text: 4. DISCUSSION large volume of continental crust combined with smaller maximal crustal thickness may lead to a planet entirely 4.1. Sensitivity Analysis § covered in continental crust. This does not greatly aﬀect As noted in , the amount of water in Earth’s man- our results, provided that the planet remains tectonically tle is poorly constrained. The gray broken lines in Fig- active and that some regions have thicker crust than oth- − ure 2 show the analytic waterwo×rld boundary if Earth’s ers (diﬀerences in crustal thickness scale as g 1 for the ma×ntle water content, x⊕, is 10 greater (dashed) and same reason as ocean basin depth). 10 smaller (dotted) than our nominal value. Note that 4.3. Homogeneity of the Mantle varying x⊕ in this model is mathematically equivalent to The homogeneity of volatiles in Earth’s lower mantle varying any of the water ﬂux parameters in (19), many of is questionable. The high 3He abundance of ocean is- which are uncertain (e.g., the subduction eﬃciency, χ). Varying x⊕ by two orders of magnitude aﬀects the wa- lands has been attributed to a poorly-mixed lower mantle terworld boundary for a 10M⊕ super-Earth by roughly (Kurz et al. 1982). By extension, this hypothesis implies that Earth’s lower mantle may hold much more water one order of magnitude; this is the dominant uncertainty than what is inferred for the upper mantle. Gonner- in our study. mann & Mukhopadhyay (2009) argue, however, that the The maximum water capacity of the mantle, x is not well known for high-mass terrestrial planets.maTxhe 3He abundance of the mantle is consistent with homoge- water storage in Earth’s mantle is thought to be concen- neous composition. trated in the transition zone (410–660 km depth). If mas- If the mantle is not well-mixed, then x represents the sive planets can only sequester water in a thin transition water fraction of those regions sampled by the mid-ocean zone, then mantle water capacity scales with planetary ridge melting and aﬀected by subduction of oceanic crust. ∝ − area rather than mass, or x g˜ 1. The dotted black Indeed, the source of mid-ocean ridge basalts (MORB) max line in Figure 2 shows the minuscule eﬀect of adopting appears to have maintained a constant water mass frac- this scaling. tion, x, for billions of years, suggesting that subduction of On the other hand, the mantle of a super-Earth should hydrated oceanic crust is depositing water in the MORB be primarily in the form of post-perovskite (Valencia source region (Hirschmann 2006). et al. 2007b), which may not have the same water capac- 4.4. Observational Constraints ity as Earth’s dominant mantle rock, perovskite. We try Our model of the deep water cycle predicts that many setting xmax = 1 and ﬁnd that the waterworld boundary super-Earths have exposed continents. It will eventu- is unchanged if post-perovskite can hold unlimited water. ally be possible to test this hypothesis by observationally Finally, the blue lines in Figure 2 show the waterworld determining the surface character of a large number of boundary for pressure-dependences of φ = 3 (dashed) high-mass terrestrial planets (see also discussion in Ab- and φ = 1 (dotted). (In both cases we use µ = σ = bot et al. 2012). φ/2 in the numerical model.) The precise strength of Disk-integrated rotational multiband photometry of the seaﬂoor pressure dependence of the deep water cycle Earth, essentially the changing colors of a pale blue dot, aﬀects the waterworld boundary by less than a factor of encode information about continents, oceans and clouds two. (Ford et al. 2001). Such “single-pixel” observations have 4.2. Plate Tectonics been used to construct coarse longitudinal color maps Our model of the deep water cycle assumes plate of Earth and Mars (Cowan et al. 2009, 2011; Fujii et al. tectonics, but it is currently unknown whether super- 2011; Hasinoﬀ et al. 2011), while simulations suggest that Earths are tectonically active. The increased heat ﬂux photometry spanning an entire planetary orbit could be of super-Earths should produce vigorous mantle convec- used to construct a rough 2D color map (Kawahara & Fu- tion (Valencia et al. 2007a; van Heck & Tackley 2011), jii 2010, 2011; Fujii & Kawahara 2012). Finally, Cowan but the increased strength and buoyancy of crust on & Strait (2013) showed that disk-integrated multiband super-Earths may prohibit plate tectonics (O’Neill & photometry of a variegated planet can be inverted to Lenardic 2007; Kite et al. 2009). Mantle convection may obtain reﬂectance spectra of its dominant surface types, even exhibit hysteresis, such that planets with identical even if the number and colors of the surfaces are not boundary conditions may or may not have plate tecton- known a priori. ics, depending on initial conditions (Lenardic & Crowley The bottom line is that a 5–10 m space telescope 2012). equipped with a coronagraph or starshade could produce Alternatively, it has been suggested that surface water a coarse surface map of an Earth-analog at a distance of is more important than planetary mass for plate tectonics 10 pc (Cowan et al. 2009; Fujii & Kawahara 2012). Such (Mian & Tozer 1990; Korenaga 2010). In a classic case low-resolution maps would be suﬃcient to identify the of chicken-and-egg, van der Lee et al. (2008) argue that a continents one expects on a tectonicaly active planet. If deep water cycle is a necessary, if not suﬃcient, condition most super-Earths exhibit the bimodal surface charac- for long-term plate tectonics. ter of Earth, it will suggest that they experience plate Continental crust formation is thought to be an in- tectonics and a deep water cycle. If, instead, large ter- evDitraabflte vbeyr-psiroondMucatyo2f6,p2la0t2e0 tectonics in the presence of restrial planets were all determined to be waterworlds, waPtreerpr(inRtutydpneiscekt u1si9n9g5LA;TAEXrnsdtytle2e0m1u3la)t,easpoj svu. p8/e1r3-/E1a0rths are it would indicate that our hypothesis is wrong. likely to have large volumes of granitic crust. In fact, the

label: section

text: DISCUSSION

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text: Super-Earths Need Not be Waterworlds 9 WATER CYCLING BETWEEN OCEAN AND MANTLE: SUPER-EARTHS NEED NOT BE WATERWORLDS Nicolas B. Cowan1 & Dorian S. Abbot2 Subject headings: planets and satellites: composition — planets and satellites: interiors — planets and satellites: oceans — planets and satellites: physical evolution — planets and satellites: surfaces — planets and satellites: tectonics 5. CONCLUSIONS they otherwise would be. Our model predicts that tectonically active 10M⊕ planets can maintain large It has long been suggested that super-Earths ought to exposed continents for water mass fractions less than be waterworlds (Stapledon 1937; Kite et al. 2009; Ab- × − 2 10 3. bot & Switzer 2011). If one accounts for the ﬁrst-order Exoplanets with suﬃciently high water content will eﬀects of gravity on ocean basin depth and water inven- tory, then a 10M⊕ planet is not expected to have exposed be water-covered regardless of the mechanism discussed here, but such “ocean planets” may betray themselves continents unless it has a water mass fraction less than 3 × 10−5, roughly ten times drier than Earth (solid red by their lower density: a planet with 10% water mass fraction will exhibit a transit depth 10% greater than line in Figure 2). an equally-massive planet with Earth-like composition We have argued, as have others (Kasting & Holm 1992; (Sotin et al. 2007). Planets with 1% water mass frac- Holm 1996), that the approximately steady-state wa- tion, however, are almost certainly waterworlds but may ter partitioning on Earth over geological time suggests have a bulk density indistinguishable from truly Earth- a seaﬂoor pressure feedback that regulates the degassing like planets. Given that simulations of water delivery at mid-ocean ridges and/or the serpentinization and sub- to habitable zone terrestrial planets predict water mass sequent subduction of oceanic crust. Although ocean vol- − − fractions of 10 5–10 2 (Raymond et al. 2004), we con- ume may change throughout a planet’s history because clude that most tectonically active planets —regardless of secular cooling, we have tackled the zeroth-order prob- of mass— will have both oceans and exposed continents, lem of steady-state solutions. enabling a silicate weathering thermostat. Notably, seaﬂoor pressure is proportional to a planet’s NBC acknowledges many insightful discussions with surface gravity. The enhanced gravity of super-Earths J.P. Townsend, S.D. Jacobsen, and C.R. Bina. The produces shallower ocean basins, but also leads to shal- authors also had useful conversations with C. Andron- lower oceans. The solid black line in Figure 2 shows the icos, H. Gilbert, R. Jeanloz, M. Manga, D. McKen- waterworld boundary if one accounts for the pressure- zie, V.S. Meadows, D.B. Rowley, J. Rudge, S. Stein, dependence of the deep water cycle. D.J. Stevenson, and R. Wordsworth. E.S. Kite provided The eﬀects of isostacy, erosion, and deposition, com- critical feedback on an early version of the manuscript. bined with a pressure-dependent deep water cycle, make × The authors thank N.H. Sleep for sharing an unpublished super-Earths 80 less susceptible to inundation than manuscript. DSA was supported by an Alfred P. Sloan research fellowship. REFERENCES Abbot, D. S., Cowan, N. B., & Ciesla, F. J. 2012, Astrophysical Ford, E. B., Seager, S., & Turner, E. L. 2001, Nature, 412, 885 Journal, 756, 178 Fujii, Y., & Kawahara, H. 2012, Astrophysical Journal, 755, 101 Abbot, D. S., & Switzer, E. R. 2011, Astrophysical Journal Fujii, Y., Kawahara, H., Suto, Y., Fukuda, S., Nakajima, T., Letters, 735, L27+ Livengood, T. A., & Turner, E. L. 2011, Astrophysical Journal, Abe, Y., Abe-Ouchi, A., Sleep, N. H., & Zahnle, K. J. 2011, 738, 184 Astrobiology, 11, 443 Fyfe, W. S. 1994, Geological Society of London Special Abe, Y., Ohtani, E., Okuchi, T., Righter, K., & Drake, M. 2000, Publications, 78, 1 Origin of the Earth and Moon, 1, 413 Gonnermann, H. M., & Mukhopadhyay, S. 2009, Nature, 459, 560 Arndt, N. T. 2013, Geochemical Perspectives, 2, 405 Harrison, T. M. 2009, Annual Review of Earth and Planetary Bounama, C., Franck, S., & von Bloh, W. 2001, Hydrology and Sciences, 37, 479 Earth System Sciences, 5, 569 Hasinoﬀ, S. W., Levin, A., Goode, P. R., & Freeman, W. T. 2011, Cowan, N. B., & Strait, T. E. 2013, Astrophysical Journal in Computer Vision (ICCV), 2011 IEEE International Letters, 765, L17 Conference on, IEEE, 185–192 Cowan, N. B., et al. 2009, Astrophysical Journal, 700, 915 Hauri, E. H., Gaetani, G. A., & Green, T. H. 2006, Earth and —. 2011, Astrophysical Journal, 731, 76 Planetary Science Letters, 248, 715 Dai, L., & Karato, S.-i. 2009, Earth and Planetary Science Hirschmann, M., & Kohlstedt, D. 2012, Physics Today, 65, 030000 Letters, 287, 277 Hirschmann, M. M. 2006, Annu. Rev. Earth Planet. Sci., 34, 629 England, P., & McKenzie, D. 1982, Geophysical Journal of the Holm, N. G. 1996, Deep Sea Research Part II: Topical Studies in Royal Astronomical Society, 70, 295 Oceanography, 43, 47 Eriksson, P. 1999, Precambrian Research, 97, 143 Huang, X., Xu, Y., & Karato, S.-i. 2005, Nature, 434, 746 Inoue, T., Wada, T., Sasaki, R., & Yurimoto, H. 2010, Physics of the Earth and Planetary Interiors, 183, 245 1 Center for Interdisciplinary Exploration and Research in As- Jarrard, R. D. 2003, Geochemistry, Geophysics, Geosystems, 4 trophysics (CIERA), Department of Earth & Planetary Sciences, Kasting, J. F., & Holm, N. G. 1992, Earth and Planetary Science Department of Physics & Astronomy, Northwestern University, Letters, 109, 507 2145 Sheridan Road, Evanston, IL, 60208, USA Kasting, J. F., Whitmire, D. P., & Reynolds, R. T. 1993, Icarus, n-cowan@northwestern.edu 101, 108 2 Department of the Geophysical Sciences, University of Kawahara, H., & Fujii, Y. 2010, Astrophysical Journal, 720, 1333 Chicago, 5734 South Ellis Avenue, Chicago, IL, 60637, USA

text: Super-Earths

label: paragraph

text: Need

text: Not

text: be

text: Waterworlds

text: WATER

text: CYCLING

text: BETWEEN

text: OCEAN

text: AND

text: MANTLE:

text: SUPER-EARTHS

text: NEED

text: NOT

text: BE

text: WATERWORLDS

text: Nicolas

label: author

text: B.

text: Dorian

text: S.

text: Subject

text: headings:

text: planets

label: abstract

text: and

text: satellites:

text: composition

text: planets

text: and

text: satellites:

text: interiors

text: planets

text: and

text: satellites:

text: oceans

text: planets

text: and

text: satellites:

text: physical

text: evolution

text: planets

text: and

text: satellites:

text: surfaces

text: planets

text: and

text: satellites:

text: tectonics

text: CONCLUSIONS

text: they

text: otherwise

text: would

text: be.

text: Our

text: model

text: predicts

text: that

text: tectonically

text: active

text: planets

text: can

text: maintain

text: large

text: It

text: has

text: long

text: been

text: suggested

text: that

text: super-Earths

text: ought

text: to

text: exposed

text: continents

text: for

text: water

text: mass

text: fractions

text: less

text: than

text: be

text: waterworlds

text: (Stapledon

text: Kite

text: et

text: al.

text: Ab-

text: bot

text: Switzer

text: If

text: one

text: accounts

text: for

text: the

text: ﬁrst-order

text: Exoplanets

text: with

text: suﬃciently

text: high

text: water

text: content

text: will

text: eﬀects

text: of

text: gravity

text: on

text: ocean

text: basin

text: depth

text: and

text: water

text: inven-

text: tory,

text: then

text: a

text: planet

text: is

text: not

text: expected

text: to

text: have

text: exposed

text: be

text: water-covered

text: regardless

text: of

text: the

text: mechanism

text: discussed

text: here,

text: but

text: such

text: “ocean

text: planets”

text: may

text: betray

text: themselves

text: continents

text: unless

text: it

text: has

text: a

text: water

text: mass

text: fraction

text: less

text: than

text: roughly

text: ten

text: times

text: drier

text: than

text: Earth

text: (solid

text: red

text: by

text: their

text: lower

text: density:

text: a

text: planet

text: with

text: water

text: mass

text: fraction

text: will

text: exhibit

text: a

text: transit

text: depth

text: greater

text: than

text: line

text: in

text: Figure

text: an

text: equally-massive

text: planet

text: with

text: Earth-like

text: composition

text: We

text: have

text: argued,

text: as

text: have

text: others

text: (Kasting

text: Holm

text: (Sotin

text: et

text: al.

text: Planets

text: with

text: water

text: mass

text: frac-

text: Holm

text: that

text: the

text: approximately

text: steady-state

text: wa-

text: tion,

text: however,

text: are

text: almost

text: certainly

text: waterworlds

text: but

text: may

text: ter

text: partitioning

text: on

text: Earth

text: over

text: geological

text: time

text: suggests

text: have

text: a

text: bulk

text: density

text: indistinguishable

text: from

text: truly

text: Earth-

text: a

text: seaﬂoor

text: pressure

text: feedback

text: that

text: regulates

text: the

text: degassing

text: like

text: planets.

text: Given

text: that

text: simulations

text: of

text: water

text: delivery

text: at

text: mid-ocean

text: ridges

text: and/or

text: the

text: serpentinization

text: and

text: sub-

text: to

text: habitable

text: zone

text: terrestrial

text: planets

text: predict

text: water

text: mass

text: sequent

text: subduction

text: of

text: oceanic

text: crust.

text: Although

text: ocean

text: vol-

text: fractions

text: of

text: (Raymond

text: et

text: al.

text: we

text: con-

text: ume

text: may

text: change

text: throughout

text: a

text: planet’s

text: history

text: because

text: clude

text: that

text: most

text: tectonically

text: active

text: planets

text: —regardless

text: of

text: secular

text: cooling,

text: we

text: have

text: tackled

text: the

text: zeroth-order

text: prob-

text: of

text: mass—

text: will

text: have

text: both

text: oceans

text: and

text: exposed

text: continents,

text: lem

text: of

text: steady-state

text: solutions.

text: enabling

text: a

text: silicate

text: weathering

text: thermostat.

text: Notably,

text: seaﬂoor

text: pressure

text: is

text: proportional

text: to

text: a

text: planet’s

text: NBC

text: acknowledges

text: many

text: insightful

text: discussions

text: with

text: surface

text: gravity.

text: The

text: enhanced

text: gravity

text: of

text: super-Earths

text: J.P.

text: Townsend,

text: S.D.

text: Jacobsen,

text: and

text: C.R.

text: Bina.

text: The

text: produces

text: shallower

text: ocean

text: basins,

text: but

text: also

text: leads

text: to

text: shal-

text: authors

text: also

text: had

text: useful

text: conversations

text: with

text: C.

text: Andron-

text: lower

text: oceans.

text: The

text: solid

text: black

text: line

text: in

text: Figure

text: shows

text: the

text: icos,

text: H.

text: Gilbert,

text: R.

text: Jeanloz,

text: M.

text: Manga,

text: D.

text: McKen-

text: waterworld

text: boundary

text: if

text: one

text: accounts

text: for

text: the

text: pressure-

text: zie,

text: V.S.

text: Meadows,

text: D.B.

text: Rowley,

text: J.

text: Rudge,

text: S.

text: Stein,

text: dependence

text: of

text: the

text: deep

text: water

text: cycle.

text: D.J.

text: Stevenson,

text: and

text: R.

text: Wordsworth.

text: E.S.

text: Kite

text: provided

text: The

text: eﬀects

text: of

text: isostacy,

text: erosion,

text: and

text: deposition,

text: com-

text: critical

text: feedback

text: on

text: an

text: early

text: version

text: of

text: the

text: manuscript.

text: bined

text: with

text: a

text: pressure-dependent

text: deep

text: water

text: cycle,

text: make

text: The

text: authors

text: thank

text: N.H.

text: Sleep

text: for

text: sharing

text: an

text: unpublished

text: super-Earths

text: less

text: susceptible

text: to

text: inundation

text: than

text: manuscript.

text: DSA

text: was

text: supported

text: by

text: an

text: Alfred

text: P.

text: Sloan

text: research

text: fellowship.

text: REFERENCES

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text: Evanston,

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text: Y.

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text: 14 C. Ku¨hn, B. A. Surya, and B. Ulbricht (cid:18)(cid:26) (cid:27) (cid:19) P { − } ∩ { ≤ − − } ( τ τ > ε τ T t ε ) t1,x t2,x t2,x 1 (cid:32)(cid:40) (cid:41) (cid:33) ≤P ∩ { ≤ − − } min Xt1,x > b(t + τ ) τ T t ε u∈[0,ε−(t2−t1)] t2+τt2,x+u 2 t2,x t2,x 1 ≤P (cid:32)(cid:40) min Xtt21+,xτt2,x+u > (cid:41)x ∩ {τ ≤ T − t (cid:33)− ε} u∈[0,ε−(t2−t1)] Xtt21+,xτt2,x Xtt21,x t2,x 1 ≤P (cid:0)(cid:8)min Xtt21+,xτt2,x(cid:9)+(cid:1)u > x ∩ {τ ≤ T − t − ε} (cid:18)(cid:26)u∈[0,ε/2](cid:26)Xt1,x(cid:18)(cid:18) x (cid:19)+ δ (cid:19)t2,(cid:27)x 1(cid:27)(cid:19) t2+τt2,x P + Xt1,x > x + δ t2 ≤P (cid:18)(cid:26) (cid:18)(cid:18) (cid:19)− σ2 (cid:19) a (cid:27)(cid:19) min exp µ u + σB > u∈[0,ε/2] 2 u a + δ P − σ2 − b + δ ∀ ∈ + exp µ (t t ) + σB − > x [a, b], (4.8) 2 2 1 t2 t1 b (cid:55)→ where the ﬁrst inequality holds because t b(t) is increasing, the second one ≤ holds by (4.7) and Xt2,x b(t + τ ), and for the fourth one, we use the t2+τt2,x 2 t2,x strong Markov property of Xt1,x. As in Lem(cid:18)ma 4.1, o(cid:26)ne ca(cid:18)n(cid:18)choose δ(cid:19)> 0 small(cid:19)e(cid:27)nough such(cid:19)that P − σ2 a ε min exp µ u + σB > < . (4.9) u∈[0,ε/2] 2 u a + δ 2 − ↓ Because B − converges stochastically to 0 for t t 0, there exists δ > 0 t2 t1 2 1 such that(cid:18)(cid:26) (cid:18)(cid:18) (cid:19) (cid:19) (cid:27)(cid:19) P − σ2 − b + δ ε exp µ (t t ) + σB − > < (4.10) 2 2 1 t2 t1 b 2 − for all t , t with t t < δ. So, it remains to show 1 2 2 1 P − ≥ − (τ τ > ε) < ε, for t t and t t small enough. (4.11) t2,x t1,x 2 1 2 1 To estimate this probability, we renew X at t + τ , where it is suﬃcient { ≥ } 1 t1,x to consider the set t + τ t . Then, we estimate Xt1,x from below and 1 t1,x 2 t2 not from above as in (4.8). But, the calculations are completely analog to the P − (cid:3) estimation of (τ τ > ε), and we are done. t1,x t2,x ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: C.

text: Ku¨hn,

text: B.

text: A.

text: Surya,

text: and

text: B.

text: Ulbricht

label: equation

text: P

text: τ

text: τ

text: ε

text: τ

text: T

text: t

text: ε

text: ≤P

text: min

text: b(t

text: τ

text: τ

text: T

text: t

text: ε

text: ≤P

text: min

text: {τ

text: T

text: t

text: ε}

text: ≤P

text: x

text: {τ

text: T

text: t

text: ε}

text: x

text: δ

text: P

text: x

text: δ

text: ≤P

text: σ2

text: a

text: min

text: exp

text: µ

text: u

text: σB

text: u

text: a

text: δ

text: P

text: σ2

text: b

text: δ

text: exp

text: µ

text: (t

text: t

text: σB

text: x

text: [a,

text: b],

text: b

text: where

text: the

text: ﬁrst

text: inequality

text: holds

text: because

text: t

text: b(t)

text: is

text: increasing,

text: the

text: second

text: one

text: holds

text: by

text: and

text: b(t

text: τ

text: and

text: for

text: the

text: fourth

text: one,

text: we

text: use

text: the

text: strong

text: Markov

text: property

text: of

text: As

text: in

text: P

text: σ2

text: a

text: ε

text: min

text: exp

text: µ

text: u

text: σB

text: u

text: a

text: δ

text: Because

text: B

text: converges

text: stochastically

text: to

text: for

text: t

text: t

text: there

text: exists

text: δ

text: such

text: P

text: σ2

text: b

text: δ

text: ε

text: exp

text: µ

text: (t

text: t

text: σB

text: b

text: for

text: all

text: t

text: t

text: with

text: t

text: t

text: δ.

text: So,

text: it

text: remains

text: to

text: show

text: P

text: (τ

text: τ

text: ε)

text: ε,

text: for

text: t

text: t

text: and

text: t

text: t

text: small

text: enough.

text: To

text: estimate

text: this

text: probability,

text: we

text: renew

text: X

text: at

text: t

text: τ

text: where

text: it

text: is

text: suﬃcient

text: to

text: consider

text: the

text: set

text: t

text: τ

text: t

text: Then,

text: we

text: estimate

text: from

text: below

text: and

text: not

text: from

text: above

text: as

text: in

text: But,

text: the

text: calculations

text: are

text: completely

text: analog

text: to

text: the

text: P

text: estimation

text: of

text: (τ

text: τ

text: ε),

text: and

text: we

text: are

text: done.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 20 C. Ku¨hn, B. A. Surya, and B. Ulbricht [1] Avellaneda, M. and Lipkin, M. D. (2003) A market-induced mechanism for stock pinning. Quantitative Finance, 3(6), 417–425. [2] Baurdoux, E. J., Chen, N, Surya, B. A. and Yamazaki, K. (2014). Dou- ble optimal stopping of a Brownian bridge. Submitted for publication, http://arxiv.org/abs/1409.2226. [3] Ben Tahar, I., Soner, M., and Touzi, N. (2007) The dynamic programming equation for the problem of optimal investment under capital gains taxes, SIAM Journal on Control and Optimization, 46(5), 1779-1801. [4] Ben Tahar, I. B., Soner, H. M., and Touzi, N. (2010) Merton Problem with taxes: characterisation, computation and approximation, SIAM Journal on Financial Mathematics, 1(1), 366-395. [5] Buescu, C., Cadenillas, A., and Pliska, S. R. (2007). A note on the eﬀects of taxes on optimal investment, Mathematical Finance, 17(4), 477-485. [6] Cadenillas, A. and Pliska, S. R. (1999). Optimal trading of a security when there are taxes and transaction costs, Finance and Stochastics, 3(2), 137– 165. [7] Constantinides, G. M. (1983). Capital market equilibrium with personal taxes, Econometrica, 51(3), 611-636. [8] Dai, M. and Zhong, Y.F. (2012). Optimal stock selling/buying strategy with reference to the ultimate average, Mathematical Finance, 22(1), 165-184. [9] DeMiguel, V. and Uppal, R. (2005). Portfolio investment with the exact tax basis via nonlinear programming, Management Science, 51(2), 277-290. [10] Du Toit, J. and Peskir, G. (2009). Selling a stock at the ultimate maximum. Annals of Applied Probability, 19(3), 983–1014. [11] Dybvig, P. and Koo, H. (1996) Investment with Taxes. Washington Univer- sity, St. Louis, MO, Working Paper. [12] Friedman, A. (1964). Partial Diﬀerential Equations of Parabolic Type, Prentice-Hall, Florida. [13] Jouini, E., Koehl, P.-F., and Touzi, N. (2000). Optimal investment with taxes: an existence result. Journal of Mathematical Economics, 33(4), 373- 388. [14] Jouini, E., Koehl, P.-F., and Touzi, N. (1999). Optimal investment with taxes: an optimal control problem with endogenous delay. Nonlinear Anal- ysis, 37, 31–56.

text: C.

text: Ku¨hn,

text: B.

text: A.

text: Surya,

text: and

text: B.

text: Ulbricht

label: reference

text: Avellaneda,

text: M.

text: and

text: Lipkin,

text: M.

text: D.

text: A

text: market-induced

text: mechanism

text: for

text: stock

text: pinning.

text: Quantitative

text: Finance,

text: Baurdoux,

text: E.

text: J.,

text: Chen,

text: N,

text: Surya,

text: B.

text: A.

text: and

text: Yamazaki,

text: K.

text: Dou-

text: ble

text: optimal

text: stopping

text: of

text: a

text: Brownian

text: bridge.

text: Submitted

text: for

text: publication,

text: Ben

text: Tahar,

text: I.,

text: Soner,

text: M.,

text: and

text: Touzi,

text: N.

text: The

text: dynamic

text: programming

text: equation

text: for

text: the

text: problem

text: of

text: optimal

text: investment

text: under

text: capital

text: gains

text: taxes,

text: SIAM

text: Journal

text: on

text: Control

text: and

text: Optimization,

text: Ben

text: Tahar,

text: I.

text: B.,

text: Soner,

text: H.

text: M.,

text: and

text: Touzi,

text: N.

text: Merton

text: Problem

text: with

text: taxes:

text: characterisation,

text: computation

text: and

text: approximation,

text: SIAM

text: Journal

text: on

text: Financial

text: Mathematics,

text: Buescu,

text: C.,

text: Cadenillas,

text: A.,

text: and

text: Pliska,

text: S.

text: R.

text: A

text: note

text: on

text: the

text: eﬀects

text: of

text: taxes

text: on

text: optimal

text: investment,

text: Mathematical

text: Finance,

text: Cadenillas,

text: A.

text: and

text: Pliska,

text: S.

text: R.

text: Optimal

text: trading

text: of

text: a

text: security

text: when

text: there

text: are

text: taxes

text: and

text: transaction

text: costs,

text: Finance

text: and

text: Stochastics,

text: Constantinides,

text: G.

text: M.

text: Capital

text: market

text: equilibrium

text: with

text: personal

text: taxes,

text: Econometrica,

text: Dai,

text: M.

text: and

text: Zhong,

text: Y.F.

text: Optimal

text: stock

text: selling/buying

text: strategy

text: with

text: reference

text: to

text: the

text: ultimate

text: average,

text: Mathematical

text: Finance,

text: DeMiguel,

text: V.

text: and

text: Uppal,

text: R.

text: Portfolio

text: investment

text: with

text: the

text: exact

text: tax

text: basis

text: via

text: nonlinear

text: programming,

text: Management

text: Science,

text: Du

text: Toit,

text: J.

text: and

text: Peskir,

text: G.

text: Selling

text: a

text: stock

text: at

text: the

text: ultimate

text: maximum.

text: Annals

text: of

text: Applied

text: Probability,

text: Dybvig,

text: P.

text: and

text: Koo,

text: H.

text: Investment

text: with

text: Taxes.

text: Washington

text: Univer-

text: sity,

text: St.

text: Louis,

text: MO,

text: Working

text: Paper.

text: Friedman,

text: A.

text: Partial

text: Diﬀerential

text: Equations

text: of

text: Parabolic

text: Type,

text: Prentice-Hall,

text: Florida.

text: Jouini,

text: E.,

text: Koehl,

text: P.-F.,

text: and

text: Touzi,

text: N.

text: Optimal

text: investment

text: with

text: taxes:

text: an

text: existence

text: result.

text: Journal

text: of

text: Mathematical

text: Economics,

text: Jouini,

text: E.,

text: Koehl,

text: P.-F.,

text: and

text: Touzi,

text: N.

text: Optimal

text: investment

text: with

text: taxes:

text: an

text: optimal

text: control

text: problem

text: with

text: endogenous

text: delay.

text: Nonlinear

text: Anal-

text: ysis,


text: 6 C. Ku¨hn, B. A. Surya, and B. Ulbricht − (b) if α > 0 and µ > (1 α)r, the value function satisﬁes the smooth ﬁt condition at the boundary, i.e., − − − ∂ V (t, x) = ∂ G(t, x) = (1 α)er(1 α)(T t) at x = b(t). (2.9) x x Figure 1: Plot of the stopping boundary in the optimization problem (2.3) with time horizon T = 3 years. The other parameters are α = 0.3, µ = 0.026, r = 0.03, σ = 0.25, P = 100, and x = 180. The boundary lies far above the purchasing 0 price P , i.e., the stock is sold with positive book proﬁts. 0 Remark 2.3 We are primarily interested in the case that µ < r, i.e., an investor who is not subject to taxation would immediatelly sell the stock. Then, Theo- ∈ − rem 2.2 tells us that the solution is nontrivial if µ ((1 α)r, r). Choosing the midpoint of the interval for µ and reasonable values for σ, T , and α, numerical calculations show that the boundary is far above the purchasing price P , see 0 Figure 1. This means that the investor always sells the stock with positive book proﬁts. Then, it is evident that there is no incentive to buy the stock back at any later time and to repeat the game (namely, a renewed investment starts with zero book proﬁt). This justiﬁes the modeling of the decision problem as a simple optimal stopping problem. On the other hand, if the stock is sold with negative book proﬁts, the model- ing is justiﬁed when wash-sales are forbidden, as, e.g., to some extent in the U.S. ##LTFigure##

text: C.

text: Ku¨hn,

text: B.

text: A.

text: Surya,

text: and

text: B.

text: Ulbricht

label: list

text: (b)

text: if

text: α

text: and

text: µ

text: α)r,

text: the

text: value

text: function

text: satisﬁes

text: the

text: smooth

text: ﬁt

text: condition

text: at

text: the

text: boundary,

text: i.e.,

text: V

text: (t,

text: x)

text: G(t,

text: x)

text: α)(T

text: t)

text: at

text: x

text: b(t).

text: x

text: x

text: Figure

text: Plot

label: caption

text: of

text: the

text: stopping

text: boundary

text: in

text: the

text: optimization

text: problem

text: with

text: time

text: horizon

text: T

text: years.

text: The

text: other

text: parameters

text: are

text: α

text: µ

text: r

text: σ

text: P

text: and

text: x

text: The

text: boundary

text: lies

text: far

text: above

text: the

text: purchasing

text: price

text: P

text: i.e.,

text: the

text: stock

text: is

text: sold

text: with

text: positive

text: book

text: proﬁts.

text: Remark

text: We

text: are

text: primarily

text: interested

text: in

text: the

text: case

text: that

text: µ

text: r,

text: i.e.,

text: an

text: investor

text: who

text: is

text: not

text: subject

text: to

text: taxation

text: would

text: immediatelly

text: sell

text: the

text: stock.

text: Then,

text: Theo-

text: rem

text: tells

text: us

text: that

text: the

text: solution

text: is

text: nontrivial

text: if

text: µ

text: α)r,

text: r).

text: Choosing

text: the

text: midpoint

text: of

text: the

text: interval

text: for

text: µ

text: and

text: reasonable

text: values

text: for

text: σ,

text: T

text: and

text: α,

text: numerical

text: calculations

text: show

text: that

text: the

text: boundary

text: is

text: far

text: above

text: the

text: purchasing

text: price

text: P

text: see

text: Figure

text: This

text: means

text: that

text: the

text: investor

text: always

text: sells

text: the

text: stock

text: with

text: positive

text: book

text: proﬁts.

text: Then,

text: it

text: is

text: evident

text: that

text: there

text: is

text: no

text: incentive

text: to

text: buy

text: the

text: stock

text: back

text: at

text: any

text: later

text: time

text: and

text: to

text: repeat

text: the

text: game

text: (namely,

text: a

text: renewed

text: investment

text: starts

text: with

text: zero

text: book

text: proﬁt).

text: This

text: justiﬁes

text: the

text: modeling

text: of

text: the

text: decision

text: problem

text: as

text: a

text: simple

text: optimal

text: stopping

text: problem.

text: On

text: the

text: other

text: hand,

text: if

text: the

text: stock

text: is

text: sold

text: with

text: negative

text: book

text: proﬁts,

text: the

text: model-

text: ing

text: is

text: justiﬁed

text: when

text: wash-sales

text: are

text: forbidden,

text: as,

text: e.g.,

text: to

text: some

text: extent

text: in

text: the

text: U.S.

text: ##LTFigure##


text: 2 edge weight and degree distribution) [49,55]. One thereby studies “mesoscale” features such as core-periphery structure and (especially) community structure. Loosely speak- ing, a community (or cluster) in a network is a set of nodes that are “more densely” connected to each other than they are to nodes in the rest of the network [23, 55]. Giving a precise deﬁnition of “densely connected” is, of course, necessary to have a method for community detection. It is important to recognize at the outset that this deﬁnition is subjective and in particular, may depend on the application in question. Correspondingly, community detection methods may need to be tailored. We restrict ourselves to hard partitions, in which each node is assigned to exactly one community, and we use the term “partition” to mean “hard partition”. It is also important, but beyond the scope of this paper, to consider “soft partitions”, in which communities can overlap [23,33,52,55]. Analysis of community structure has been very useful in a wide range of applica- tions; many of which are described in [23,25,49,55]. In social networks, communities can reveal groups of people with common interests, places of residence, or other sim- ilarities [50,66]. In biological systems, communities can reveal functional groups that are responsible for synthesizing or regulating an important chemical product [29,40]. In the present paper, we use ﬁnancial-asset correlation networks as examples [8,15]. Despite the diversity of markets, ﬁnancial products, and geographical locations, ﬁnan- cial assets can exhibit strong time-dependent correlations, both within and between asset classes. It is a primary concern for market practitioners (e.g., for portfolio di- versiﬁcation) to estimate the strengths of these correlations and to identify sets of assets that are highly correlated [43,67]. Most methods for detecting communities are designed for static networks. How- ever, in many applications, entities and/or interactions between entities evolve in time. In such applications, one can use the formalism of temporal networks, where nodes and/or their edges weights vary in time [31]. This is important for numerous applica- tions, including person-to-person communication [68], one-to-many information dis- semination (e.g., Twitter networks [27] and Facebook networks [70]), cell biology [31], neuroscience [7], ecology [31], ﬁnance [20–22,51], and more. Two main approaches have been adopted to detect communities in time-dependent networks. The ﬁrst entails constructing a static network by aggregating snapshots of the evolving network at diﬀerent points in time into a single network (e.g., by taking the mean or total edge weight for each edge across all time points, which can be prob- lematic if the set of nodes varies in time and which also makes restrictive assumptions on the interaction dynamics between entities [30]). One can then use standard network techniques. The second approach entails using static community-detection techniques on each element of a time-ordered sequence of networks at diﬀerent times or on each element of a time-ordered sequence of network aggregations1 (computed as above) over diﬀerent time intervals (which can be either overlapping or nonoverlapping) and then tracking the communities across the sequence [3,20,21,32,42,52]. A third approach consists of embedding a time-ordered sequence of networks in a larger network [18,44,54]. Each element of the sequence is a network layer, and nodes at diﬀerent time points are joined by inter-layer edges. This approach was introduced 1One needs to distinguish between this kind of aggregation and the averaging of a set of time series over a moving window to construct a correlation matrix, which one can then interpret as a ﬁxed-time snapshot of a time-evolving network. Although both involve averaging over a time window, the former situation entails averaging a network, and the latter situation entails averaging over a collection of time series (one for each node) with no directly observable edge weights. ##LTLine##

text: edge

text: weight

text: and

text: degree

text: distribution)

text: One

text: thereby

text: studies

text: “mesoscale”

text: features

text: such

text: as

text: core-periphery

text: structure

text: and

text: (especially)

text: community

text: structure.

text: Loosely

text: speak-

text: ing,

text: a

text: community

text: (or

text: cluster)

text: in

text: a

text: network

text: is

text: a

text: set

text: of

text: nodes

text: that

text: are

text: “more

text: densely”

text: connected

text: to

text: each

text: other

text: than

text: they

text: are

text: to

text: nodes

text: in

text: the

text: rest

text: of

text: the

text: network

text: Giving

text: a

text: precise

text: deﬁnition

text: of

text: “densely

text: connected”

text: is,

text: of

text: course,

text: necessary

text: to

text: have

text: a

text: method

text: for

text: community

text: detection.

text: It

text: is

text: important

text: to

text: recognize

text: at

text: the

text: outset

text: that

text: this

text: deﬁnition

text: is

text: subjective

text: and

text: in

text: particular,

text: may

text: depend

text: on

text: the

text: application

text: in

text: question.

text: Correspondingly,

text: community

text: detection

text: methods

text: may

text: need

text: to

text: be

text: tailored.

text: We

text: restrict

text: ourselves

text: to

text: hard

text: partitions,

text: in

text: which

text: each

text: node

text: is

text: assigned

text: to

text: exactly

text: one

text: community,

text: and

text: we

text: use

text: the

text: term

text: “partition”

text: to

text: mean

text: “hard

text: partition”.

text: It

text: is

text: also

text: important,

text: but

text: beyond

text: the

text: scope

text: of

text: this

text: paper,

text: to

text: consider

text: “soft

text: partitions”,

text: in

text: which

text: communities

text: can

text: overlap

text: Analysis

text: of

text: community

text: structure

text: has

text: been

text: very

text: useful

text: in

text: a

text: wide

text: range

text: of

text: applica-

text: tions;

text: many

text: of

text: which

text: are

text: described

text: in

text: In

text: social

text: networks,

text: communities

text: can

text: reveal

text: groups

text: of

text: people

text: with

text: common

text: interests,

text: places

text: of

text: residence,

text: or

text: other

text: sim-

text: ilarities

text: In

text: biological

text: systems,

text: communities

text: can

text: reveal

text: functional

text: groups

text: that

text: are

text: responsible

text: for

text: synthesizing

text: or

text: regulating

text: an

text: important

text: chemical

text: product

text: In

text: the

text: present

text: paper,

text: we

text: use

text: ﬁnancial-asset

text: correlation

text: networks

text: as

text: examples

text: Despite

text: the

text: diversity

text: of

text: markets,

text: ﬁnancial

text: products,

text: and

text: geographical

text: locations,

text: ﬁnan-

text: cial

text: assets

text: can

text: exhibit

text: strong

text: time-dependent

text: correlations,

text: both

text: within

text: and

text: between

text: asset

text: classes.

text: It

text: is

text: a

text: primary

text: concern

text: for

text: market

text: practitioners

text: (e.g.,

text: for

text: portfolio

text: di-

text: versiﬁcation)

text: to

text: estimate

text: the

text: strengths

text: of

text: these

text: correlations

text: and

text: to

text: identify

text: sets

text: of

text: assets

text: that

text: are

text: highly

text: correlated

text: Most

text: methods

text: for

text: detecting

text: communities

text: are

text: designed

text: for

text: static

text: networks.

text: How-

text: ever,

text: in

text: many

text: applications,

text: entities

text: and/or

text: interactions

text: between

text: entities

text: evolve

text: in

text: time.

text: In

text: such

text: applications,

text: one

text: can

text: use

text: the

text: formalism

text: of

text: temporal

text: networks,

text: where

text: nodes

text: and/or

text: their

text: edges

text: weights

text: vary

text: in

text: time

text: This

text: is

text: important

text: for

text: numerous

text: applica-

text: tions,

text: including

text: person-to-person

text: communication

text: one-to-many

text: information

text: dis-

text: semination

text: (e.g.,

text: Twitter

text: networks

text: and

text: Facebook

text: networks

text: cell

text: biology

text: neuroscience

text: ecology

text: ﬁnance

text: and

text: more.

text: Two

text: main

text: approaches

text: have

text: been

text: adopted

text: to

text: detect

text: communities

text: in

text: time-dependent

text: networks.

text: The

text: ﬁrst

text: entails

text: constructing

text: a

text: static

text: network

text: by

text: aggregating

text: snapshots

text: of

text: the

text: evolving

text: network

text: at

text: diﬀerent

text: points

text: in

text: time

text: into

text: a

text: single

text: network

text: (e.g.,

text: by

text: taking

text: the

text: mean

text: or

text: total

text: edge

text: weight

text: for

text: each

text: edge

text: across

text: all

text: time

text: points,

text: which

text: can

text: be

text: prob-

text: lematic

text: if

text: the

text: set

text: of

text: nodes

text: varies

text: in

text: time

text: and

text: which

text: also

text: makes

text: restrictive

text: assumptions

text: on

text: the

text: interaction

text: dynamics

text: between

text: entities

text: One

text: can

text: then

text: use

text: standard

text: network

text: techniques.

text: The

text: second

text: approach

text: entails

text: using

text: static

text: community-detection

text: techniques

text: on

text: each

text: element

text: of

text: a

text: time-ordered

text: sequence

text: of

text: networks

text: at

text: diﬀerent

text: times

text: or

text: on

text: each

text: element

text: of

text: a

text: time-ordered

text: sequence

text: of

text: network

text: (computed

text: as

text: above)

text: over

text: diﬀerent

text: time

text: intervals

text: (which

text: can

text: be

text: either

text: overlapping

text: or

text: nonoverlapping)

text: and

text: then

text: tracking

text: the

text: communities

text: across

text: the

text: sequence

text: A

text: third

text: approach

text: consists

text: of

text: embedding

text: a

text: time-ordered

text: sequence

text: of

text: networks

text: in

text: a

text: larger

text: network

text: Each

text: element

text: of

text: the

text: sequence

text: is

text: a

text: network

text: layer,

text: and

text: nodes

text: at

text: diﬀerent

text: time

text: points

text: are

text: joined

text: by

text: inter-layer

text: edges.

text: This

text: approach

text: was

text: introduced

text: needs

text: to

text: distinguish

text: between

text: this

text: kind

text: of

text: aggregation

text: and

text: the

text: averaging

text: of

text: a

text: set

text: of

text: time

text: series

text: over

text: a

text: moving

text: window

text: to

text: construct

text: a

text: correlation

text: matrix,

text: which

text: one

text: can

text: then

text: interpret

text: as

text: a

text: ﬁxed-time

text: snapshot

text: of

text: a

text: time-evolving

text: network.

text: Although

text: both

text: involve

text: averaging

text: over

text: a

text: time

text: window,

text: the

text: former

text: situation

text: entails

text: averaging

text: a

text: network,

text: and

text: the

text: latter

text: situation

text: entails

text: averaging

text: over

text: a

text: collection

text: of

text: time

text: series

text: (one

text: for

text: each

text: node)

text: with

text: no

text: directly

text: observable

text: edge

text: weights.

text: ##LTLine##


text: 11 (cid:18) (cid:19) (cid:88) (cid:2) (cid:3) N − − rˆ(C, t) = π δ + tΛ (M δ ) π ρ | δ(c , c ) , (2.14) i ij ii ij ij i i j i j i,j=1 where the term in brackets on the left-hand side is a linearization of the exponen- tial term in (2.12), M and π are as deﬁned in (2.12) on a network with adjacency | | − i matrix A := A+ + A , and ρ | is the probability of jumping from node i to node j i j at stationarity conditional on the network structure [44]. If the network is unipartite, unsigned, and undirected, then ρ | reduces to the stationary probability π . i j j (cid:0)(cid:80) (cid:1) A third null network that we consider is a uniform (U) null network, with adjacency- (cid:104) (cid:105) (cid:104) (cid:105) matrix entries P = k 2/(2m), where k := N k /N denotes the mean strength ij i=1 i in a network. W(cid:32)e thereby ob(cid:33)tain the equivalent m(cid:20) axi(cid:18)mization prob(cid:19)lem(cid:21)s (cid:88) (cid:104) (cid:105) (cid:104) (cid:105) N − k 2 ⇔ − k 2 max A δ(c , c ) max Tr ST A 1 S , (2.15) C∈C ij 2m i j S∈S 2m N i,j=1 × where A is an unsigned adjacency matrix and 1 is an N N matrix in which every entry is 1.7 The expected(cid:16)e(cid:80)dge wei(cid:14)ght(cid:17)in (2.15)Nis constant and satisﬁes (cid:80) (cid:104)k(cid:105)2 Ni=1 ki N 2 2m (cid:104) (cid:105) = = = A , 2m N k N2 i=1 i (cid:104) (cid:105) where A denotes the mean value of the adjacency matrix.8 One way to generate (cid:104) (cid:105) an unweighted network with adjacency matrix A 1 is to generate each edge with (cid:104) (cid:105) (cid:104) (cid:105) ≤ N probability A (provided A 1). That is, the presence and absence of an edge (cid:0) (cid:80) (cid:1) (including self-edges) are independent and identically distributed (i.i.d.) Bernoulli (cid:104) (cid:105) random variables with probability A . More generally, any probability distribution on the set of adjacency matrices that satisﬁes E N W = 2m and E(W ) = E (cid:48) (cid:48) i,j=1 ijE (cid:104) (cid:105) ij (W (cid:48) (cid:48)) for all i, j, i , j has an expected adjacency matrix (W ) = A 1 . The i j N adjacency matrix of the U null network is symmetric and positive semideﬁnite. One can derive the multiscale formulation in (2.6) for the U null network from the stability quality function in precisely the same way as it is derived for the NG null network, except that one needs to consider exponentially distributed waiting times at each node (cid:104) (cid:105) with rates proportional to node strength (i.e., Λ = δ(i, j)k / k ) [37]. ij i model approach in [64]. This multiscale formulation results in one resolution− p−aramete−r γ1 for the term (ki+kj+)/(2m+) and a second resolution parameter γ2 for the term (ki kj )/(2m ) in (2.13) (see [42] for an application of this multiscale formulation to the United Nations General Assembly resolution networks). Without an application-driven justiﬁcation for how to choose these parameters, this increases the parameter space substantially, so we only consider the case γ1 = γ2 in this paper. 7For a network in which all nodes have the same strength, the uniform and Newman-Girvan null ⇔ (cid:104) (cid:105) networks are equivalent because ki = kj for all i,j ki = 2m/N = k for all i. This was pointed out for an application to foreign exchange markets in [20,21]. 8Although we use the uniform null network on unsigned adjacency matrices in this paper, the expected edge weight in the uniform null network is always nonnegative for correlation matrices, as (cid:104) (cid:105) ≥ positive semideﬁniteness guarantees that A = 1TA1/(N2) 0. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: N

text: rˆ(C,

text: t)

text: π

text: δ

text: tΛ

text: (M

text: δ

text: π

text: ρ

text: δ(c

text: c

text: i

text: ij

text: ii

text: ij

text: ij

text: i

text: i

text: j

text: i

text: j

text: where

text: the

text: term

text: in

text: brackets

text: on

text: the

text: left-hand

text: side

text: is

text: a

text: linearization

text: of

text: the

text: exponen-

text: tial

text: term

text: in

text: M

text: and

text: π

text: are

text: as

text: deﬁned

text: in

text: on

text: a

text: network

text: with

text: adjacency

text: i

text: matrix

text: A

text: A+

text: A

text: and

text: ρ

text: is

text: the

text: probability

text: of

text: jumping

text: from

text: node

text: i

text: to

text: node

text: j

text: i

text: j

text: at

text: stationarity

text: conditional

text: on

text: the

text: network

text: structure

text: If

text: the

text: network

text: is

text: unipartite,

text: unsigned,

text: and

text: undirected,

text: then

text: ρ

text: reduces

text: to

text: the

text: stationary

text: probability

text: π

text: i

text: j

text: j

text: A

text: third

text: network

text: that

text: we

text: consider

text: is

text: a

text: uniform

text: (U)

text: network,

text: with

text: adjacency-

text: matrix

text: entries

text: P

text: k

text: where

text: k

text: N

text: k

text: /N

text: denotes

text: the

text: mean

text: strength

text: ij

text: i

text: in

text: a

text: network.

text: thereby

text: the

text: equivalent

text: N

text: k

text: k

text: max

text: A

text: δ(c

text: c

text: max

text: Tr

text: ST

text: A

text: S

text: C∈C

text: ij

text: i

text: j

text: S∈S

text: N

text: where

text: A

text: is

text: an

text: unsigned

text: adjacency

text: matrix

text: and

text: is

text: an

text: N

text: N

text: matrix

text: in

text: which

text: every

text: entry

text: is

text: The

text: constant

text: and

text: satisﬁes

text: ki

text: N

text: A

text: N

text: k

text: i

text: where

text: A

text: denotes

text: the

text: mean

text: value

text: of

text: the

text: adjacency

text: One

text: way

text: to

text: generate

text: an

text: unweighted

text: network

text: with

text: adjacency

text: matrix

text: A

text: is

text: to

text: generate

text: each

text: edge

text: with

text: N

text: probability

text: A

text: (provided

text: A

text: That

text: is,

text: the

text: presence

text: and

text: absence

text: of

text: an

text: edge

text: (including

text: self-edges)

text: are

text: independent

text: and

text: identically

text: distributed

text: (i.i.d.)

text: Bernoulli

text: random

text: variables

text: with

text: probability

text: A

text: More

text: generally,

text: any

text: probability

text: distribution

text: on

text: the

text: set

text: of

text: adjacency

text: matrices

text: that

text: satisﬁes

text: E

text: N

text: W

text: and

text: E(W

text: E

text: ijE

text: ij

text: (W

text: for

text: all

text: i,

text: j,

text: i

text: j

text: has

text: an

text: expected

text: adjacency

text: matrix

text: (W

text: A

text: The

text: i

text: j

text: N

text: adjacency

text: matrix

text: of

text: the

text: U

text: network

text: is

text: symmetric

text: and

text: positive

text: semideﬁnite.

text: One

text: can

text: derive

text: the

text: multiscale

text: formulation

text: in

text: for

text: the

text: U

text: network

text: from

text: the

text: stability

text: quality

text: function

text: in

text: precisely

text: the

text: same

text: way

text: as

text: it

text: is

text: derived

text: for

text: the

text: NG

text: network,

text: except

text: that

text: one

text: needs

text: to

text: consider

text: exponentially

text: distributed

text: waiting

text: times

text: at

text: each

text: node

text: with

text: rates

text: proportional

text: to

text: node

text: strength

text: (i.e.,

text: Λ

text: δ(i,

text: j)k

text: k

text: ij

text: i

text: model

text: approach

text: in

text: This

text: multiscale

text: formulation

text: results

text: in

text: one

text: resolution−

text: p−aramete−r

text: γ1

text: for

text: the

text: term

text: and

text: a

text: second

text: resolution

text: parameter

text: γ2

text: for

text: the

text: term

text: (ki

text: kj

text: in

text: (see

text: for

text: an

text: application

text: of

text: this

text: multiscale

text: formulation

text: to

text: the

text: United

text: Nations

text: General

text: Assembly

text: resolution

text: networks).

text: Without

text: an

text: application-driven

text: justiﬁcation

text: for

text: how

text: to

text: choose

text: these

text: parameters,

text: this

text: increases

text: the

text: parameter

text: space

text: substantially,

text: so

text: we

text: only

text: consider

text: the

text: case

text: γ1

text: γ2

text: in

text: this

text: paper.

text: a

text: network

text: in

text: which

text: all

text: nodes

text: have

text: the

text: same

text: strength,

text: the

text: uniform

text: and

text: Newman-Girvan

text: networks

text: are

text: equivalent

text: because

text: ki

text: kj

text: for

text: all

text: i,j

text: ki

text: k

text: for

text: all

text: i.

text: This

text: was

text: pointed

text: out

text: for

text: an

text: application

text: to

text: foreign

text: exchange

text: markets

text: in

text: we

text: use

text: the

text: uniform

text: network

text: on

text: unsigned

text: adjacency

text: matrices

text: in

text: this

text: paper,

text: the

text: expected

text: edge

text: weight

text: in

text: the

text: uniform

text: network

text: is

text: always

text: nonnegative

text: for

text: correlation

text: matrices,

text: as

text: positive

text: semideﬁniteness

text: guarantees

text: that

text: A

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 12        0 1 1 ω 0 0 0 0 0     1 0 0 0 ω 0 0 0 0  11 21  1 0 0 0 0 ω 0 0 0    31 12 22 ←→  ω0 ω0 00 01 10 11 ω0 ω0 00  Layer 1 32 13 23  0 0 ω 1 1 0 0 0 ω  0 0 0 ω 0 0 0 1 0 Layer 2 33 0 0 0 0 ω 0 1 0 1 0 0 0 0 0 ω 0 1 0 Layer 3 Fig. 3.1. Example of (left) a multilayer network with unweighted intra-layer connections (solid lines) and uniformly weighted inter-layer connections (dashed curves) and (right) its corresponding adjacency matrix. (The adjacency matrix that corresponds to a multilayer network is sometimes called a “supra-adjacency matrix” in the network-science literature [35].) We restrict our attention to temporal networks in which only edges vary in time. (Thus, each node is present in all layers.) We use the notation A for a layer in a T { } s sequence of adjacency matrices = A , . . . , A|T | , and we denote node i in layer s 1 by i . We use the term multilayer network for a network deﬁned on the set of nodes { s } 1 , . . . , N ; 1 , . . . , N ; . . . ; 1|T |, . . . , N|T | [35]. 1 1 2 2 Thus far, computations that have used a multilayer framework for temporal net- works have almost always assumed (1) that inter-layer connections exist only between nodes that correspond to the same entity (i.e., between nodes i and i for some i (cid:54) s r and s = r) and (2) that the network layers are “ordinal” (i.e., inter-layer edges exist only between consecutive layers) [7,35,44,45,57]. It is also typically assumed that (3) inter-layer connections are uniform (i.e., inter-layer edges have the same weight). In a recent review article on multilayer networks [35], condition (1) was called “diagonal” coupling, and condition (2) implies that a network is “layer-coupled”. We refer to the type of coupling deﬁned by (1), (2), and (3) as ordinal diagonal and uniform inter- ∈ R layer coupling and we denote the value of the inter-layer edge weight by ω . We show a simple illustration of a multilayer network with ordinal diagonal and uniform inter-layer coupling in Fig. 3.1. One can consider more general inter-layer connections (e.g., nonuniform ones). Although we restrict our attention to uniform coupling in our theoretical and computational discussions, we give an example of a nonuniform choice of inter-layer coupling in Section . Results similar to those of subsection also apply in this more general case. The authors of [44] generalized the single-layer multiscale modularity-maximization problem in (2.6) to a multilayer network using a similar approach as the one used to derive the NGS null network from a stochastic Markov process on the observed net- |T | work. For simplicity, we express intra-layer and inter-layer connections in an N - |T | × |T | node multilayer network using a single N N matrix. Each node i in layer (cid:48) − A s s has the unique index i := i + (s 1)N, and we use to denote the multilayer A | − | adjacency matrix, which has entries (cid:48) (cid:48) = A δ(s, r)+ωδ( s r , 1) when the inter- i j ijs layer coupling is ordinal diagonal and uniform. (As discussed in [35], one can use either an adjacency tensor or an adjacency matrix to represent a multilayer network.) |T | The generalization in [44] consists of applying the function in (2.14) to the N -node ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: ω

text: ω

text: ω

text: ω0

text: ω0

text: ω0

text: ω0

text: Layer

text: ω

text: ω

text: ω

text: Layer

text: ω

text: ω

text: Layer

text: Fig.

text: Example

text: of

text: (left)

text: a

text: multilayer

text: network

text: with

text: unweighted

text: intra-layer

text: connections

text: (solid

text: lines)

text: and

text: uniformly

text: weighted

text: inter-layer

text: connections

text: (dashed

text: curves)

text: and

text: (right)

text: its

text: corresponding

text: adjacency

text: matrix.

text: (The

text: adjacency

text: matrix

text: that

text: corresponds

text: to

text: a

text: multilayer

text: network

text: is

text: sometimes

text: called

text: a

text: “supra-adjacency

text: matrix”

text: in

text: the

text: network-science

text: literature

text: We

text: restrict

text: our

text: attention

text: to

text: temporal

text: networks

text: in

text: which

text: only

text: edges

text: vary

text: in

text: time.

text: (Thus,

text: each

text: node

text: is

text: present

text: in

text: all

text: layers.)

text: We

text: use

text: the

text: notation

text: A

text: for

text: a

text: layer

text: in

text: a

text: T

text: s

text: sequence

text: of

text: adjacency

text: matrices

text: A

text: A|T

text: and

text: we

text: denote

text: node

text: i

text: in

text: layer

text: s

text: by

text: i

text: We

text: use

text: the

text: term

text: multilayer

text: network

text: for

text: a

text: network

text: deﬁned

text: on

text: the

text: set

text: of

text: nodes

text: s

text: N

text: N

text: N|T

text: Thus

text: far,

text: computations

text: that

text: have

text: used

text: a

text: multilayer

text: framework

text: for

text: temporal

text: net-

text: works

text: have

text: almost

text: always

text: assumed

text: that

text: inter-layer

text: connections

text: exist

text: only

text: between

text: nodes

text: that

text: correspond

text: to

text: the

text: same

text: entity

text: (i.e.,

text: between

text: nodes

text: i

text: and

text: i

text: for

text: some

text: i

text: s

text: r

text: and

text: s

text: r)

text: and

text: that

text: the

text: network

text: layers

text: are

text: “ordinal”

text: (i.e.,

text: inter-layer

text: edges

text: exist

text: only

text: between

text: consecutive

text: layers)

text: It

text: is

text: also

text: typically

text: assumed

text: that

text: inter-layer

text: connections

text: are

text: uniform

text: (i.e.,

text: inter-layer

text: edges

text: have

text: the

text: same

text: weight).

text: In

text: a

text: recent

text: review

text: article

text: on

text: multilayer

text: networks

text: condition

text: was

text: called

text: “diagonal”

text: coupling,

text: and

text: condition

text: implies

text: that

text: a

text: network

text: is

text: “layer-coupled”.

text: We

text: refer

text: to

text: the

text: type

text: of

text: coupling

text: deﬁned

text: by

text: and

text: as

text: ordinal

text: diagonal

text: and

text: uniform

text: inter-

text: R

text: layer

text: coupling

text: and

text: we

text: denote

text: the

text: value

text: of

text: the

text: inter-layer

text: edge

text: weight

text: by

text: ω

text: We

text: show

text: a

text: simple

text: illustration

text: of

text: a

text: multilayer

text: network

text: with

text: ordinal

text: diagonal

text: and

text: uniform

text: inter-layer

text: coupling

text: in

text: Fig.

text: One

text: can

text: consider

text: more

text: general

text: inter-layer

text: connections

text: (e.g.,

text: nonuniform

text: ones).

text: Although

text: we

text: restrict

text: our

text: attention

text: to

text: uniform

text: coupling

text: in

text: our

text: theoretical

text: and

text: computational

text: discussions,

text: we

text: give

text: an

text: example

text: of

text: a

text: nonuniform

text: choice

text: of

text: inter-layer

text: coupling

text: in

text: Section

text: Results

text: similar

text: to

text: those

text: of

text: subsection

text: also

text: apply

text: in

text: this

text: more

text: general

text: case.

text: The

text: authors

text: of

text: generalized

text: the

text: single-layer

text: multiscale

text: modularity-maximization

text: problem

text: in

text: to

text: a

text: multilayer

text: network

text: using

text: a

text: similar

text: approach

text: as

text: the

text: one

text: used

text: to

text: derive

text: the

text: NGS

text: network

text: from

text: a

text: stochastic

text: Markov

text: process

text: on

text: the

text: observed

text: net-

text: |T

text: work.

text: For

text: simplicity,

text: we

text: express

text: intra-layer

text: and

text: inter-layer

text: connections

text: in

text: an

text: N

text: |T

text: |T

text: node

text: multilayer

text: network

text: using

text: a

text: single

text: N

text: N

text: matrix.

text: Each

text: node

text: i

text: in

text: layer

text: A

text: s

text: s

text: has

text: the

text: unique

text: index

text: i

text: i

text: (s

text: and

text: we

text: use

text: to

text: denote

text: the

text: multilayer

text: A

text: adjacency

text: matrix,

text: which

text: has

text: entries

text: A

text: δ(s,

text: r)+ωδ(

text: s

text: r

text: when

text: the

text: inter-

text: i

text: j

text: ijs

text: layer

text: coupling

text: is

text: ordinal

text: diagonal

text: and

text: uniform.

text: (As

text: discussed

text: in

text: one

text: can

text: use

text: either

text: an

text: adjacency

text: tensor

text: or

text: an

text: adjacency

text: matrix

text: to

text: represent

text: a

text: multilayer

text: network.)

text: |T

text: The

text: generalization

text: in

text: consists

text: of

text: applying

text: the

text: function

text: in

text: to

text: the

text: N

text: -node

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

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text: 3 in [44] and the resulting network is a type of multilayer network [11, 35]. The main diﬀerence between this approach and the previous approach is that the presence of nonzero inter-layer edges introduces a dependence between communities identiﬁed in one layer and connectivity patterns in other layers. Thus far, most computations that have used a multilayer representation of temporal networks have assumed that inter-layer connections are “diagonal” (i.e., they exist only between copies of the same node) and “ordinal” (i.e., they exist only between consecutive layers) [35]. Diagonal is a natural model of the persistence of node identity in time, while ordinal preserves the time ordering. The authors of [44] derived a generalization of modularity maximization, a popular clustering method for static networks, to multilayer networks. Modularity is a function that measures the “quality” of a network partition into disjoint sets of nodes by computing the diﬀerence between the total edge weight in sets in the observed network and the total expected edge weight in the same sets in a “null network” generated from some “null model” [23, 55]. Modularity maximization consists of maximizing the modularity quality function over the space of network partitions. (In practice, given the combinatorial complexity of this maximization problem, one uses some computational heuristic and ﬁnds a local maximum [28].) Intuitively, the null model controls for connectivity patterns that one anticipates ﬁnding in a network, and one uses modularity maximization to identify connectivity patterns in an observed network that are stronger than anticipated. We give a precise deﬁnition of the modularity function for single-layer networks in Section , where (importantly) we distinguish between a “null network” and a “null model” in modularity maximization. In Section , we discuss the choice of null network for a given application. In Section , we describe the generalization of single-layer modularity to multilayer networks proposed in [44]. To date, almost no theory has explained how a multilayer partition obtained with zero inter-layer coupling (which reduces to single-layer mod- ularity maximization on each layer independently) diﬀers from a multilayer partition obtained with nonzero inter-layer coupling. In Section , we prove several theoretical properties of an optimal solution for the multilayer maximization problem to better understand how such partitions diﬀer and how one can exploit this diﬀerence in prac- tice. We also describe two implementation issues that arise when using the popular Louvain heuristic [10] to solve the multilayer maximization problem, and we suggest ways to mitigate them. The results of Section are independent of the choice of qual- ity function on individual layers and only depend on the form of the maximization problem. Section contains a concluding discussion. G Consider an N-node network and let the edge weights between pairs of nodes { | ∈ { }} ∈ R × G be A i, j 1, . . . , N , so that A = (A ) N N is the adjacency matrix of . ij ij In this paper, we only consider symmetric adjacency matrices (and hence undirected networks), so A = A for all i and j. The strength of a node i is ij ji (cid:88) (cid:88) N N k = A = A , (2.1) i ij ji j=1 j=1 and it is given by the ith row (or column) sum of A.

text: in

text: and

text: the

text: resulting

text: network

text: is

text: a

text: type

text: of

text: multilayer

text: network

text: The

text: main

text: diﬀerence

text: between

text: this

text: approach

text: and

text: the

text: previous

text: approach

text: is

text: that

text: the

text: presence

text: of

text: nonzero

text: inter-layer

text: edges

text: introduces

text: a

text: dependence

text: between

text: communities

text: identiﬁed

text: in

text: one

text: layer

text: and

text: connectivity

text: patterns

text: in

text: other

text: layers.

text: Thus

text: far,

text: most

text: computations

text: that

text: have

text: used

text: a

text: multilayer

text: representation

text: of

text: temporal

text: networks

text: have

text: assumed

text: that

text: inter-layer

text: connections

text: are

text: “diagonal”

text: (i.e.,

text: they

text: exist

text: only

text: between

text: copies

text: of

text: the

text: same

text: node)

text: and

text: “ordinal”

text: (i.e.,

text: they

text: exist

text: only

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text: consecutive

text: layers)

text: Diagonal

text: is

text: a

text: natural

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text: the

text: persistence

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text: node

text: identity

text: in

text: time,

text: while

text: ordinal

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text: the

text: time

text: ordering.

text: The

text: authors

text: of

text: derived

text: a

text: generalization

text: of

text: modularity

text: maximization,

text: a

text: popular

text: clustering

text: method

text: for

text: static

text: networks,

text: to

text: multilayer

text: networks.

text: Modularity

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text: a

text: function

text: that

text: measures

text: the

text: “quality”

text: of

text: a

text: network

text: partition

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text: disjoint

text: sets

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text: the

text: diﬀerence

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text: the

text: observed

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text: total

text: expected

text: edge

text: weight

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text: the

text: same

text: sets

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text: a

text: “null

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text: some

text: “null

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text: Modularity

text: maximization

text: consists

text: of

text: maximizing

text: the

text: modularity

text: quality

text: function

text: over

text: the

text: space

text: of

text: network

text: partitions.

text: (In

text: practice,

text: given

text: the

text: combinatorial

text: complexity

text: of

text: this

text: maximization

text: problem,

text: one

text: uses

text: some

text: computational

text: heuristic

text: and

text: ﬁnds

text: a

text: local

text: maximum

text: Intuitively,

text: the

text: model

text: controls

text: for

text: connectivity

text: patterns

text: that

text: one

text: anticipates

text: ﬁnding

text: in

text: a

text: network,

text: and

text: one

text: uses

text: modularity

text: maximization

text: to

text: identify

text: connectivity

text: patterns

text: in

text: an

text: observed

text: network

text: that

text: are

text: stronger

text: than

text: anticipated.

text: We

text: give

text: a

text: precise

text: deﬁnition

text: of

text: the

text: modularity

text: function

text: for

text: single-layer

text: networks

text: in

text: Section

text: where

text: (importantly)

text: we

text: distinguish

text: between

text: a

text: “null

text: network”

text: and

text: a

text: “null

text: model”

text: in

text: modularity

text: maximization.

text: In

text: Section

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text: discuss

text: the

text: choice

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text: network

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text: a

text: given

text: application.

text: In

text: Section

text: we

text: describe

text: the

text: generalization

text: of

text: single-layer

text: modularity

text: to

text: multilayer

text: networks

text: proposed

text: in

text: To

text: date,

text: almost

text: no

text: theory

text: has

text: explained

text: how

text: a

text: multilayer

text: partition

text: obtained

text: with

text: zero

text: inter-layer

text: coupling

text: (which

text: reduces

text: to

text: single-layer

text: mod-

text: ularity

text: maximization

text: on

text: each

text: layer

text: independently)

text: diﬀers

text: from

text: a

text: multilayer

text: partition

text: obtained

text: with

text: nonzero

text: inter-layer

text: coupling.

text: In

text: Section

text: we

text: prove

text: several

text: theoretical

text: properties

text: of

text: an

text: optimal

text: solution

text: for

text: the

text: multilayer

text: maximization

text: problem

text: to

text: better

text: understand

text: how

text: such

text: partitions

text: diﬀer

text: and

text: how

text: one

text: can

text: exploit

text: this

text: diﬀerence

text: in

text: prac-

text: tice.

text: We

text: also

text: describe

text: two

text: implementation

text: issues

text: that

text: arise

text: when

text: using

text: the

text: popular

text: Louvain

text: heuristic

text: to

text: solve

text: the

text: multilayer

text: maximization

text: problem,

text: and

text: we

text: suggest

text: ways

text: to

text: mitigate

text: them.

text: The

text: results

text: of

text: Section

text: are

text: independent

text: of

text: the

text: choice

text: of

text: qual-

text: ity

text: function

text: on

text: individual

text: layers

text: and

text: only

text: depend

text: on

text: the

text: form

text: of

text: the

text: maximization

text: problem.

text: Section

text: contains

text: a

text: concluding

text: discussion.

text: G

text: Consider

text: an

text: N-node

text: network

text: and

text: let

text: the

text: edge

text: weights

text: between

text: pairs

text: of

text: nodes

text: R

text: G

text: be

text: A

text: i,

text: j

text: N

text: so

text: that

text: A

text: (A

text: N

text: N

text: is

text: the

text: adjacency

text: matrix

text: of

text: ij

text: ij

text: In

text: this

text: paper,

text: we

text: only

text: consider

text: symmetric

text: adjacency

text: matrices

text: (and

text: hence

text: undirected

text: networks),

text: so

text: A

text: A

text: for

text: all

text: i

text: and

text: j.

text: The

text: strength

text: of

text: a

text: node

text: i

text: is

text: ij

text: ji

text: N

text: N

text: k

text: A

text: A

text: i

text: ij

text: ji

text: and

text: it

text: is

text: given

text: by

text: the

text: ith

text: row

text: (or

text: column)

text: sum

text: of

text: A.