text: University Press, Cambridge, 2000). [20] C. Y. Tseng and M. B. Wise, “Inﬂaton Two-Point Correlation in the Presence of a Cosmic String,” Phys. Rev. D 80, 103512 (2009) [arXiv:0908.0543 [astro-ph.CO]]. [21] S. Jazayeri, Y. Akrami, H. Firouzjahi, A. R. Solomon and Y. Wang, “Inﬂationary power asymmetry from primordial domain walls,” arXiv:1408.3057 [astro-ph.CO]. [22] T. W. B. Kibble, G. Lazarides and Q. Shaﬁ, “Walls Bounded by Strings,” Phys. Rev. D 26, 435 (1982). [23] A. Vilenkin and A. E. Everett, “Cosmic Strings and Domain Walls in Models with Goldstone and PseudoGoldstone Bosons,” Phys. Rev. Lett. 48, 1867 (1982). [24] M. Barriola and A. Vilenkin, “Gravitational Field of a Global Monopole,” Phys. Rev. Lett. 63, 341 (1989). 43

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text: Y.

text: Tseng

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text: B.

text: Wise,

text: “Inﬂaton

text: Two-Point

text: Correlation

text: in

text: the

text: Presence

text: of

text: a

text: Cosmic

text: String,”

text: Phys.

text: Rev.

text: D

text: [astro-ph.CO]].

text: S.

text: Jazayeri,

text: Y.

text: Akrami,

text: H.

text: Firouzjahi,

text: A.

text: R.

text: Solomon

text: and

text: Y.

text: Wang,

text: “Inﬂationary

text: power

text: asymmetry

text: from

text: primordial

text: domain

text: walls,”

text: [astro-ph.CO].

text: T.

text: W.

text: B.

text: Kibble,

text: G.

text: Lazarides

text: and

text: Q.

text: Shaﬁ,

text: “Walls

text: Bounded

text: by

text: Strings,”

text: Phys.

text: Rev.

text: D

text: A.

text: Vilenkin

text: and

text: A.

text: E.

text: Everett,

text: “Cosmic

text: Strings

text: and

text: Domain

text: Walls

text: in

text: Models

text: with

text: Goldstone

text: and

text: PseudoGoldstone

text: Bosons,”

text: Phys.

text: Rev.

text: Lett.

text: M.

text: Barriola

text: and

text: A.

text: Vilenkin,

text: “Gravitational

text: Field

text: of

text: a

text: Global

text: Monopole,”

text: Phys.

text: Rev.

text: Lett.


text: (cid:80) 11 • ≥ The bank is solvent if and only if the capital buﬀer w can absorb the losses, i.e., w M m . α=1 α • (cid:48) ∈ { } ≥ (cid:48) The bank is in default at layer α 1, 2, . . . , M but not in any layer α + 1 if and only if (cid:88) (cid:88) (cid:88) (cid:48)− (cid:48) α 1 M ≤ α w + b < m w + b . (A1) α α α α=1 α=1 α=1 • The bank is in complete default, meaning that it cannot pay oﬀ any of its interbank debts nor any of its liabilities to external creditors, if and only if (cid:88) (cid:88) M M m > w + b . α α α=1 α=1 Here, we focus on cascades in the interbank network; for this purpose, we do not need to distinguish between default (cid:80) at the most senior level (M) from complete default because both have the same eﬀect on the banks that had lent to the defaulted bank. Rearranging the strict inequality in Eq. (A1) and dividing by the total lending M l > 0 gives the response α=1 α function in Eq. (10). Appendix B: Relationship with “multistage complex contagion” Our model is related to the “multistage complex contagion” model of Melnik et al. [25]. In their model, nodes exist in one of three states: inactive, active, and hyper-active. Their hyper-active nodes are active nodes with greater inﬂuence on their neighbors. To see the commonality between the models, consider our model with every node having the same junior threshold R and with the multiplex network consisting of two identical layers (i.e., all the J multiplex edges are “fully overlapped” [16, 37]). Then every neighbor in junior-default (respectively, in senior-default) contributes m + m = 1 (respectively, m + m = 2) in the numerator of the response function. Thus, this model J S J S is equivalent to a special case of the model in [25] in which i) the “bonus inﬂuence parameter” β (which captures the extra inﬂuence of hyper-active nodes) takes the value 1; ii) the network is a directed single-layer graph; and iii) the thresholds for activation and hyper-activation are R and R + b /(l + l ) = R + b /(2l ), respectively. To model J J J J S J J J real ﬁnancial systems, however, edges should be only partially overlapped, a phenomenon studied in the contexts of statistical mechanics in [37] and of percolation in [16]. Our multilevel contagion model could be generalized to such a partially overlapped case by adapting techniques from the study of mutual percolation on multiplex networks with arbitrary amounts of overlap [16], a nontrivial task left for future work. Appendix C: Derivation of the ﬁrst-order cascade condition Now we obtain a simple expression that approximately captures for which parameters a vanishingly small seed leads to a cascade that results in a ﬁnite fraction of banks in junior- or senior-default. Note that if the threshold R > 0, ≡ ≡ J then (0, 0) is a ﬁxed point of the recursion (4) [because Bmk (0) 1m=0 and Fα((cid:126)l,(cid:126)b,(cid:126)0) 0 provided that RJ > 0]. The ﬁrst-order cascade condition is the linear instability of Eq. (4) at this ﬁxed point at the origin (φJ, φS) = (0, 0) in the → → t t double limit φJ 0 and φS 0. This linear instability provides a suﬃcient condition for the numbers of junior- and 0 0 senior-defaulted ba(cid:18)nks to grow(cid:19) (at least initially). The Jacobian matrix of the right-hand side of Eq. (4) is J J J (cid:18) (cid:19) = JJJ JJS (C1) SJ SS E − E − = E [lJ(11>RJ(lJ+l−S) 10>RJ)] E [lS(11>RJ(lJ+l−S) 10>RJ)] [lJ(11−bJ>RJ(lJ+lS) 1−bJ>RJ(lJ+lS))] [lS(11−bJ>RJ(lJ+lS) 1−bJ>RJ(lJ+lS))] (C2) (The expectations E are over the degree distributions pJ,loan, pS,loan, pJ,borrow.) Assume that R > 0, so that the lJ lS ≥ bJ J ≥ factors 10>RJ in the top row of (C2) vanish. Also, note that bJ 0 and that RJ > 0 and lJ + lS 0, so the

label: list

label: paragraph

text: The

text: bank

text: is

text: solvent

text: if

text: and

text: only

text: if

text: the

text: capital

text: buﬀer

text: w

text: can

text: absorb

text: the

text: losses,

text: i.e.,

text: w

text: M

text: m

text: α

text: The

text: bank

text: is

text: in

text: default

text: at

text: layer

text: α

text: M

text: but

text: not

text: in

text: any

text: layer

text: α

text: if

text: and

text: only

text: if

label: equation

text: α

text: M

text: α

text: w

text: b

text: m

text: w

text: b

text: α

text: α

text: α

text: The

text: bank

text: is

text: in

text: complete

text: default,

text: meaning

text: that

text: it

text: cannot

text: pay

text: oﬀ

text: any

text: of

text: its

text: interbank

text: debts

text: nor

text: any

text: of

text: its

text: liabilities

text: to

text: external

text: creditors,

text: if

text: and

text: only

text: if

text: M

text: M

text: m

text: w

text: b

text: α

text: α

text: Here,

text: we

text: focus

text: on

text: cascades

text: in

text: the

text: interbank

text: network;

text: for

text: this

text: purpose,

text: we

text: do

text: not

text: need

text: to

text: distinguish

text: between

text: default

text: at

text: the

text: most

text: senior

text: level

text: (M)

text: from

text: complete

text: default

text: because

text: both

text: have

text: the

text: same

text: eﬀect

text: on

text: the

text: banks

text: that

text: had

text: lent

text: to

text: the

text: defaulted

text: bank.

text: Rearranging

text: the

text: strict

text: inequality

text: in

text: Eq.

text: and

text: dividing

text: by

text: the

text: total

text: lending

text: M

text: l

text: gives

text: the

text: response

text: α

text: function

text: in

text: Eq.

text: Appendix

text: B:

text: Relationship

text: with

text: “multistage

text: complex

text: contagion”

text: Our

text: model

text: is

text: related

text: to

text: the

text: “multistage

text: complex

text: contagion”

text: model

text: of

text: Melnik

text: et

text: al.

text: In

text: their

text: model,

text: nodes

text: exist

text: in

text: one

text: of

text: three

text: states:

text: inactive,

text: active,

text: and

text: hyper-active.

text: Their

text: hyper-active

text: nodes

text: are

text: active

text: nodes

text: with

text: greater

text: inﬂuence

text: on

text: their

text: neighbors.

text: To

text: see

text: the

text: commonality

text: between

text: the

text: models,

text: consider

text: our

text: model

text: with

text: every

text: node

text: having

text: the

text: same

text: junior

text: threshold

text: R

text: and

text: with

text: the

text: multiplex

text: network

text: consisting

text: of

text: two

text: identical

text: layers

text: (i.e.,

text: all

text: the

text: J

text: multiplex

text: edges

text: are

text: “fully

text: overlapped”

text: Then

text: every

text: neighbor

text: in

text: junior-default

text: (respectively,

text: in

text: senior-default)

text: contributes

text: m

text: m

text: (respectively,

text: m

text: m

text: in

text: the

text: numerator

text: of

text: the

text: response

text: function.

text: Thus,

text: this

text: model

text: J

text: S

text: J

text: S

text: is

text: equivalent

text: to

text: a

text: special

text: case

text: of

text: the

text: model

text: in

text: in

text: which

text: i)

text: the

text: “bonus

text: inﬂuence

text: parameter”

text: β

text: (which

text: captures

text: the

text: extra

text: inﬂuence

text: of

text: hyper-active

text: nodes)

text: takes

text: the

text: value

text: ii)

text: the

text: network

text: is

text: a

text: directed

text: single-layer

text: graph;

text: and

text: iii)

text: the

text: thresholds

text: for

text: activation

text: and

text: hyper-activation

text: are

text: R

text: and

text: R

text: b

text: /(l

text: l

text: R

text: b

text: respectively.

text: To

text: model

text: J

text: J

text: J

text: J

text: S

text: J

text: J

text: J

text: real

text: ﬁnancial

text: systems,

text: however,

text: edges

text: should

text: be

text: only

text: partially

text: overlapped,

text: a

text: phenomenon

text: studied

text: in

text: the

text: contexts

text: of

text: statistical

text: mechanics

text: in

text: and

text: of

text: percolation

text: in

text: Our

text: multilevel

text: contagion

text: model

text: could

text: be

text: generalized

text: to

text: such

text: a

text: partially

text: overlapped

text: case

text: by

text: adapting

text: techniques

text: from

text: the

text: study

text: of

text: mutual

text: percolation

text: on

text: multiplex

text: networks

text: with

text: arbitrary

text: amounts

text: of

text: overlap

text: a

text: nontrivial

text: task

text: left

text: for

text: future

text: work.

text: Appendix

text: C:

text: Derivation

text: of

text: the

text: ﬁrst-order

text: cascade

text: condition

text: Now

text: we

text: obtain

text: a

text: simple

text: expression

text: that

text: approximately

text: captures

text: for

text: which

text: parameters

text: a

text: vanishingly

text: small

text: seed

text: leads

text: to

text: a

text: cascade

text: that

text: results

text: in

text: a

text: ﬁnite

text: fraction

text: of

text: banks

text: in

text: junior-

text: or

text: senior-default.

text: Note

text: that

text: if

text: the

text: threshold

text: R

text: J

text: then

text: is

text: a

text: ﬁxed

text: point

text: of

text: the

text: recursion

text: [because

text: Bmk

text: and

text: provided

text: that

text: RJ

text: The

text: ﬁrst-order

text: cascade

text: condition

text: is

text: the

text: linear

text: instability

text: of

text: Eq.

text: at

text: this

text: ﬁxed

text: point

text: at

text: the

text: origin

text: (φJ,

text: φS)

text: in

text: the

text: t

text: t

text: double

text: limit

text: φJ

text: and

text: φS

text: This

text: linear

text: instability

text: provides

text: a

text: suﬃcient

text: condition

text: for

text: the

text: numbers

text: of

text: junior-

text: and

text: senior-defaulted

text: to

text: (at

text: least

text: initially).

text: The

text: Jacobian

text: matrix

text: of

text: the

text: right-hand

text: side

text: of

text: Eq.

text: is

text: J

text: J

text: J

text: JJJ

text: JJS

text: SJ

text: SS

text: E

text: E

text: E

text: E

text: (The

text: expectations

text: E

text: are

text: over

text: the

text: degree

text: distributions

text: pJ,loan,

text: pS,loan,

text: pJ,borrow.)

text: Assume

text: that

text: R

text: so

text: that

text: the

text: lJ

text: lS

text: bJ

text: J

text: factors

text: in

text: the

text: top

text: row

text: of

text: vanish.

text: Also,

text: note

text: that

text: bJ

text: and

text: that

text: RJ

text: and

text: lJ

text: lS

text: so

text: the


text: 3 are two types of defaults (bankruptcies): junior-defaults defaulted. This bank is in junior-default if and only if and senior-defaults. A bank is in junior-default if and its losses m + m exceeds the bank’s equity w, and this J S only if its capital (a.k.a. equity or net worth; see Fig. 1) bank is in senior-default if and only if its losses m +m J S is negative, meaning that it cannot pay all of its junior exceed the sum of its equity w and its junior liabilities debts. A bank is in senior-default if and only if its capital b . J is so far below zero that it cannot pay its all of its senior To exploit the solution technique for threshold cas- debts nor any of its junior debts. Note that banks in cades in networks [18–20, 22, 23], we express these in- senior-default are necessarily in junior-default as well. equalities in terms of losses relative to total out-degree ∈ { } We assume that when a bank goes into junior-default, l + l : the bank is in default at layer α J, S if and J S it cannot repay any of its junior-level liabilities, and when only if (cid:40) a bank goes into senior-default, it cannot pay any of its junior liabilities nor any of its senior liabilities. Said dif- ferently, we assume that loss given default of seniority m + m ≡ w if α = J αlowiser1.00T%hifsorasisnutmerpbtainonk ldiaebpialrittisesfroofmsepnrieovriiotyusofcaαscaandde lJJ + lSS > Rα llwJJ+++bllJSS if α = S (1) models of ﬁnancial contagion, such as that of Gai and [In Appendix , we generalize the thresholds R in Kapadia [1], in which loss given default is 100% for all ≥ α Eq. (1) to the general case of M 1 layers.] Figure 2 interbank liabilities, no matter their seniority. Table I illustrates a bank in junior-default and a bank in senior- makes this assumption concrete: the value of a bank’s default in two ways: as losses on a balance sheet (top debt is equal to some positive value (say, 1 unit of cur- row) and as contagion in a multiplex network (bottom rency) if the bank is solvent (i.e., if its equity is nonneg- row). ative); the value of the bank’s junior debt falls to zero The dynamics of multilevel default contagion can when the bank becomes insolvent, and the value of its be described as a multiplex version of the threshold senior debt falls to zero when the bank’s equity passes a − model [21, 24, 33, 36]. (Other multiplex generalizations nonpositive threshold value (derived later to be b ). J were studied in [18–20].) The “response functions” for each layer are (cid:40) TABLE I. Relationship between the equity of a borrower and the value of its junior debts and senior debts held by its credi- tvoarlus.abHleerwe,hweenatshseumboertrhowaterjuinsiosorlavnendts,esnoiovradlueebstsoaf rdeeebqtusaalrlye FJ((cid:126)l,(cid:126)b, m(cid:126) ) ≡ (cid:40)1 if mlJJ++mlSS > RJ (2a) 0 otherwise either zero or $1. If the borrower’s equity is negative but suf- ≥ − − ﬁncoineentolyf iltasrgjuen(ior debbJt)s,;tthheen bitorcraonweprayis iitns “sejunnioiorrd-deebftasubltu”t. FS((cid:126)l,(cid:126)b, m(cid:126) ) ≡ 1 if mJl+Jm+SlS bJ > RJ , (2b) 0 otherwise For suﬃciently negative equity of the borrower, the borrower ≡ ≡ ≡ cannot pay neither its junior debts nor its senior debts; the where (cid:126)l (l , l ),(cid:126)b (b , b ) and m(cid:126) (m , m ). If borrower is in “senior-default”, and all its debts have value J S J S J S the outputs of F and F are both 0, then the bank is zero. J S solvent; if F gives 1, then the bank is in junior-default; J equity status junior debt value senior debt value if F gives 1, then the bank is in senior-default (and in ≥ S 0 solvent $1 $1 junior-default). <−0 junior-default 0 $1 We note in passing that a special case of this model < bJ senior-default 0 0 coincides with a special case of the “multistage complex contagion” model of Melnik et al. [25]; for details, see Appendix . II. MULTILEVEL CASCADES ON 2-LAYER MULTIPLEX NETWORKS OF JUNIOR AND 3. Thresholds for junior- and senior-default SENIOR LOANS Initially, all banks are solvent, and a small fraction This multiplex model adds a new dimension of com- of banks are chosen to be in junior-default or in senior- plexity to previous threshold models [1–3, 21, 22, 24] by default. These initial defaults could be caused by the considering multiple levels of seniority of debt, so to un- loss of external assets. These bank failures cause losses derstand the model we start by simplifying in two other for their creditors, who may default as a result, which dimensions, the distribution of networks and the distri- may cause one or more of their creditors to default, and bution of thresholds. so on, resulting in a cascade of default. First, each layer is modeled as an independent, lo- Consider some point in time during a cascade. For a cally treelike, random directed network generated by the certain bank, let m denote the number of its junior- conﬁguration model. The sparsity and independence J borrowers that have junior-defaulted, and let m de- of the layers implies that the number of “overlapping S note the number of its senior-borrowers that have senior- edges” (i.e., junior and senior loans between the same ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: are

text: two

text: types

text: of

text: defaults

text: (bankruptcies):

text: junior-defaults

text: defaulted.

text: This

text: bank

text: is

text: in

text: junior-default

text: if

text: and

text: only

text: if

text: and

text: senior-defaults.

text: A

text: bank

text: is

text: in

text: junior-default

text: if

text: and

text: its

text: losses

text: m

text: m

text: exceeds

text: the

text: bank’s

text: equity

text: w,

text: and

text: this

text: J

text: S

text: only

text: if

text: its

text: capital

text: (a.k.a.

text: equity

text: or

text: net

text: worth;

text: see

text: Fig.

text: bank

text: is

text: in

text: senior-default

text: if

text: and

text: only

text: if

text: its

text: losses

text: m

text: +m

text: J

text: S

text: is

text: negative,

text: meaning

text: that

text: it

text: cannot

text: pay

text: all

text: of

text: its

text: junior

text: exceed

text: the

text: sum

text: of

text: its

text: equity

text: w

text: and

text: its

text: junior

text: liabilities

text: debts.

text: A

text: bank

text: is

text: in

text: senior-default

text: if

text: and

text: only

text: if

text: its

text: capital

text: b

text: J

text: is

text: so

text: far

text: below

text: zero

text: that

text: it

text: cannot

text: pay

text: its

text: all

text: of

text: its

text: senior

text: To

text: exploit

text: the

text: solution

text: technique

text: for

text: threshold

text: cas-

text: debts

text: nor

text: any

text: of

text: its

text: junior

text: debts.

text: Note

text: that

text: banks

text: in

text: cades

text: in

text: networks

text: we

text: express

text: these

text: in-

text: senior-default

text: are

text: necessarily

text: in

text: junior-default

text: as

text: well.

text: equalities

text: in

text: terms

text: of

text: losses

text: relative

text: to

text: total

text: out-degree

text: We

text: assume

text: that

text: when

text: a

text: bank

text: goes

text: into

text: junior-default,

text: l

text: l

text: the

text: bank

text: is

text: in

text: default

text: at

text: layer

text: α

text: J,

text: S

text: if

text: and

text: J

text: S

text: it

text: cannot

text: repay

text: any

text: of

text: its

text: junior-level

text: liabilities,

text: and

text: when

text: only

text: if

text: a

text: bank

text: goes

text: into

text: senior-default,

text: it

text: cannot

text: pay

text: any

text: of

text: its

text: junior

text: liabilities

text: nor

text: any

text: of

text: its

text: senior

text: liabilities.

text: Said

text: dif-

text: ferently,

text: we

text: assume

text: that

text: loss

text: given

text: default

text: of

text: seniority

text: m

text: m

text: w

text: if

text: α

text: J

text: lJJ

text: lSS

text: Rα

text: llwJJ+++bllJSS

text: if

text: α

text: S

text: models

text: of

text: ﬁnancial

text: contagion,

text: such

text: as

text: that

text: of

text: Gai

text: and

text: [In

text: Appendix

text: we

text: generalize

text: the

text: thresholds

text: R

text: in

text: Kapadia

text: in

text: which

text: loss

text: given

text: default

text: is

text: for

text: all

text: α

text: Eq.

text: to

text: the

text: general

text: case

text: of

text: M

text: layers.]

text: Figure

text: interbank

text: liabilities,

text: no

text: matter

text: their

text: seniority.

text: Table

text: I

text: illustrates

text: a

text: bank

text: in

text: junior-default

text: and

text: a

text: bank

text: in

text: senior-

text: makes

text: this

text: assumption

text: concrete:

text: the

text: value

text: of

text: a

text: bank’s

text: default

text: in

text: two

text: ways:

text: as

text: losses

text: on

text: a

text: balance

text: sheet

text: (top

text: debt

text: is

text: equal

text: to

text: some

text: positive

text: value

text: (say,

text: unit

text: of

text: cur-

text: row)

text: and

text: as

text: contagion

text: in

text: a

text: multiplex

text: network

text: (bottom

text: rency)

text: if

text: the

text: bank

text: is

text: solvent

text: (i.e.,

text: if

text: its

text: equity

text: is

text: nonneg-

text: row).

text: ative);

text: the

text: value

text: of

text: the

text: bank’s

text: junior

text: debt

text: falls

text: to

text: zero

text: The

text: dynamics

text: of

text: multilevel

text: default

text: contagion

text: can

text: when

text: the

text: bank

text: becomes

text: insolvent,

text: and

text: the

text: value

text: of

text: its

text: be

text: described

text: as

text: a

text: multiplex

text: version

text: of

text: the

text: threshold

text: senior

text: debt

text: falls

text: to

text: zero

text: when

text: the

text: bank’s

text: equity

text: passes

text: a

text: model

text: (Other

text: multiplex

text: generalizations

text: nonpositive

text: threshold

text: value

text: (derived

text: later

text: to

text: be

text: b

text: J

text: were

text: studied

text: in

text: The

text: “response

text: functions”

text: for

text: each

text: layer

text: are

text: TABLE

text: I.

text: Relationship

text: between

text: the

text: equity

text: of

text: a

text: borrower

text: and

text: the

text: value

text: of

text: its

text: junior

text: debts

text: and

text: senior

text: debts

text: held

text: by

text: its

text: credi-

text: rdeeebqtusaalrlye

text: if

text: mlJJ++mlSS

text: RJ

text: otherwise

text: either

text: zero

text: or

text: If

text: the

text: borrower’s

text: equity

text: is

text: negative

text: but

text: suf-

text: ﬁncoineentolyf

text: iltasrgjuen(ior

text: debbJt)s,;tthheen

text: bitorcraonweprayis

text: iitns

text: “sejunnioiorrd-deebftasubltu”t.

text: if

text: mJl+Jm+SlS

text: bJ

text: RJ

text: otherwise

text: For

text: suﬃciently

text: negative

text: equity

text: of

text: the

text: borrower,

text: the

text: borrower

text: cannot

text: pay

text: neither

text: its

text: junior

text: debts

text: nor

text: its

text: senior

text: debts;

text: the

text: where

text: (l

text: l

text: (b

text: b

text: and

text: (m

text: m

text: If

text: borrower

text: is

text: in

text: “senior-default”,

text: and

text: all

text: its

text: debts

text: have

text: value

text: J

text: S

text: J

text: S

text: J

text: S

text: the

text: outputs

text: of

text: F

text: and

text: F

text: are

text: both

text: then

text: the

text: bank

text: is

text: zero.

text: J

text: S

text: solvent;

text: if

text: F

text: gives

text: then

text: the

text: bank

text: is

text: in

text: junior-default;

text: J

text: equity

text: status

text: junior

text: debt

text: value

text: senior

text: debt

text: value

text: if

text: F

text: gives

text: then

text: the

text: bank

text: is

text: in

text: senior-default

text: (and

text: in

text: S

text: solvent

text: junior-default).

text: junior-default

text: We

text: note

text: in

text: passing

text: that

text: a

text: special

text: case

text: of

text: this

text: model

text: bJ

text: senior-default

text: coincides

text: with

text: a

text: special

text: case

text: of

text: the

text: “multistage

text: complex

text: contagion”

text: model

text: of

text: Melnik

text: et

text: al.

text: for

text: details,

text: see

text: Appendix

text: II.

text: MULTILEVEL

text: CASCADES

text: ON

text: MULTIPLEX

text: NETWORKS

text: OF

text: JUNIOR

text: AND

text: Thresholds

text: for

text: junior-

text: and

text: senior-default

text: SENIOR

text: LOANS

text: Initially,

text: all

text: banks

text: are

text: solvent,

text: and

text: a

text: small

text: fraction

text: This

text: multiplex

text: model

text: adds

text: a

text: new

text: dimension

text: of

text: com-

text: of

text: banks

text: are

text: chosen

text: to

text: be

text: in

text: junior-default

text: or

text: in

text: senior-

text: plexity

text: to

text: previous

text: threshold

text: models

text: by

text: default.

text: These

text: initial

text: defaults

text: could

text: be

text: caused

text: by

text: the

text: considering

text: multiple

text: levels

text: of

text: seniority

text: of

text: debt,

text: so

text: to

text: un-

text: loss

text: of

text: external

text: assets.

text: These

text: bank

text: failures

text: cause

text: losses

text: derstand

text: the

text: model

text: we

text: start

text: by

text: simplifying

text: in

text: two

text: other

text: for

text: their

text: creditors,

text: who

text: may

text: default

text: as

text: a

text: result,

text: which

text: dimensions,

text: the

text: distribution

text: of

text: networks

text: and

text: the

text: distri-

text: may

text: cause

text: one

text: or

text: more

text: of

text: their

text: creditors

text: to

text: default,

text: and

text: bution

text: of

text: thresholds.

text: so

text: on,

text: resulting

text: in

text: a

text: cascade

text: of

text: default.

text: First,

text: each

text: layer

text: is

text: modeled

text: as

text: an

text: independent,

text: lo-

text: Consider

text: some

text: point

text: in

text: time

text: during

text: a

text: cascade.

text: For

text: a

text: cally

text: treelike,

text: random

text: directed

text: network

text: generated

text: by

text: the

text: certain

text: bank,

text: let

text: m

text: denote

text: the

text: number

text: of

text: its

text: junior-

text: conﬁguration

text: model.

text: The

text: sparsity

text: and

text: independence

text: J

text: borrowers

text: that

text: have

text: junior-defaulted,

text: and

text: let

text: m

text: de-

text: of

text: the

text: layers

text: implies

text: that

text: the

text: number

text: of

text: “overlapping

text: S

text: note

text: the

text: number

text: of

text: its

text: senior-borrowers

text: that

text: have

text: senior-

text: edges”

text: (i.e.,

text: junior

text: and

text: senior

text: loans

text: between

text: the

text: same

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ( ) 6 λ () > � capturing defaults caused by two bankrupt creditors  > (using the second-order cascade condition) was not nec- 〉 7 ������-��� ������� ������� ��� > � essary to approximate simulations well, as shown in 〈�� ������-���� ������� ������� �� � Appendix and in Fig. 6. =�� 6 ������ ���� ������� ������� �� � The parameters satisfying this senior-only cascade con- � � dition (8) are shown in green in Fig. 3(a(cid:104)) f(cid:105)or unc(cid:104)or(cid:105)related ��� 5 〈〈 〉〉 ≈ Erergdio˝ons-Rsh´erninyiksnewtiwthortkhseodfemnseiatny dofegjureneiorlJloaannsd(cid:104)llS(cid:105) f.orTthhies ���� �� �����〈�� 〉 =��〈��〉 same reason that the junior-only cascade regioJn shrinks ��� 4 �� (cid:104) (cid:105) �� � � with lS . �� � � 3. Multiplex cascades �� 3 �� � � The blue region in Fig. 3(a) is the multiplex cascade re- � 2 (cid:104) (cid:105) (cid:104) (cid:105) � gion, the set of average out-degrees ( l , l ) such that � the ﬁrst-order cascade condition [inequJalityS(6)] is satis- ���� 1 � ﬁed. NJot≥e that,{aJs poJinted} out in [18, page 3], we have �� = 〈 〉 λmax( ) max JJ, SS . Thus, a suﬃcient condition 0 0 1 2 3 4 5 6 7 for the multiplex cascade condition to hold is that at least ( ) ∈ { } �������� ������ �� ������ ����� ��� ���� � one kind of default α J, S is supercritical (meaning � J that αα > 1) even in the absence of the other kind of � default. Thus, the blue region in Fig. 3(a) contains both the orange and green regions. Importantly, however, the blue region contains pairs of 6 (cid:104) (cid:105) (cid:104) (cid:105) � ionntleyrmaneddiasetneieodr-goendlyencsaitsiceasd(e lrJeg,iolnSs.) oInuttshidiseptharetjuonf ipoar-- ��� ≈ �� 4 〈 〉 raJalomneetesruﬃspcaecet,onceaituhseergjluonbiaolr-cdaesfcaaudletss n(io.er.s,eJnJioJr-<de1fauanltds ����� ��������� �� 〈���〉 ���� SS < 1), but a large cascade of defaults can neverthe- �� 2 � less occur. These cascades, akin to the “multiplexity- facilitated cascades” in [18, 20], result from the interac- 〈 〉 / 〈 〉 0 tion of junior- and senior-defaults. 0 2 4 6 8 C. Optimal seniority ratio ��������� ����� � � � � Note that the multiplex cascade region, show in blue FIG. 3. (Color online) (a) The multiplex cascade region (cid:104) (cid:105) (cid:104) (cid:105) in Fig. 3(a), is not symmetric w(cid:104)ith(cid:105) respe(cid:104)ct t(cid:105)o the average [the edge-density parameters ( lJ , lS ) satisfying Eq. (6), out-degrees in the two layers, l and l . This asym- drawn as a blue region with a thick, solid boundary] con- J S metry of the cascade region results from the asymmetric tains the junior-only and senior-only cascade conditions given role that junior and senior loans play in the response by Eqs. (7) and (8) and drawn as an orange region (with a function [Eq. (1)]. Given this asymmetry, what ratio of dashed boundary) and green region (with a dotted boundary), respectively. Moreover, the multiplex cascade region is asym- junior and senior loans is optimal in some sense? (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) metric in the densities of junior and senior loans, ( l , l ). Suppose that the average total degree l + l is J S J S The optimal seniority ratio, deﬁned in Eq. (9), is drawn as a kept constant and that we tune the fraction of loans (cid:104) (cid:105) (cid:104) (cid:105) ≈ (cid:104) (cid:105) (cid:104) (cid:105) black line. This ratio l / l 1.79 minimizes the cascade S J that are junior. For some loan densities l + l , J S window, the set of mean degrees satisfying the ﬁrst-order cas- the multiplex cascade condition (6) holds no matter the cade condition, Eq. (6). The case of equally many junior and f{ra(cid:104)cti(cid:105)on(cid:104) o(cid:105)f junior lo(cid:104)an(cid:105)s. F(cid:104)or(cid:105)e}xample, the line segment senior loans (on average) is shown with a dashed line, which ( l , l ) : 4 = l + l is contained within the has a slightly larger cascade window. (b) The cascade win- J S J S multiplex cascade region in Fig. 3(a), so defaults spread dow as a function of the ratio of senior loans to junior loans, (cid:104) (cid:105) (cid:104) (cid:105) ≈ widely no matter the composition of seniority levels. In lS / lJ , achieves a minimum size at 1.79. other cases, such as the line segment of mean degrees (cid:104) (cid:105) (cid:104) (cid:105) satisfying 7 = l + l in Fig. 3(a), the cascade condi- J S tion (6) holds if and only if the fraction of junior loans is very small or somewhat large. In cases like this one, economic boom and decrease during a recession, yet the having an intermediate fraction of junior debts prevents fraction of loans that are senior does not change. Thus, defaults from spreading widely. we are considering rays through the origin in Fig. 3(a). (cid:104) (cid:105) (cid:104) (cid:105) ≥ Now suppose that we can set the ratio l / l of For “seniority ratios” σ 0, deﬁne the cascade win- S J (cid:104) (cid:105) (cid:104) (cid:105) the average numbers of senior and junior loans per bank, dow to be the set of average degrees ( l , l ) such that (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) S J but the total density of loans l + l can vary. For l / l = σ and such that the ﬁrst-order cascade condi- S S S J instance, the number of loans may increase during an tion (6) holds. Figure 3(b) shows this cascade window as ##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## ##LTLine## ##LTLine##

font: default

text: λ

text: capturing

text: defaults

text: caused

text: by

text: two

text: bankrupt

text: creditors

text: (using

text: the

text: second-order

text: cascade

text: condition)

text: was

text: not

text: nec-

text: essary

text: to

text: approximate

text: simulations

text: well,

text: as

text: shown

text: in

text: Appendix

text: and

text: in

text: Fig.

text: The

text: parameters

text: satisfying

text: this

text: senior-only

text: cascade

text: con-

text: dition

text: are

text: shown

text: in

text: green

text: in

text: Fig.

text: f.orTthhies

text: same

text: reason

text: that

text: the

text: junior-only

text: cascade

text: regioJn

text: shrinks

text: with

text: lS

text: Multiplex

text: cascades

text: The

text: blue

text: region

text: in

text: Fig.

text: is

text: the

text: multiplex

text: cascade

text: re-

text: gion,

text: the

text: set

text: of

text: average

text: out-degrees

text: l

text: l

text: such

text: that

text: the

text: ﬁrst-order

text: cascade

text: condition

text: is

text: satis-

text: ﬁed.

text: NJot≥e

text: that,{aJs

text: poJinted}

text: out

text: in

text: page

text: we

text: have

text: λmax(

text: max

text: JJ,

text: SS

text: Thus,

text: a

text: suﬃcient

text: condition

text: for

text: the

text: multiplex

text: cascade

text: condition

text: to

text: hold

text: is

text: that

text: at

text: least

text: one

text: kind

text: of

text: default

text: α

text: J,

text: S

text: is

text: supercritical

text: (meaning

text: J

text: that

text: αα

text: even

text: in

text: the

text: absence

text: of

text: the

text: other

text: kind

text: of

text: default.

text: Thus,

text: the

text: blue

text: region

text: in

text: Fig.

text: contains

text: both

text: the

text: orange

text: and

text: green

text: regions.

text: Importantly,

text: however,

text: the

text: blue

text: region

text: contains

text: pairs

text: of

text: lrJeg,iolnSs.)

text: oInuttshidiseptharetjuonf

text: ipoar--

text: SS

text: but

text: a

text: large

text: cascade

text: of

text: defaults

text: can

text: neverthe-

text: less

text: occur.

text: These

text: cascades,

text: akin

text: to

text: the

text: “multiplexity-

text: facilitated

text: cascades”

text: in

text: result

text: from

text: the

text: interac-

text: tion

text: of

text: junior-

text: and

text: senior-defaults.

text: C.

text: Optimal

text: seniority

text: ratio

text: Note

text: that

text: the

text: multiplex

text: cascade

text: region,

text: show

text: in

text: blue

text: FIG.

text: (Color

text: online)

text: (a)

text: The

text: multiplex

text: cascade

text: region

text: in

text: Fig.

text: is

text: not

text: symmetric

text: the

text: average

text: [the

text: edge-density

text: parameters

text: lJ

text: lS

text: satisfying

text: Eq.

text: out-degrees

text: in

text: the

text: two

text: layers,

text: l

text: and

text: l

text: This

text: asym-

text: drawn

text: as

text: a

text: blue

text: region

text: with

text: a

text: thick,

text: solid

text: boundary]

text: con-

text: J

text: S

text: metry

text: of

text: the

text: cascade

text: region

text: results

text: from

text: the

text: asymmetric

text: tains

text: the

text: junior-only

text: and

text: senior-only

text: cascade

text: conditions

text: given

text: role

text: that

text: junior

text: and

text: senior

text: loans

text: play

text: in

text: the

text: response

text: by

text: Eqs.

text: and

text: and

text: drawn

text: as

text: an

text: orange

text: region

text: (with

text: a

text: function

text: [Eq.

text: Given

text: this

text: asymmetry,

text: what

text: ratio

text: of

text: dashed

text: boundary)

text: and

text: green

text: region

text: (with

text: a

text: dotted

text: boundary),

text: respectively.

text: Moreover,

text: the

text: multiplex

text: cascade

text: region

text: is

text: asym-

text: junior

text: and

text: senior

text: loans

text: is

text: optimal

text: in

text: some

text: sense?

text: metric

text: in

text: the

text: densities

text: of

text: junior

text: and

text: senior

text: loans,

text: l

text: l

text: Suppose

text: that

text: the

text: average

text: total

text: degree

text: l

text: l

text: is

text: J

text: S

text: J

text: S

text: The

text: optimal

text: seniority

text: ratio,

text: deﬁned

text: in

text: Eq.

text: is

text: drawn

text: as

text: a

text: kept

text: constant

text: and

text: that

text: we

text: tune

text: the

text: fraction

text: of

text: loans

text: black

text: line.

text: This

text: ratio

text: l

text: l

text: minimizes

text: the

text: cascade

text: S

text: J

text: that

text: are

text: junior.

text: For

text: some

text: loan

text: densities

text: l

text: l

text: J

text: S

text: window,

text: the

text: set

text: of

text: mean

text: degrees

text: satisfying

text: the

text: ﬁrst-order

text: cas-

text: the

text: multiplex

text: cascade

text: condition

text: holds

text: no

text: matter

text: the

text: cade

text: condition,

text: Eq.

text: The

text: case

text: of

text: equally

text: many

text: junior

text: and

text: junior

text: the

text: line

text: segment

text: senior

text: loans

text: (on

text: average)

text: is

text: shown

text: with

text: a

text: dashed

text: line,

text: which

text: l

text: l

text: l

text: l

text: is

text: contained

text: within

text: the

text: has

text: a

text: slightly

text: larger

text: cascade

text: window.

text: (b)

text: The

text: cascade

text: win-

text: J

text: S

text: J

text: S

text: multiplex

text: cascade

text: region

text: in

text: Fig.

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text: 7 (cid:104) (cid:105) (cid:104) (cid:105) a function of the seniority ratio l / l . Note that the How does the cascade region (and its optimal senior- S J size of the cascade window is minimized at a unique value ity ratio) change as the banks’ threshold R changes? (cid:104) (cid:105) (cid:104) (cid:105) J of l / l , which is approximately 1.79 in Fig. 3. We Here we keep the assumption that banks all have the S J deﬁne the optimal seniority ratio to be this minimizer, same junior-default threshold R , but we vary R . The J J or more precisely, multiplex network is still a pair of uncorrelated Erd˝os- (cid:104) (cid:105) (cid:104) (cid:105) R´enyi random graphs with mean degrees l and l . J S (cid:104) (cid:105) If banks have a smaller capital ratio R , then default be- J l |{ ≥ J ≥ }| (cid:104) S(cid:105) := arg min r 0 : λ ( ) 1 (9a) comes more likely, so the multiplex cascade region grows, lJ optimal σ≥0 max as shown in Fig. 4(a). Conversely, as RJ increases, the cascade region shrinks, and it can even splinter into two | · | regions [see the purple regions for R = 0.25 in Fig. 4(a)]. where r is the distance from the origin, denotes J R J In cases like this one with two disjoint cascade regions, Lebesgue measure on , and λ ( ) is evaluated at max the optimal ratio deﬁned in Eq. (9) does not exist because { ≥ J ≥ } the set r 0 : λ ( ) 1 is empty for some σ. [A (cid:104) (cid:105) √ r (cid:104) (cid:105) √ rσ max l = , l = , (9b) sensible deﬁnition for the optimal seniority ratio could J S 1 + σ2 1 + σ2 be the slope that maximizes the distance from the ray through the origin (with that slope) to the two cascade where we have changed from polar to Cartesian coor- regions.] (cid:55)→ (cid:104) (cid:105) (cid:104) (cid:105) dinates via (r, arctan σ) ( l , l ). If this optimal The asymmetry between junior and senior loans also (cid:104) (cid:105) (cid:104) (cid:105) J S seniority ratio ( lS / lJ )optimal is greater than one (as changes with the junior-default threshold RJ. In particu- in Fig. 3), then senior debts should be issued more than lar, the optimal seniority ratio increases as RJ decreases, junior debts (on average) for the sake of minimizing the as shown in Fig. 4(b). size of the set of network densities such that bankrupt- In some ﬁnancial crises, many banks ﬁnd themselves cies spread widely. That is, if a ﬁnancial network keeps in ﬁnancial distress, because of losses in external assets, approximately the same ratio of senior to junior loans for instance. The model studied here approximates this (cid:104) (cid:105) (cid:104) (cid:105) ( l / l ) but changes the total number of loans (say, scenario as a small junior-default threshold R for every S J J over the course of a business cycle), then there should be bank (and a small fraction of banks initially in default). more senior loans than junior loans in order to minimize Decreasing R enlarges the cascade region, as illustrated J the risk of a large cascade of defaults. in Fig. 4(a). However, the banks can reduce systemic Intuitively, senior loans are stabilizing because they are risk if they increase, on average, the fraction of debts less signiﬁcant early in a cascade: a node in junior-default that are senior. Figure 4(b) shows the optimal amount causes losses (of assets) for its lenders (in-neighbors) in of that increase. the junior layer only. Thus, changing some loans from ju- III. MORE THAN TWO SENIORITY LEVELS nior to senior makes large cascades less likely, and hence it shrinks the cascade window. However, not all loans Now we show that mixing among several seniority should be senior, because that case is identical to all levels can further reduce systemic risk (in the sense of loans being junior, which has a dangerously large cas- shrinking the cascade region). cade window [look at the horizontal and vertical axes of ≥ We generalize the model to M 1 seniority levels. Fig. 3(a)]. In between those extremes lies an optimal ra- Let (1, 2, . . . , M) denote the indices of the seniority lev- tio of senior to junior loans. That optimal ratio exceeds els in increasing order of seniority, and let R denote one [the dashed black 45-degree line in Fig. 3(a)] because 1 the threshold for default at the most junior layer. The senior loans are less contagious, in the sense that nodes response function for default at level i, derived in Ap- in junior-default can still repay their senior debts. pendix , is In Fig. 7 of Appendix , we show that the three types of (cid:40) (cid:80) (cid:80) (cid:80) cascade regions shown in Fig. 3(a) (junior-only, senior- only, and multiplex) closely agree with the ﬁxed point − − of the recursion equations (4) and with numerical sim- F ((cid:126)l,(cid:126)b, m(cid:126) ) = 1 if Ms=1 ms ik=11 bk > R1 Ms=1 ls . i ulations of the model for uncorrelated Erd˝os-R´enyi 0 otherwise multiplex networks. These comparisons also show that (cid:80) (10) within the multiplex cascade region many nodes end up in junior-default, but many nodes are in senior-default only when senior loans greatly outnumber junior loans. By convention, sums of the form 0 are zero. k=1 In practice, the number of defaults of any type is of pri- A. First-order cascade condition mary concern, so our deﬁnition of the optimal seniority ratio in Eq. (9) ignores types of defaults. The cascade condition, a straightforward generaliza- D. Eﬀect of varying the thresholds (ratio of equity tion of the one for M = 2 layers in Sec. , is that the to interbank assets) Jacobian matrix J M ∈ RM×M has largest eigenvalue ex- J ceeding one. The (i, j) entry of M is ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: a

text: function

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text: the

text: seniority

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text: cascade

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text: banks’

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text: optimal

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text: or

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text: sense

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text: In

text: Fig.

text: of

text: Appendix

text: we

text: show

text: that

text: the

text: three

text: types

text: of

text: cascade

text: regions

text: shown

text: in

text: Fig.

text: (junior-only,

text: senior-

text: only,

text: and

text: multiplex)

text: closely

text: agree

text: with

text: the

text: ﬁxed

text: point

text: of

text: the

text: recursion

text: equations

text: and

text: with

text: numerical

text: sim-

text: F

text: if

text: ms

text: bk

text: ls

text: i

text: ulations

text: of

text: the

text: model

text: for

text: uncorrelated

text: Erd˝os-R´enyi

text: otherwise

text: multiplex

text: networks.

text: These

text: comparisons

text: also

text: show

text: that

text: within

text: the

text: multiplex

text: cascade

text: region

text: many

text: nodes

text: end

text: up

text: in

text: junior-default,

text: but

text: many

text: nodes

text: are

text: in

text: senior-default

text: only

text: when

text: senior

text: loans

text: greatly

text: outnumber

text: junior

text: loans.

text: By

text: convention,

text: sums

text: of

text: the

text: form

text: are

text: zero.

text: In

text: practice,

text: the

text: number

text: of

text: defaults

text: of

text: any

text: type

text: is

text: of

text: pri-

text: A.

text: First-order

text: cascade

text: condition

text: mary

text: concern,

text: so

text: our

text: deﬁnition

text: of

text: the

text: optimal

text: seniority

text: ratio

text: in

text: Eq.

text: ignores

text: types

text: of

text: defaults.

text: The

text: cascade

text: condition,

text: a

text: straightforward

text: generaliza-

text: D.

text: Eﬀect

text: of

text: varying

text: the

text: thresholds

text: (ratio

text: of

text: equity

text: tion

text: of

text: the

text: one

text: for

text: M

text: layers

text: in

text: Sec.

text: is

text: that

text: the

text: to

text: interbank

text: assets)

text: Jacobian

text: matrix

text: J

text: M

text: RM×M

text: has

text: largest

text: eigenvalue

text: ex-

text: J

text: ceeding

text: one.

text: The

text: (i,

text: j)

text: entry

text: of

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text: - ������ ��� ����� ���� ϕ 10 ��������� � as the default of a large bank. We instead take an 10 ��� approach typical in physics of studying ensembles of ��� ﬁnancial networks [1, 21]. We show that cascades on 8 ��� networks with diﬀerent seniority may be analyzed with 6 ��� a variant of multiplex threshold cascade models studied ��� in physics [18–20]. 4 � ϕ The seniority of debt is just one dimension of multi- 0.0 0.2 0.4 0.6 0.8 1.0 plexity in ﬁnancial networks. Other dimensions of debt ��������� � include the maturity of loans and whether loans are 10 ��� collateralized [8, 9]. Furthermore, other important in- 8 ��� teractions include the trade of ﬁnancial assets [43] and ��� common exposures (i.e., multiple institutions holding the 6 ��� same ﬁnancial assets) [9, 44]. 4 ��� We call our model “multilevel contagion” because a � node can become “contagious” in one layer without nec- 0.0 0.2 0.4 0.6 0.8 1.0 �������� �� ����� ���� ��� ������ essarily becoming contagious on another layer (e.g., a bank can be in junior-default but not in senior-default). By contrast, in other models of threshold contagion in FIG. 6. (Color online) For heavy-tailed degree dis- multiplex networks [18–20] and in variants of percola- tributions like those in the Brazilian interbank lending net- ≈ tion on multiplex networks [13–17], when a node changes work [28], having 30% of loans be junior (and the rest state in one layer, it necessarily changes state in other senior) reduces systemic risk, in that it minimizes the size layers. Multilevel contagion might give insight into other of the set of mean total out-degrees such that bankruptcy kinds of social contagion. For example, one might use it spreads widely. The gray-scale background shows the average fraction of banks in junior-default (top panel) and in senior- to model the adoption of a new product if “fans” of the default (bottom panel) in simulations on 2400 banks, begun product spread inﬂuence to neighbors of one type but from 10 banks initially in senior-default, averaged over 30 “super-fans” spread inﬂuence to neighbors of two types simulations. The green lines mark the boundary of the cas- (a multiplex version of the multistage complex contagion J J cade region [λmax( ) = 1 with computed from Eq. (C4)]. model [25]). We hope our model will stimulate further The junior-default threshold of every bank is RJ = 0.18, research on these many types of cascades in social, ﬁnan- as in Fig. 3. Directed graphs are generated with the static cial, and other multiplex networks. model [39, 40] with out-degree distributions having tail be- havior ∝ k−γ with γ = 2.83, which approximates the value ACKNOWLEDGMENTS found in the Brazilian interbank lending network [28]. A frac- tion of loans (horizontal axis) chosen uniformly at random C.D.B. is supported by the James S. McDonnell Post- are junior, and the rest are senior. The “optimal” fraction doctoral Fellowship in Studying Complex Systems. T.K. of junior loans (that minimizes the height of the green line) ≈ ≈ is supported by KAKENHI 25780203 and 24243044. The is 0.30. For the mean out-degree (vertical axis) of 8.5 authors thank the Isaac Newton Institute for Mathemat- observed for Brazilian banks [28], the expected cascade size is small if and only if an intermediate fraction of loans are ical Sciences (Cambridge, UK) for support and hospital- junior. ity during the programme Systemic Risk: Mathematical Modelling and Interdisciplinary Approaches, where work on this paper was undertaken. Appendix A: Thresholds of default for M levels of seniority ≥ Here we consider the general case of M 1 levels of seniority of interbank debt (rather than just two, junior and ∈ { } senior), and we label the levels of seniority by α 1, 2, . . . , M . We choose the convention that debts of seniority α (cid:48) (cid:48) are more senior than debts of seniority α whenever α > α . Recall from Fig. 1 the stylized balance sheet of a bank: a bank’s assets consist of external assets e and l many ∈ { } α interbank loans of seniority α 1, 2, . . . , M , while its liabilities consist of external liabilities d and b many interbank ∈ { } α liabilities of seniority α 1, 2, . . . , M . We assume that the external liabilities d (which could include, for example, demand deposits) are more senior than debts to other ﬁnancial institutions. The equity w is the diﬀerence between (cid:88) assets and liabilities, − M − w = e d + (l (cid:80) b ). α α α=1 (cid:80) Now suppose that this bank has some interbank assets (i.e., M l > 0) and that it suﬀers losses on m of its ∈ { } α=1 α α− l many interbank assets (for α 1, 2, . . . , M ), so that its interbank assets now have total value M (l m ). α α=1 α α Then exactly one of the following events occurs: ##LTFigure## ##LTFigure## ##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##

text: ϕ

text: as

text: the

text: default

text: of

text: a

text: large

text: bank.

text: We

text: instead

text: take

text: an

text: approach

text: typical

text: in

text: physics

text: of

text: studying

text: ensembles

text: of

text: ﬁnancial

text: networks

text: We

text: show

text: that

text: cascades

text: on

text: networks

text: with

text: diﬀerent

text: seniority

text: may

text: be

text: analyzed

text: with

text: a

text: variant

text: of

text: multiplex

text: threshold

text: cascade

text: models

text: studied

text: in

text: physics

text: ϕ

text: The

text: seniority

text: of

text: debt

text: is

text: just

text: one

text: dimension

text: of

text: multi-

text: plexity

text: in

text: ﬁnancial

text: networks.

text: Other

text: dimensions

text: of

text: debt

text: include

text: the

text: maturity

text: of

text: loans

text: and

text: whether

text: loans

text: are

text: collateralized

text: Furthermore,

text: other

text: important

text: in-

text: teractions

text: include

text: the

text: trade

text: of

text: ﬁnancial

text: assets

text: and

text: common

text: exposures

text: (i.e.,

text: multiple

text: institutions

text: holding

text: the

text: same

text: ﬁnancial

text: assets)

text: We

text: call

text: our

text: model

text: “multilevel

text: contagion”

text: because

text: a

text: node

text: can

text: become

text: “contagious”

text: in

text: one

text: layer

text: without

text: nec-

text: essarily

text: becoming

text: contagious

text: on

text: another

text: layer

text: (e.g.,

text: a

text: bank

text: can

text: be

text: in

text: junior-default

text: but

text: not

text: in

text: senior-default).

text: By

text: contrast,

text: in

text: other

text: models

text: of

text: threshold

text: contagion

text: in

text: FIG.

text: (Color

text: online)

text: For

text: heavy-tailed

text: degree

text: dis-

text: multiplex

text: networks

text: and

text: in

text: variants

text: of

text: percola-

text: tributions

text: like

text: those

text: in

text: the

text: Brazilian

text: interbank

text: lending

text: net-

text: tion

text: on

text: multiplex

text: networks

text: when

text: a

text: node

text: changes

text: work

text: having

text: of

text: loans

text: be

text: junior

text: (and

text: the

text: rest

text: state

text: in

text: one

text: layer,

text: it

text: necessarily

text: changes

text: state

text: in

text: other

text: senior)

text: reduces

text: systemic

text: risk,

text: in

text: that

text: it

text: minimizes

text: the

text: size

text: layers.

text: Multilevel

text: contagion

text: might

text: give

text: insight

text: into

text: other

text: of

text: the

text: set

text: of

text: mean

text: total

text: out-degrees

text: such

text: that

text: bankruptcy

text: kinds

text: of

text: social

text: contagion.

text: For

text: example,

text: one

text: might

text: use

text: it

text: spreads

text: widely.

text: The

text: gray-scale

text: background

text: shows

text: the

text: average

text: fraction

text: of

text: banks

text: in

text: junior-default

text: (top

text: panel)

text: and

text: in

text: senior-

text: to

text: model

text: the

text: adoption

text: of

text: a

text: new

text: product

text: if

text: “fans”

text: of

text: the

text: default

text: (bottom

text: panel)

text: in

text: simulations

text: on

text: banks,

text: begun

text: product

text: spread

text: inﬂuence

text: to

text: neighbors

text: of

text: one

text: type

text: but

text: from

text: banks

text: initially

text: in

text: senior-default,

text: averaged

text: over

text: “super-fans”

text: spread

text: inﬂuence

text: to

text: neighbors

text: of

text: two

text: types

text: simulations.

text: The

text: green

text: lines

text: mark

text: the

text: boundary

text: of

text: the

text: cas-

text: (a

text: multiplex

text: version

text: of

text: the

text: multistage

text: complex

text: contagion

text: J

text: J

text: cade

text: region

text: [λmax(

text: with

text: computed

text: from

text: Eq.

text: model

text: We

text: hope

text: our

text: model

text: will

text: stimulate

text: further

text: The

text: junior-default

text: threshold

text: of

text: every

text: bank

text: is

text: RJ

text: research

text: on

text: these

text: many

text: types

text: of

text: cascades

text: in

text: social,

text: ﬁnan-

text: as

text: in

text: Fig.

text: Directed

text: graphs

text: are

text: generated

text: with

text: the

text: static

text: cial,

text: and

text: other

text: multiplex

text: networks.

text: model

text: with

text: out-degree

text: distributions

text: having

text: tail

text: be-

text: havior

text: k−γ

text: with

text: γ

text: which

text: approximates

text: the

text: value

text: ACKNOWLEDGMENTS

text: found

text: in

text: the

text: Brazilian

text: interbank

text: lending

text: network

text: A

text: frac-

text: tion

text: of

text: loans

text: (horizontal

text: axis)

text: chosen

text: uniformly

text: at

text: random

text: C.D.B.

text: is

text: supported

text: by

text: the

text: James

text: S.

text: McDonnell

text: Post-

text: are

text: junior,

text: and

text: the

text: rest

text: are

text: senior.

text: The

text: “optimal”

text: fraction

text: doctoral

text: Fellowship

text: in

text: Studying

text: Complex

text: Systems.

text: T.K.

text: of

text: junior

text: loans

text: (that

text: minimizes

text: the

text: height

text: of

text: the

text: green

text: line)

text: is

text: supported

text: by

text: KAKENHI

text: and

text: The

text: is

text: For

text: the

text: mean

text: out-degree

text: (vertical

text: axis)

text: of

text: authors

text: thank

text: the

text: Isaac

text: Newton

text: Institute

text: for

text: Mathemat-

text: observed

text: for

text: Brazilian

text: banks

text: the

text: expected

text: cascade

text: size

text: is

text: small

text: if

text: and

text: only

text: if

text: an

text: intermediate

text: fraction

text: of

text: loans

text: are

text: ical

text: Sciences

text: (Cambridge,

text: UK)

text: for

text: support

text: and

text: hospital-

text: junior.

text: ity

text: during

text: the

text: programme

text: Systemic

text: Risk:

text: Mathematical

text: Modelling

text: and

text: Interdisciplinary

text: Approaches,

text: where

text: work

text: on

text: this

text: paper

text: was

text: undertaken.

text: Appendix

text: A:

text: Thresholds

text: of

text: default

text: for

text: M

text: levels

text: of

text: seniority

text: Here

text: we

text: consider

text: the

text: general

text: case

text: of

text: M

text: levels

text: of

text: seniority

text: of

text: interbank

text: debt

text: (rather

text: than

text: just

text: two,

text: junior

text: and

text: senior),

text: and

text: we

text: label

text: the

text: levels

text: of

text: seniority

text: by

text: α

text: M

text: We

text: choose

text: the

text: convention

text: that

text: debts

text: of

text: seniority

text: α

text: are

text: more

text: senior

text: than

text: debts

text: of

text: seniority

text: α

text: whenever

text: α

text: α

text: Recall

text: from

text: Fig.

text: the

text: stylized

text: balance

text: sheet

text: of

text: a

text: bank:

text: a

text: bank’s

text: assets

text: consist

text: of

text: external

text: assets

text: e

text: and

text: l

text: many

text: α

text: interbank

text: loans

text: of

text: seniority

text: α

text: M

text: while

text: its

text: liabilities

text: consist

text: of

text: external

text: liabilities

text: d

text: and

text: b

text: many

text: interbank

text: α

text: liabilities

text: of

text: seniority

text: α

text: M

text: We

text: assume

text: that

text: the

text: external

text: liabilities

text: d

text: (which

text: could

text: include,

text: for

text: example,

text: demand

text: deposits)

text: are

text: more

text: senior

text: than

text: debts

text: to

text: other

text: ﬁnancial

text: institutions.

text: The

text: equity

text: w

text: is

text: the

text: diﬀerence

text: between

text: assets

text: and

text: liabilities,

text: M

text: w

text: e

text: d

text: (l

text: b

text: α

text: α

text: Now

text: suppose

text: that

text: this

text: bank

text: has

text: some

text: interbank

text: assets

text: (i.e.,

text: M

text: l

text: and

text: that

text: it

text: suﬀers

text: losses

text: on

text: m

text: of

text: its

text: α

text: α−

text: l

text: many

text: interbank

text: assets

text: (for

text: α

text: M

text: so

text: that

text: its

text: interbank

text: assets

text: now

text: have

text: total

text: value

text: M

text: (l

text: m

text: α

text: α

text: α

text: Then

text: exactly

text: one

text: of

text: the

text: following

text: events

text: occurs:

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

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text: · · · { · · · } ˆ ˆ with (O , O , ) the quantum averages for the corresponding observables O , O , . 1 2 1 2 Then if we measure only the energy of the system, what is the number of microstates for the energy value E? This often happens for macroscopic systems whose complete set of observables or macro-quantities cannot be accessible easily. According to the previous analysis, macrostates are completely determined by the system’s (initial) quantum states. Thus, although the other macro-quantities are not known to us, the number of microstates is still determined by the number of non-vanishing coeﬃcients in the initial state. In ordinary statistical mechanics [1], a closed system is usually described by a micro- canonical ensemble, in which each “copied” system is at some speciﬁc state in the relevant phase space, with a priori equal probability. Then according to the ergodic hypothesis, the ensemble average of any physical quantity is identical to the long-time average of that quantity through a series of measurements on the system. In a complete quantum me- chanics description, no presumed ensemble is necessary, thus microcanonical ensemble is not suitable for a quantum closed system. However, according to the discussions around Eq. (22), the relevant Hilbert subspace is ergodic under the closed system’s evolution. This {| (cid:105)} indicates that the collection of states φ(t) for all instants may serves as some “micro- canonical quasi-ensemble” for a quantum closed system. Then, according to the above average identiﬁcation, for a quantum closed system, there may be a following relation for (cid:90) ˆ an observable O 1 T (cid:104) | | (cid:105) ˆ ˆ T r(ρˆ O) = lim dt φ(t) O φ(t) . (28) en →∞ T T 0 (cid:90) ρˆ is the density matrix for the “microcanonical (cid:88)quasi-ensemble” en 1 T | (cid:105)(cid:104) | → | | | (cid:105)(cid:104) | ρˆ = lim dt φ(t) φ(t) α 2 E E , (29) e(cid:80)n →∞ n n n T T 0 n | (cid:105) | (cid:105) where φ(0) = α E in Eq. (21) is used. Easily to see, the factor 1/T plays n n n N the role of equal probability, just like 1/ in Eq. (1). In addition to the long-time av- erage, there is also a quantum average that should be done for each instant state. For the macro-quantity energy (E), the above identiﬁcation is applicable, in particular the reduced density matrix in Eq. (29) is identical to the one in Eq. (21) for the measurement ˆ outcomes. However, if we measure a quantity with an observable O noncommutative with (cid:54) ˆ ˆ the Hamiltonian, i.e. [O, H] = 0, the evolution of the closed system will be destroyed 18 ##LTLine## ##LTLine##

text: ˆ

text: ˆ

text: with

text: (O

text: O

text: the

text: quantum

text: averages

text: for

text: the

text: corresponding

text: observables

text: O

text: O

text: Then

text: if

text: we

text: measure

text: only

text: the

text: energy

text: of

text: the

text: system,

text: what

text: is

text: the

text: number

text: of

text: microstates

text: for

text: the

text: energy

text: value

text: E?

text: This

text: often

text: happens

text: for

text: macroscopic

text: systems

text: whose

text: complete

text: set

text: of

text: observables

text: or

text: macro-quantities

text: cannot

text: be

text: accessible

text: easily.

text: According

text: to

text: the

text: previous

text: analysis,

text: macrostates

text: are

text: completely

text: determined

text: by

text: the

text: system’s

text: (initial)

text: quantum

text: states.

text: Thus,

text: although

text: the

text: other

text: macro-quantities

text: are

text: not

text: known

text: to

text: us,

text: the

text: number

text: of

text: microstates

text: is

text: still

text: determined

text: by

text: the

text: number

text: of

text: non-vanishing

text: coeﬃcients

text: in

text: the

text: initial

text: state.

text: In

text: ordinary

text: statistical

text: mechanics

text: a

text: closed

text: system

text: is

text: usually

text: described

text: by

text: a

text: micro-

text: canonical

text: ensemble,

text: in

text: which

text: each

text: “copied”

text: system

text: is

text: at

text: some

text: speciﬁc

text: state

text: in

text: the

text: relevant

text: phase

text: space,

text: with

text: a

text: priori

text: equal

text: probability.

text: Then

text: according

text: to

text: the

text: ergodic

text: hypothesis,

text: the

text: ensemble

text: average

text: of

text: any

text: physical

text: quantity

text: is

text: identical

text: to

text: the

text: long-time

text: average

text: of

text: that

text: quantity

text: through

text: a

text: series

text: of

text: measurements

text: on

text: the

text: system.

text: In

text: a

text: complete

text: quantum

text: me-

text: chanics

text: description,

text: no

text: presumed

text: ensemble

text: is

text: necessary,

text: thus

text: microcanonical

text: ensemble

text: is

text: not

text: suitable

text: for

text: a

text: quantum

text: closed

text: system.

text: However,

text: according

text: to

text: the

text: discussions

text: around

text: Eq.

text: the

text: relevant

text: Hilbert

text: subspace

text: is

text: ergodic

text: under

text: the

text: closed

text: system’s

text: evolution.

text: This

text: indicates

text: that

text: the

text: collection

text: of

text: states

text: φ(t)

text: for

text: all

text: instants

text: may

text: serves

text: as

text: some

text: “micro-

text: canonical

text: quasi-ensemble”

text: for

text: a

text: quantum

text: closed

text: system.

text: Then,

text: according

text: to

text: the

text: above

text: average

text: identiﬁcation,

text: for

text: a

text: quantum

text: closed

text: system,

text: there

text: may

text: be

text: a

text: following

text: relation

text: for

text: ˆ

text: an

text: observable

text: O

text: T

text: ˆ

text: ˆ

text: T

text: r(ρˆ

text: O)

text: lim

text: dt

text: φ(t)

text: O

text: φ(t)

text: en

text: T

text: T

text: ρˆ

text: is

text: the

text: density

text: matrix

text: for

text: the

text: “microcanonical

text: en

text: T

text: ρˆ

text: lim

text: dt

text: φ(t)

text: φ(t)

text: α

text: E

text: E

text: n

text: n

text: n

text: T

text: T

text: n

text: where

text: α

text: E

text: in

text: Eq.

text: is

text: used.

text: Easily

text: to

text: see,

text: the

text: factor

text: plays

text: n

text: n

text: n

text: N

text: the

text: role

text: of

text: equal

text: probability,

text: just

text: like

text: in

text: Eq.

text: In

text: addition

text: to

text: the

text: long-time

text: av-

text: erage,

text: there

text: is

text: also

text: a

text: quantum

text: average

text: that

text: should

text: be

text: done

text: for

text: each

text: instant

text: state.

text: For

text: the

text: macro-quantity

text: energy

text: (E),

text: the

text: above

text: identiﬁcation

text: is

text: applicable,

text: in

text: particular

text: the

text: reduced

text: density

text: matrix

text: in

text: Eq.

text: is

text: identical

text: to

text: the

text: one

text: in

text: Eq.

text: for

text: the

text: measurement

text: ˆ

text: outcomes.

text: However,

text: if

text: we

text: measure

text: a

text: quantity

text: with

text: an

text: observable

text: O

text: noncommutative

text: with

text: ˆ

text: ˆ

text: the

text: Hamiltonian,

text: i.e.

text: [O,

text: H]

text: the

text: evolution

text: of

text: the

text: closed

text: system

text: will

text: be

text: destroyed

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