peaked proﬁle in terms of indirect detection is a G (cid:64) much brighter gamma-ray emission relative to the case of EΓ syst. an Einasto or NFW proﬁle. For the extended gamma-ray (cid:144)d stat. excess, it is found th(cid:18)at a(cid:19)gen(cid:20)eralis(cid:18)ed N(cid:19)F(cid:21)W proﬁle (cid:70) 2EdΓ m(cid:88)ΣDvM(cid:92)(cid:61)(cid:61)33(cid:137)01G0e(cid:45)2V6cm3 s(cid:45)1 r −γ r γ−3 ρ(r) = ρ 1 + . (1) 10(cid:45)8 s rs rs 0.3 1.0 (cid:64) 3.0(cid:68) 10.0 with γ = 1.2 gives the best ﬁt [16]. E GeV Γ The following simpliﬁed model gives a good ﬁt to the extended gamma-ray excess shown in ﬁg. 1. We take the dark matter χ to be a Dirac fermion with mass m FIG. 1. The data points show the extended gamma-ray excess DM ◦× ◦ from a 7 7 region centred on the galactic centre (from [16]). which interacts with a pseudoscalar a with mass ma (cid:88) The red and black error bars show the systematic and sta- through the coupling gDM: tistical uncertainties respectively. The blue solid line shows the photon spectrum corresponding to 30 GeV dark matter L ⊃ − g√ − √g i DM aχ¯γ5χ i f af¯γ5f + h.c. (2) with an annihilation cross-section that gives the observed relic 2 2 density. The branching ratios are determined by the Yukawa f couplings y . f The pseudoscalar couples to the Standard Model fermions with g , which we assume is equal to the Stan- f ≡ dard model Yukawa coupling g = y m /174 GeV. logically similar excess has also been reported at more f f f This relation is common for pseudoscalars, motivated extended distances from the galactic plane [48, 49]. from the minimal ﬂavour violation (MFV) ansatz [47]. In addition to dark matter annihilation, it has been The photon ﬂux Φ at Earth from a region ∆Ω, assum- suggested that interactions between cosmic rays and ing prompt photon emission arising from annihilation of gas [50–52] or an unresolved population of millisecond Dirac dark matte(cid:18)r, is [60(cid:19)] (cid:88) pulsars [12, 14, 15, 53] can explain the excess. However, more detailed studies have raised problems with both of these explanations [17, 54]. It is also possible that a new dΦ = 1 r(cid:12) ρ(cid:12) 2 (cid:104)J(cid:105)∆Ω (cid:104)σv(cid:105) dNγf , (3) mechanism not proposed is responsible, since the galactic dE 4 4π m f dE γ DM γ f centre is a complex astrophysical environment [55]. For the purpose of this work, we assume that all of the excess where r(cid:12) = 8.25 kpc is the distance from the galac- is a result of dark matter annihilation. We use the results − tic centre to the Earth, ρ(cid:12) = 0.42 GeVcm 3 is the local (cid:104) (cid:105) from the analysis of [16] (listed in their Appendix A), ◦ × ◦ dark matter density [61, 62], σv is the annihilation f who considered all events within a 7 7 regi∗on centred cross-section to f¯f and dNf/dE is the energy spectrum on the galactic centre (the position of Sgr A ). Galac- γ γ of photons produced per annihilation to f¯f. We use the tic backgrounds were modelled with the standard LAT tabulated values of dNf/dE from [60, 63], which are diﬀuse model, with isotropic residuals assumed for in- γ γ generated with PYTHIA 8.135 [64] and disregard any con- strumental and extragalactic sources. After background tribution to the ﬂux that is not prompt i.e. we neglect subtraction the extended emission component that they all photons generated by the propagation of cosmic rays. ﬁnd is shown in ﬁg. 1, where the red and black error bars (cid:90) The average J factor over a region of size ∆Ω is correspond to systematic and statistical uncertainties respectively. (cid:104) (cid:105) 1 To proceed with the dark matter interpretation, it is J = cos b J(b, l) db dl , (4) necessary to specify the dark matter halo proﬁle. While ∆Ω it is well determined far from the galactic centre, the (cid:12) slope is uncertain at small radii; typically there are no where (cid:90) (cid:18) (cid:19) (cid:12) (cid:12) observations below 1 kpc and the resolution of numerical (cid:12) ∼ (cid:12) simulations is 0.1 kpc. The Einasto [56] and Navarro, ds ρ(r) 2 Frenk and White (NFW) [57] proﬁles are traditionally J(b, l) = √ (5) r(cid:12) ρ(cid:12) used as benchmark proﬁles as they provide good ﬁts to l.o.s r= r(cid:12)2 +s2−2r(cid:12)scosbcosl ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: dark

text: matter

text: numerical

text: simulations

text: However,

text: it

text: is

text: possible

text: that

text: the

text: dark

text: matter

text: halo

text: proﬁle

text: remains

text: diver-

text: ge∝nt

text: c−lose

text: to

text: the

text: centre

text: such

text: that

text: proﬁles

text: may

text: behave

text: as

text: s

text: ρ

text: r

text: γ

text: with

text: γ

text: (γ

text: in

text: the

text: NFW

text: proﬁle).

text: As

text: an

text: example,

text: the

text: Via

text: Lactea

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text: a

text: proﬁle

text: m

text: c

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text: γ

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text: that

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text: γ-ray

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text: the

text: morphology

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text: the

text: proﬁle,

text: the

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text: G

text: much

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text: emission

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text: to

text: the

text: case

text: of

text: EΓ

text: syst.

text: an

text: Einasto

text: or

text: NFW

text: proﬁle.

text: For

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text: extended

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text: it

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text: found

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text: r

text: −γ

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text: the

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text: GeV

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text: following

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text: ﬁg.

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text: take

text: the

text: dark

text: matter

text: χ

text: to

text: be

text: a

text: Dirac

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text: mass

text: m

text: FIG.

text: The

text: data

text: points

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text: the

text: extended

text: gamma-ray

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text: a

text: region

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text: on

text: the

text: galactic

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text: which

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text: pseudoscalar

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text: mass

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text: red

text: and

text: black

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text: the

text: systematic

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text: through

text: the

text: coupling

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text: The

text: blue

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text: the

text: photon

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text: GeV

text: dark

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text: g√

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text: i

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text: aχ¯γ5χ

text: i

text: f

text: h.c.

text: with

text: an

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text: that

text: gives

text: the

text: observed

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text: density.

text: The

text: branching

text: ratios

text: are

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