text: 5 (cid:88) (cid:89) network size, but because the density of triangles decreases for larger L. Moreover, the results of bond percolation on the S =1 − N1 (cid:88)i (cid:89)j Aijuij, (2) wL−it3h-ctlhoenendumveerrsiicoanlscoufrvγe(3fo, 3r )thneetowriogrikna(lnontetswhoowrknr)ecgoairndcliedses of the number of layers. This is due to the fact that trian- − 1 S = p 1 N i j Aijuij . (3) gLl−e3s-calroentihneg okneelypsshtroiartnlgoloespsinptarecst.enHteinncγe,(3L,−33)-cnleotnweodrkγs(3a,n3d) networks have eﬀectively the same structure as the original γ(3, 3) networks, i.e., triangles randomly connected via sin- Equation (2) resembles Eq. (24) of Ref. [19], which is de- gle edges (see Fig. 3). These observations conﬁrm that elim- rived for the late time behavior of the susceptible-infected re- inating the short loops in the γ(3, 3) network, by L-cloning, covered model using the message passing approach and is the transforms its structure to a treelike network for which the same as Eq. (7) of Ref. [14]. This theory is an example of theory is designed, and that the diﬀerence between the theory an A theory which uses the information on the exact con- ij and the numerics on the original γ(3, 3) network is mainly due nectivity of pairs of nodes represented in the network adja- to the triangles present in the network. cency matrix. If we assume that the edges leaving nodes with Figure 4(b) shows the bond percolation results for the degree k have approximately the same probability u of not k power grid network of the western United States [18] and its leading to the GCC, then Eqs. (1) and (2) can be reduced to a L-cloned versions. It has been previously observed that mean- degree-based approximation as described by Eqs. (1) and (2) ﬁeld theories are very inaccurate for this network [10, 11], an of Ref. [21]: (cid:88) example of which is P (cid:48) theory for bond percolation which k,k employs Eqs. (4) and (5) [10]. Figure 4(b) shows that A ij uk = 1 − (cid:88)p + p P(k(cid:48)|k)(uk(cid:48))k(cid:48)−1 , (4) theory is also inaccurate for this network; however, as L is (cid:48) increased, the numerical result for the L-cloned version of the k − S = 1 P(k)(uk)k , (5) k 1 (cid:48)| (a) where P(k) is the degree distribution, and P(k k) is the con- 0.8 ditional probability that a neighbor of a degree-k node is a L=1 (cid:48) 1 degree-k node. This assumption is correct when the degree L=2 0.6 oimf anteeidghubsoinrgs othfeacnoonddeitiwointahlapnryobdaebgirlietey;ctahnisbieswtheellcaapspe,rofxo-r S LL==1500 00..68 NN==110044,, LL==13 example, for conﬁguration model and Pk,k(cid:48)-rewired networks 0.4 Aij theory S 0.4 NN==15×014,0 L4,= L5=01 [h1a0v]e. thIne sthaemaebpsreonbcaeboilfitydeogfreleea-ddeinggreteoctohrereGlaCtiCona,nadllEeqd.g(e1s) 0.2 0.02 NN==55××110044,, LL==350 0.5 0.6 0.7 0.8 0.9 1 is further reduced to Eq. (16.4) or (17.27) of Ref. [1] and p 0 Eqs. (2) and (3) are reduced to Eqs. (16.2) and (17.28) of [1] 0.5 0.6 0.7 0.8 0.9 1 p respectively. − Although Eqs. (1) (3) are more general and precise than 1 their aforementioned reductions, they still employ the as- (b) sumption that the network is locally treelike. Hence, they can 0.8 be inaccurate for clustered networks [10, 11]. L=1 L=2 To inspect the role of short loops in the accuracy of the 0.6 L=20 aforementioned A theory we consider synthetic and real- S ij P theory world clustered networks introduced in Sec. and apply L- 0.4 kk’ A theory cloning to reduce clustering while preserving degree distribu- ij tion and degree-degree correlation between and beyond near- 0.2 est neighbors. In Fig. 4(a), we show how the size of the GCC, in bond percolation, depends on p for the γ(3, 3) network and 00. 2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 its L-cloned versions with diﬀerent L, and compare the nu- p merical results with the theoretical predictions of the A the- ij ory from Eqs. (1) and (2). The theory does not make an ac- FIG. 4. (Color online) Bond percolation on (a) a γ(3,3) network with approximately 104 nodes, and (b) the US power grid network curate prediction for the original network (L = 1). However, and their respective L-cloned networks. Here L = 1 indicates the when L is increased the numerical curves approach the result result of numerical simulations on the original network. The result of the A theory for the original network. It is worth mention- ij of the A theory is shown by the dashed line. The result of numerical ing that the result of the A theory is the same for the original ij ij simulation on L-cloned networks approaches the A theory as L is ij network and any of its L-cloned versions. increased. On the US power grid network the prediction made by In the inset of Fig. 4(a), we illustrate the eﬀect of L-cloning the A theory is diﬀerent from and more accurate than that of the ij on two γ(3, 3) networks with diﬀerent sizes. Since the curves Pk,k(cid:48) theory studied in Refs. [10, 21]. Nevertheless, the two theories in the inset that correspond to the same value of L match each have the same result on γ(3,3) networks. The numerical results were other, we conclude that the shift towards the A theory in the derived from averaging over at least 100 realizations of the bond ij main Fig. 4(a) with increasing L is not due to the change in the percolation process. ##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## ##LTLine## ##LTLine##

font: VMJLKA+txex

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text: 6 network approaches the prediction of the Aij theory. We con- 1 clude that the Aij theory for bond percolation will be accurate L=1 for L-cloned networks with suﬃciently large L. 0.8 L =2 −3 We also run numerical simulations of bond percolation on L =100 −3,−4 L−3- and L−3,−4-cloned versions of the power grid network. 0.6 L−3=50 In Fig. 5 we show that the diﬀerence between the Aij theory S L =100 −3 and numerics on L− -cloned networks decreases as we start 0.4 3 L =100 to increase the number of layers L− . However, the numerical f(3,4) 3 A theory results on L− -cloned networks do not approach the A the- 0.2 ij 3 ij ory curve even for very large L− . Moreover, the numerical 3 results on L− − -cloned networks diﬀer from the A theory 0 3, 4 ij 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 even more than the results on L− -cloned networks. p 3 Due to the fact that in L - and L -cloned networks short 3 3,4 FIG. 5. (Color online) For a large number of layers, the simulation loops of speciﬁed lengths are preserved, to create these net- results of bond percolation on L− -cloned versions of the western US works only the edges that are not part of such loops (hence 3 power grid [18] network converges to a curve that has an apprecia- only a fraction of the total number of edges) are reassigned ble diﬀerence from the A theory. This indicates that the presence (rewired), while in L-cloned networks all edges are rewired. ij of triangles causes a signiﬁcant error in the theoretical prediction. For example, in the power grid network, 79% of links are Similarly the numerics for L− − -cloned networks have an apprecia- not part of loops of length 3 and 67% of links are not part ble deviation from the theory3;,th4is deviation is even larger than that of loops of length 3 or 4. By running simulations on Lf - of L−3-cloned networks, indicating that squares have also an appre- cloned networks, we show that if the same fraction of links, ciable eﬀect on the accuracy of the theory. These deviations are not as in L− − -cloned networks, for example, are randomly se- due to the smaller number of links rewired in L− and L− − cloned 3, 4 3 3, 4 lected and rewired between the layers, the numerical results networks; since the numerics on Lf(3,4) cloned networks (in which are appreciably closer to the theoretical prediction, compared the fraction of rewired links is as low as that of L−3,−4-cloned net- works) with a large number of layers matches the theory very well. to the numerical results for the L− − -cloned networks. This 3, 4 The number of layers is indicated by L for each curve and the index implies that the preserved short loops, and not the smaller of L denotes the type of cloning. fraction of rewired links, is the reason that the numerics for L− − -cloned (and similarly for L− -cloned) networks do not 3, 4 3 match the A theory even for large number of layers. ij a pair approximation method and assuming that the network In Fig. 5 the numerical results for L -cloned power grid f(3,4) is treelike [see Eqs. (17.5(cid:88)4) and (17.55) of Ref. [1]]: are compared with the results for other variants of L-cloning. It is observed that the result for L -cloned networks, with a f dS − loarryg.eTnhuims binedriocfatleasyetrhsaLt efv, menatwchitehsfveewryerwreelwl iwreitdhltihneksA,isj itnhcee- dti = βSi Aijpij , (cid:88) (6) j the clustered structure is eﬀectively destroyed for large L , f dp − − the numerical results approach the adjacency tree-based the- dtij = β(1 pij)[ pij + Ajkpjk] , (7) ory. Similar results (not shown) were observed for site perco- k(cid:44)i lation. (cid:80) These results demonstrate that triangles and also squares where p is the conditional probability that node j is in- ij play a signiﬁcant role in the inaccurate prediction of the A fected given that node i is susceptible. Then the fraction of ij − theory for the power grid network. On the other hand, as the the population that is infected is just I = 1 S . Since i i numerical results for L− − -cloned networks are closer to the Eq. (7) assumes that the network is locally treelike, we ex- 3, 4 theory than the numerics for the original network, there must pect the predictions made by Eqs. (6) and (7) to be inaccu- exist other sources of error too. One source is the ﬁnite size rate for clustered networks. Although there are cases where eﬀect,2 as the original network has a size smaller than any of the theoretical results and actual behavior match quite well, its cloned versions. Other short loops with lengths larger than such as in Fig. (17.5) of Ref. [1], the accuracy of prediction 4 may also have an eﬀect on the accuracy of the A theory. of Eq. (6) is not generally guaranteed for clustered networks. ij B. SI epidemic model We demonstrate this in Fig. 6 by simulations of SI dynamics on the γ(3, 3) and the US power grid networks and their cor- In the Susceptible-Infected (SI) model of spread of disease, responding L-cloned versions: The theoretical predictions for two possible states are considered for a node: It is either in- these two clustered networks are only accurate for suﬃciently fected or susceptible [1]. The quantity of interest is the frac- large L. tion of network nodes that are in the infected population I at In order to examine the contribution of short loops of cer- time t after the disease begins spreading from a very small tain length against any other possible source of inaccuracy we fraction of infected nodes. The probability S that node i is in employ the extensions of L-cloning introduced in Secs. and . i the susceptible population at time t can be obtained by using The numerical results of SI dynamics on L−3-cloned versions of the γ(3, 3) networks (not shown) coincide with the results on the original network. This indicates that the presence of triangles is the source of inaccuracy of Aij theory for γ(3, 3) 2 The ﬁnite size of the networks is known to cause deviations from the the- networks. oretical predictions; for example, the ﬁnite-size eﬀect is shown to cause The numerics for the SI dynamics on L− - or L− − -cloned error in the Pi,i(cid:48)(cid:48) theory for binary-state dynamics [22]. versions of the power grid network are clos3er to the3,A4 theory k,k ij ##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##

label: paragraph

text: network

text: approaches

text: the

text: prediction

text: of

text: the

text: Aij

text: theory.

text: We

text: con-

font: default

text: clude

text: that

text: the

text: Aij

text: theory

text: for

text: bond

text: percolation

text: will

text: be

text: accurate

text: for

text: L-cloned

text: networks

text: with

text: suﬃciently

text: large

text: L.

text: L

text: −3

text: We

text: also

text: run

text: numerical

text: simulations

text: of

text: bond

text: percolation

text: on

text: L

text: and

text: versions

text: of

text: the

text: power

text: grid

text: network.

text: In

text: Fig.

text: we

text: show

text: that

text: the

text: diﬀerence

text: between

text: the

text: Aij

text: theory

text: S

text: L

text: −3

text: and

text: numerics

text: on

text: L−

text: -cloned

text: networks

text: decreases

text: as

text: we

text: start

text: L

text: to

text: increase

text: the

text: number

text: of

text: layers

text: L−

text: However,

text: the

text: numerical

text: A

text: theory

text: results

text: on

text: L−

text: -cloned

text: networks

text: do

text: not

text: approach

text: the

text: A

text: the-

text: ij

text: ij

text: ory

text: curve

text: even

text: for

text: very

text: large

text: L−

text: Moreover,

text: the

text: numerical

text: results

text: on

text: L−

text: -cloned

text: networks

text: diﬀer

text: from

text: the

text: A

text: theory

text: ij

text: even

text: more

text: than

text: the

text: results

text: on

text: L−

text: -cloned

text: networks.

text: p

text: Due

text: to

text: the

text: fact

text: that

text: in

text: L

text: and

text: L

text: -cloned

text: networks

text: short

text: FIG.

text: (Color

text: online)

text: For

text: a

text: large

text: number

text: of

text: layers,

text: the

text: simulation

text: loops

text: of

text: speciﬁed

text: lengths

text: are

text: preserved,

text: to

text: create

text: these

text: net-

text: results

text: of

text: bond

text: percolation

text: on

text: L−

text: -cloned

text: versions

text: of

text: the

text: western

text: US

text: works

text: only

text: the

text: edges

text: that

text: are

text: not

text: part

text: of

text: such

text: loops

text: (hence

text: power

text: grid

text: network

text: converges

text: to

text: a

text: curve

text: that

text: has

text: an

text: apprecia-

text: only

text: a

text: fraction

text: of

text: the

text: total

text: number

text: of

text: edges)

text: are

text: reassigned

text: ble

text: diﬀerence

text: from

text: the

text: A

text: theory.

text: This

text: indicates

text: that

text: the

text: presence

text: (rewired),

text: while

text: in

text: L-cloned

text: networks

text: all

text: edges

text: are

text: rewired.

text: ij

text: of

text: triangles

text: causes

text: a

text: signiﬁcant

text: error

text: in

text: the

text: theoretical

text: prediction.

text: For

text: example,

text: in

text: the

text: power

text: grid

text: network,

text: of

text: links

text: are

text: Similarly

text: the

text: numerics

text: for

text: L−

text: -cloned

text: networks

text: have

text: an

text: apprecia-

text: not

text: part

text: of

text: loops

text: of

text: length

text: and

text: of

text: links

text: are

text: not

text: part

text: ble

text: deviation

text: from

text: the

text: deviation

text: is

text: even

text: larger

text: than

text: that

text: of

text: loops

text: of

text: length

text: or

text: By

text: running

text: simulations

text: on

text: Lf

text: of

text: networks,

text: indicating

text: that

text: squares

text: have

text: also

text: an

text: appre-

text: cloned

text: networks,

text: we

text: show

text: that

text: if

text: the

text: same

text: fraction

text: of

text: links,

text: ciable

text: eﬀect

text: on

text: the

text: accuracy

text: of

text: the

text: theory.

text: These

text: deviations

text: are

text: not

text: as

text: in

text: L−

text: -cloned

text: networks,

text: for

text: example,

text: are

text: randomly

text: se-

text: due

text: to

text: the

text: smaller

text: number

text: of

text: links

text: rewired

text: in

text: L−

text: and

text: L−

text: cloned

text: lected

text: and

text: rewired

text: between

text: the

text: layers,

text: the

text: numerical

text: results

text: networks;

text: since

text: the

text: numerics

text: on

text: cloned

text: networks

text: (in

text: which

text: are

text: appreciably

text: closer

text: to

text: the

text: theoretical

text: prediction,

text: compared

text: the

text: fraction

text: of

text: rewired

text: links

text: is

text: as

text: low

text: as

text: that

text: of

text: net-

text: works)

text: with

text: a

text: large

text: number

text: of

text: layers

text: matches

text: the

text: theory

text: very

text: well.

text: to

text: the

text: numerical

text: results

text: for

text: the

text: L−

text: -cloned

text: networks.

text: This

text: The

text: number

text: of

text: layers

text: is

text: indicated

text: by

text: L

text: for

text: each

text: curve

text: and

text: the

text: index

text: implies

text: that

text: the

text: preserved

text: short

text: loops,

text: and

text: not

text: the

text: smaller

text: of

text: L

text: denotes

text: the

text: type

text: of

text: cloning.

text: fraction

text: of

text: rewired

text: links,

text: is

text: the

text: reason

text: that

text: the

text: numerics

text: for

text: L−

text: -cloned

text: (and

text: similarly

text: for

text: L−

text: -cloned)

text: networks

text: do

text: not

text: match

text: the

text: A

text: theory

text: even

text: for

text: large

text: number

text: of

text: layers.

text: ij

text: a

text: pair

text: approximation

text: method

text: and

text: assuming

text: that

text: the

text: network

text: In

text: Fig.

text: the

text: numerical

text: results

text: for

text: L

text: -cloned

text: power

text: grid

text: is

text: treelike

text: [see

text: Eqs.

text: and

text: of

text: Ref.

text: are

text: compared

text: with

text: the

text: results

text: for

text: other

text: variants

text: of

text: L-cloning.

text: It

text: is

text: observed

text: that

text: the

text: result

text: for

text: L

text: -cloned

text: networks,

text: with

text: a

text: f

text: dS

text: loarryg.eTnhuims

text: binedriocfatleasyetrhsaLt

text: efv,

text: menatwchitehsfveewryerwreelwl

text: iwreitdhltihneksA,isj

text: itnhcee-

text: dti

text: βSi

text: Aijpij

text: j

text: the

text: clustered

text: structure

text: is

text: eﬀectively

text: destroyed

text: for

text: large

text: L

text: f

text: dp

text: the

text: numerical

text: results

text: approach

text: the

text: adjacency

text: tree-based

text: the-

text: dtij

text: pij)[

text: pij

text: Ajkpjk]

text: ory.

text: Similar

text: results

text: (not

text: shown)

text: were

text: observed

text: for

text: site

text: perco-

text: lation.

text: These

text: results

text: demonstrate

text: that

text: triangles

text: and

text: also

text: squares

text: where

text: p

text: is

text: the

text: conditional

text: probability

text: that

text: node

text: j

text: is

text: in-

text: ij

text: play

text: a

text: signiﬁcant

text: role

text: in

text: the

text: inaccurate

text: prediction

text: of

text: the

text: A

text: fected

text: given

text: that

text: node

text: i

text: is

text: susceptible.

text: Then

text: the

text: fraction

text: of

text: ij

text: theory

text: for

text: the

text: power

text: grid

text: network.

text: On

text: the

text: other

text: hand,

text: as

text: the

text: the

text: population

text: that

text: is

text: infected

text: is

text: just

text: I

text: S

text: Since

text: i

text: i

text: numerical

text: results

text: for

text: L−

text: -cloned

text: networks

text: are

text: closer

text: to

text: the

text: Eq.

text: assumes

text: that

text: the

text: network

text: is

text: locally

text: treelike,

text: we

text: ex-

text: theory

text: than

text: the

text: numerics

text: for

text: the

text: original

text: network,

text: there

text: must

text: pect

text: the

text: predictions

text: made

text: by

text: Eqs.

text: and

text: to

text: be

text: inaccu-

text: exist

text: other

text: sources

text: of

text: error

text: too.

text: One

text: source

text: is

text: the

text: ﬁnite

text: size

text: rate

text: for

text: clustered

text: networks.

text: Although

text: there

text: are

text: cases

text: where

text: as

text: the

text: original

text: network

text: has

text: a

text: size

text: smaller

text: than

text: any

text: of

text: the

text: theoretical

text: results

text: and

text: actual

text: behavior

text: match

text: quite

text: well,

text: its

text: cloned

text: versions.

text: Other

text: short

text: loops

text: with

text: lengths

text: larger

text: than

text: such

text: as

text: in

text: Fig.

text: of

text: Ref.

text: the

text: accuracy

text: of

text: prediction

text: may

text: also

text: have

text: an

text: eﬀect

text: on

text: the

text: accuracy

text: of

text: the

text: A

text: theory.

text: of

text: Eq.

text: is

text: not

text: generally

text: guaranteed

text: for

text: clustered

text: networks.

text: ij

text: B.

text: SI

text: epidemic

text: model

text: We

text: demonstrate

text: this

text: in

text: Fig.

text: by

text: simulations

text: of

text: SI

text: dynamics

text: on

text: the

text: and

text: the

text: US

text: power

text: grid

text: networks

text: and

text: their

text: cor-

text: In

text: the

text: Susceptible-Infected

text: (SI)

text: model

text: of

text: spread

text: of

text: disease,

text: responding

text: L-cloned

text: versions:

text: The

text: theoretical

text: predictions

text: for

text: two

text: possible

text: states

text: are

text: considered

text: for

text: a

text: node:

text: It

text: is

text: either

text: in-

text: these

text: two

text: clustered

text: networks

text: are

text: only

text: accurate

text: for

text: suﬃciently

text: fected

text: or

text: susceptible

text: The

text: quantity

text: of

text: interest

text: is

text: the

text: frac-

text: large

text: L.

text: tion

text: of

text: network

text: nodes

text: that

text: are

text: in

text: the

text: infected

text: population

text: I

text: at

text: In

text: order

text: to

text: examine

text: the

text: contribution

text: of

text: short

text: loops

text: of

text: cer-

text: time

text: t

text: after

text: the

text: disease

text: begins

text: spreading

text: from

text: a

text: very

text: small

text: tain

text: length

text: against

text: any

text: other

text: possible

text: source

text: of

text: inaccuracy

text: we

text: fraction

text: of

text: infected

text: nodes.

text: The

text: probability

text: S

text: that

text: node

text: i

text: is

text: in

text: employ

text: the

text: extensions

text: of

text: L-cloning

text: introduced

text: in

text: Secs.

text: and

text: i

text: the

text: susceptible

text: population

text: at

text: time

text: t

text: can

text: be

text: obtained

text: by

text: using

text: The

text: numerical

text: results

text: of

text: SI

text: dynamics

text: on

text: versions

text: of

text: the

text: networks

text: (not

text: shown)

text: coincide

text: with

text: the

text: results

text: on

text: the

text: original

text: network.

text: This

text: indicates

text: that

text: the

text: presence

text: of

text: triangles

text: is

text: the

text: source

text: of

text: inaccuracy

text: of

text: Aij

text: theory

text: for

text: The

text: ﬁnite

text: size

text: of

text: the

text: networks

text: is

text: known

text: to

text: cause

text: deviations

text: from

text: the

text: the-

text: networks.

text: oretical

text: predictions;

text: for

text: example,

text: the

text: ﬁnite-size

text: eﬀect

text: is

text: shown

text: to

text: cause

text: The

text: numerics

text: for

text: the

text: SI

text: dynamics

text: on

text: L−

text: or

text: L−

text: -cloned

text: error

text: in

text: the

text: theory

text: for

text: binary-state

text: dynamics

text: versions

text: of

text: the

text: power

text: grid

text: network

text: are

text: to

text: theory

text: k,k

text: ij

text: ##LTFigure##

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text: 8 1 squares are preserved. We used these properties of network cloning to investigate 0.8 the eﬀect of short loops on dynamical processes running on L=1 networks and to inspect the applicability of tree-based the- 0.6 L=2 ories to clustered networks (i.e., networks with appreciable M L=5 density of short loops). In this regard the accuracy of theo- 0.4 L=20 ries for percolation, SI epidemics, and the Ising model were A theory investigated by comparing the theoretical predictions against ij 0.2 numerical simulations on examples of clustered networks and their cloned versions. 00. 5 0.6 0.7 0.8 0.9 1 1.1 We demonstrated that L-cloned networks with suﬃciently 1/T large L are the ensemble of networks for which the A theo- ij ries are designed. Hence, by running numerical simulations FIG. 8. (Color online) Magnetization M versus inverse temperature for the Ising model on a γ(3,3) network and its L-cloned versions. on L-cloned networks one can predict the outcome and po- Here L = 1 indicates the result of numerical simulations on the orig- tential beneﬁts of Aij theories for dynamical processes where inal γ(3,3) network with approximately 106 nodes, averaged over 20 such theories do not yet exist. In addition, by comparing the realizations of the Ising model. The A theory is inaccurate on this numerics for the dynamics on L− and L− − -cloned networks ij 3 3, 4 network; however its performance improves appreciably with L on with the numerics on L-cloned networks we highlighted the the L-cloned versions of this network. eﬀect of triangles and squares on the deviation of numerical results from the theoretical predictions. Using L -cloned net- f works we ruled out any possible eﬀect due to the diﬀerent − fraction of reassigned (rewired) links in L-cloned networks which transforms Eqs. (8) (10) to the Bethe-Peierls equa- and in L− - or L− − -cloned networks and demonstrated the tions of Ref. [24]. 3 3, 4 signiﬁcant eﬀect of short loops on the accuracy of the A the- These two methods are A theories; hence, as in the pre- ij ij ories. vious examples, their predictions for clustered networks are Nevertheless, it was previously shown [11] that mean-ﬁeld prone to errors. Figure 8 illustrates the behavior of the mag- theories can perform well on clustered networks with high netization M versus inverse temperature 1/T (for J = 1 and values of mean nearest-neighbor degree. Moreover, the exis- k = 1) in a γ(3, 3) network and its L-cloned versions; the tence of a double percolation phase transition on some clus- results are compared to the theoretical prediction from the tered networks has recently been reported [25], an eﬀect that above equations. As expected, the tree-based approach does is not considered in the tree-based theories. These ﬁndings not show the correct behavior for the original γ(3, 3) network. imply that although network clustering can cause inaccuracy However, for suﬃciently large L, the prediction of A theo- ij in tree-based theories, its net eﬀect and strength depend also ries matches the numerical simulations on the corresponding on other factors. In this regard, inspecting the organization L-cloned versions of the network. It is worth mentioning that, of clustered structures in networks (e.g., as in Ref. [26]) can as expected, the numerical results on L−3-cloned γ(3, 3) net- contribute to the understanding of the eﬀect of network clus- work (not shown) match the numerics on the original γ(3, 3) tering on dynamical processes, and its mitigation by, for ex- network. This implies that the improved agreement between ample, high values of mean nearest-neighbor degree. the numerical results on L-cloned networks and the A the- ij It is worth noting that the belief propagation and Bethe- ory is not due to the larger size of L-cloned networks. Hence, Peierls methods used for deriving analytical results for the the presence of triangles is the source of inaccuracy in the A ij Ising model have also been employed in many other prob- theory for the Ising model on this network. IV. CONCLUSION lems in various areas [15]. According to the tree-based nature of these approximations, the models for which they are used are subject to inaccuracies due to network clustering (short We introduced the so-called L-cloning of networks, a new loops). This suggests that network L-cloning and its exten- technique to create random ensembles of networks with cer- sions can be applied to gain a better understanding of the ac- tain properties based on real-world or synthetic complex net- curacy of these models as well. works. We demonstrated in Sec. that L-cloned networks have Another potential application of cloned networks is the degree distribution and degree-degree correlation between analysis of the ﬁnite size eﬀect on real systems, where a and beyond nearest neighbors that are identical to those in the small sample of a large network is available. By inspecting original network from which they are constructed. However, the result of numerical simulations on the sample and on the density of triangles in L-cloned networks, and hence the its L-cloned versions, one can examine the sensitivity of re- clustering coeﬃcient C, is reduced by a factor of L. More- sults to changes in network size. If the network of interest is over, the expected density of any short loops approaches clustered, the changes made in network clustering should be zero for suﬃciently large L. Some useful extensions of L- considered in addition to the ﬁnite size eﬀect. The extensions cloned networks, i.e., L− -, L− − -, and L -cloned networks, 3 3, 4 f of L-cloning that preserve the density of loops of speciﬁed were also introduced; similar to L-cloned networks, these lengths can help achieve this goal. networks have degree distribution and degree-degree correla- ACKNOWLEDGEMENTS tions between and beyond nearest neighbors identical to those of the original network from which they are constructed. On the other hand, in L− -cloned networks the triangles are A.F., S.M., and J.P.G. acknowledge funding provided by 3 preserved, and in L− − -cloned networks both triangles and Science Foundation Ireland (SFI) under programme 11/PI/1026. 3, 4 ##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##

text: squares

text: are

text: preserved.

text: We

text: used

text: these

text: properties

text: of

text: network

text: cloning

text: to

text: investigate

text: the

text: eﬀect

text: of

text: short

text: loops

text: on

text: dynamical

text: processes

text: running

text: on

text: networks

text: and

text: to

text: inspect

text: the

text: applicability

text: of

text: tree-based

text: the-

text: ories

text: to

text: clustered

text: networks

text: (i.e.,

text: networks

text: with

text: appreciable

text: M

text: density

text: of

text: short

text: loops).

text: In

text: this

text: regard

text: the

text: accuracy

text: of

text: theo-

text: ries

text: for

text: percolation,

text: SI

text: epidemics,

text: and

text: the

text: Ising

text: model

text: were

text: A

text: theory

text: investigated

text: by

text: comparing

text: the

text: theoretical

text: predictions

text: against

text: ij

text: numerical

text: simulations

text: on

text: examples

text: of

text: clustered

text: networks

text: and

text: their

text: cloned

text: versions.

text: We

text: demonstrated

text: that

text: L-cloned

text: networks

text: with

text: suﬃciently

text: large

text: L

text: are

text: the

text: ensemble

text: of

text: networks

text: for

text: which

text: the

text: A

text: theo-

text: ij

text: ries

text: are

text: designed.

text: Hence,

text: by

text: running

text: numerical

text: simulations

text: FIG.

text: (Color

text: online)

text: Magnetization

text: M

text: versus

text: inverse

text: temperature

text: for

text: the

text: Ising

text: model

text: on

text: a

text: network

text: and

text: its

text: L-cloned

text: versions.

text: on

text: L-cloned

text: networks

text: one

text: can

text: predict

text: the

text: outcome

text: and

text: po-

text: Here

text: L

text: indicates

text: the

text: result

text: of

text: numerical

text: simulations

text: on

text: the

text: orig-

text: tential

text: beneﬁts

text: of

text: Aij

text: theories

text: for

text: dynamical

text: processes

text: where

text: inal

text: network

text: with

text: approximately

text: nodes,

text: averaged

text: over

text: such

text: theories

text: do

text: not

text: yet

text: exist.

text: In

text: addition,

text: by

text: comparing

text: the

text: realizations

text: of

text: the

text: Ising

text: model.

text: The

text: A

text: theory

text: is

text: inaccurate

text: on

text: this

text: numerics

text: for

text: the

text: dynamics

text: on

text: L−

text: and

text: L−

text: -cloned

text: networks

text: ij

text: network;

text: however

text: its

text: performance

text: improves

text: appreciably

text: with

text: L

text: on

text: with

text: the

text: numerics

text: on

text: L-cloned

text: networks

text: we

text: highlighted

text: the

text: the

text: L-cloned

text: versions

text: of

text: this

text: network.

text: eﬀect

text: of

text: triangles

text: and

text: squares

text: on

text: the

text: deviation

text: of

text: numerical

text: results

text: from

text: the

text: theoretical

text: predictions.

text: Using

text: L

text: -cloned

text: net-

text: f

text: works

text: we

text: ruled

text: out

text: any

text: possible

text: eﬀect

text: due

text: to

text: the

text: diﬀerent

text: fraction

text: of

text: reassigned

text: (rewired)

text: links

text: in

text: L-cloned

text: networks

text: which

text: transforms

text: Eqs.

text: to

text: the

text: Bethe-Peierls

text: equa-

text: and

text: in

text: L−

text: or

text: L−

text: -cloned

text: networks

text: and

text: demonstrated

text: the

text: tions

text: of

text: Ref.

text: signiﬁcant

text: eﬀect

text: of

text: short

text: loops

text: on

text: the

text: accuracy

text: of

text: the

text: A

text: the-

text: These

text: two

text: methods

text: are

text: A

text: theories;

text: hence,

text: as

text: in

text: the

text: pre-

text: ij

text: ij

text: ories.

text: vious

text: examples,

text: their

text: predictions

text: for

text: clustered

text: networks

text: are

text: Nevertheless,

text: it

text: was

text: previously

text: shown

text: that

text: mean-ﬁeld

text: prone

text: to

text: errors.

text: Figure

text: illustrates

text: the

text: behavior

text: of

text: the

text: mag-

text: theories

text: can

text: perform

text: well

text: on

text: clustered

text: networks

text: with

text: high

text: netization

text: M

text: versus

text: inverse

text: temperature

text: (for

text: J

text: and

text: values

text: of

text: mean

text: nearest-neighbor

text: degree.

text: Moreover,

text: the

text: exis-

text: k

text: in

text: a

text: network

text: and

text: its

text: L-cloned

text: versions;

text: the

text: tence

text: of

text: a

text: double

text: percolation

text: phase

text: transition

text: on

text: some

text: clus-

text: results

text: are

text: compared

text: to

text: the

text: theoretical

text: prediction

text: from

text: the

text: tered

text: networks

text: has

text: recently

text: been

text: reported

text: an

text: eﬀect

text: that

text: above

text: equations.

text: As

text: expected,

text: the

text: tree-based

text: approach

text: does

text: is

text: not

text: considered

text: in

text: the

text: tree-based

text: theories.

text: These

text: ﬁndings

text: not

text: show

text: the

text: correct

text: behavior

text: for

text: the

text: original

text: network.

text: imply

text: that

text: although

text: network

text: clustering

text: can

text: cause

text: inaccuracy

text: However,

text: for

text: suﬃciently

text: large

text: L,

text: the

text: prediction

text: of

text: A

text: theo-

text: ij

text: in

text: tree-based

text: theories,

text: its

text: net

text: eﬀect

text: and

text: strength

text: depend

text: also

text: ries

text: matches

text: the

text: numerical

text: simulations

text: on

text: the

text: corresponding

text: on

text: other

text: factors.

text: In

text: this

text: regard,

text: inspecting

text: the

text: organization

text: L-cloned

text: versions

text: of

text: the

text: network.

text: It

text: is

text: worth

text: mentioning

text: that,

text: of

text: clustered

text: structures

text: in

text: networks

text: (e.g.,

text: as

text: in

text: Ref.

text: can

text: as

text: expected,

text: the

text: numerical

text: results

text: on

text: net-

text: contribute

text: to

text: the

text: understanding

text: of

text: the

text: eﬀect

text: of

text: network

text: clus-

text: work

text: (not

text: shown)

text: match

text: the

text: numerics

text: on

text: the

text: original

text: tering

text: on

text: dynamical

text: processes,

text: and

text: its

text: mitigation

text: by,

text: for

text: ex-

text: network.

text: This

text: implies

text: that

text: the

text: improved

text: agreement

text: between

text: ample,

text: high

text: values

text: of

text: mean

text: nearest-neighbor

text: degree.

text: the

text: numerical

text: results

text: on

text: L-cloned

text: networks

text: and

text: the

text: A

text: the-

text: ij

text: It

text: is

text: worth

text: noting

text: that

text: the

text: belief

text: propagation

text: and

text: Bethe-

text: ory

text: is

text: not

text: due

text: to

text: the

text: larger

text: size

text: of

text: L-cloned

text: networks.

text: Hence,

text: Peierls

text: methods

text: used

text: for

text: deriving

text: analytical

text: results

text: for

text: the

text: the

text: presence

text: of

text: triangles

text: is

text: the

text: source

text: of

text: inaccuracy

text: in

text: the

text: A

text: ij

text: Ising

text: model

text: have

text: also

text: been

text: employed

text: in

text: many

text: other

text: prob-

text: theory

text: for

text: the

text: Ising

text: model

text: on

text: this

text: network.

text: IV.

text: CONCLUSION

text: lems

text: in

text: various

text: areas

text: According

text: to

text: the

text: tree-based

text: nature

text: of

text: these

text: approximations,

text: the

text: models

text: for

text: which

text: they

text: are

text: used

text: are

text: subject

text: to

text: inaccuracies

text: due

text: to

text: network

text: clustering

text: (short

text: We

text: introduced

text: the

text: so-called

text: L-cloning

text: of

text: networks,

text: a

text: new

text: loops).

text: This

text: suggests

text: that

text: network

text: L-cloning

text: and

text: its

text: exten-

text: technique

text: to

text: create

text: random

text: ensembles

text: of

text: networks

text: with

text: cer-

text: sions

text: can

text: be

text: applied

text: to

text: gain

text: a

text: better

text: understanding

text: of

text: the

text: ac-

text: tain

text: properties

text: based

text: on

text: real-world

text: or

text: synthetic

text: complex

text: net-

text: curacy

text: of

text: these

text: models

text: as

text: well.

text: works.

text: We

text: demonstrated

text: in

text: Sec.

text: that

text: L-cloned

text: networks

text: have

text: Another

text: potential

text: application

text: of

text: cloned

text: networks

text: is

text: the

text: degree

text: distribution

text: and

text: degree-degree

text: correlation

text: between

text: analysis

text: of

text: the

text: ﬁnite

text: size

text: eﬀect

text: on

text: real

text: systems,

text: where

text: a

text: and

text: beyond

text: nearest

text: neighbors

text: that

text: are

text: identical

text: to

text: those

text: in

text: the

text: small

text: sample

text: of

text: a

text: large

text: network

text: is

text: available.

text: By

text: inspecting

text: original

text: network

text: from

text: which

text: they

text: are

text: constructed.

text: However,

text: the

text: result

text: of

text: numerical

text: simulations

text: on

text: the

text: sample

text: and

text: on

text: the

text: density

text: of

text: triangles

text: in

text: L-cloned

text: networks,

text: and

text: hence

text: the

text: its

text: L-cloned

text: versions,

text: one

text: can

text: examine

text: the

text: sensitivity

text: of

text: re-

text: clustering

text: coeﬃcient

text: C,

text: is

text: reduced

text: by

text: a

text: factor

text: of

text: L.

text: More-

text: sults

text: to

text: changes

text: in

text: network

text: size.

text: If

text: the

text: network

text: of

text: interest

text: is

text: over,

text: the

text: expected

text: density

text: of

text: any

text: short

text: loops

text: approaches

text: clustered,

text: the

text: changes

text: made

text: in

text: network

text: clustering

text: should

text: be

text: zero

text: for

text: suﬃciently

text: large

text: L.

text: Some

text: useful

text: extensions

text: of

text: L-

text: considered

text: in

text: addition

text: to

text: the

text: ﬁnite

text: size

text: eﬀect.

text: The

text: extensions

text: cloned

text: networks,

text: i.e.,

text: L−

text: L−

text: and

text: L

text: -cloned

text: networks,

text: f

text: of

text: L-cloning

text: that

text: preserve

text: the

text: density

text: of

text: loops

text: of

text: speciﬁed

text: were

text: also

text: introduced;

text: similar

text: to

text: L-cloned

text: networks,

text: these

text: lengths

text: can

text: help

text: achieve

text: this

text: goal.

text: networks

text: have

text: degree

text: distribution

text: and

text: degree-degree

text: correla-

text: ACKNOWLEDGEMENTS

text: tions

text: between

text: and

text: beyond

text: nearest

text: neighbors

text: identical

text: to

text: those

text: of

text: the

text: original

text: network

text: from

text: which

text: they

text: are

text: constructed.

text: On

text: the

text: other

text: hand,

text: in

text: L−

text: -cloned

text: networks

text: the

text: triangles

text: are

text: A.F.,

text: S.M.,

text: and

text: J.P.G.

text: acknowledge

text: funding

text: provided

text: by

text: preserved,

text: and

text: in

text: L−

text: -cloned

text: networks

text: both

text: triangles

text: and

text: Science

text: Foundation

text: Ireland

text: (SFI)

text: under

text: programme

text: ##LTFigure##

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text: 3 m 2.5 c − Ω M 2 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 83 s T M 82 P t Hi e 81 g a r e v A 80 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Date in October 2012 Figure 6: EH2 pool resistivity (measured at the recirculation input to the PL, marked as # in Fig. 3) and the average number of triggered photomultipliers per cosmic ray, during a period in October 2012, after repairs were made to the polishing loop. The correlation indicates an increasing attenuation length in the water with rise in resistivity. 14 ##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##

text: m

text: c

text: Ω

text: M

text: s

text: T

text: M

text: P

text: t

text: Hi

text: e

text: g

text: a

text: r

text: e

text: v

text: A

text: Date

text: in

text: October

text: Figure

text: pool

text: resistivity

text: (measured

text: at

text: the

text: recirculation

text: input

text: to

text: the

text: PL,

text: marked

text: as

text: in

text: Fig.

text: and

text: the

text: average

text: number

text: of

text: triggered

text: photomultipliers

text: per

text: cosmic

text: ray,

text: during

text: a

text: period

text: in

text: October

text: after

text: repairs

text: were

text: made

text: to

text: the

text: polishing

text: loop.

text: The

text: correlation

text: indicates

text: an

text: increasing

text: attenuation

text: length

text: in

text: the

text: water

text: with

text: rise

text: in

text: resistivity.

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text: 1 10 Ag s) Fe Zn Zn t ni u 0 10 y ar Fe Cu tr Ag Co Ni bi r A ( Ca Cr y −1 t 10 Al Co si Mg Cr n e Ca t n I −2 10 0 2 4 6 8 10 X−Ray Energy (keV) Figure 8: XRF analysis of EH1 sludge sample taken March 2012. at Rensselaer Polytechnic Institute via gas chromatography with methanization and ﬂame ionization detection (FID). Unfortunately, these samples failed to provide any conclusive results. Measured Dissolved Inorganic Carbon (DIC) had signiﬁcant variation within samples from the same sample port, and no conclusions or trends could be deduced from the data over all. This is most likely the result of a combination of low concentrations and imperfect sampling technique. Indeed, a theoretical calculation [24] of the resistivity attributed solely ◦ to aqueous CO at 22.5 suggests that concentrations as low as 10-100 ppb (by weight) 2 would be enough to aﬀect resistivity at our observed levels. Furthermore, this method does not distinguish between CO from atmospheric dissolution and microbial metabolism. Both 2 sources may have contributed to the concentrations necessary to lower the resistivity of the pools. Table 2 summarizes data gathered from additional sensors (detailed in section g) that pro- vides insight into the resistivity drop that seems to occur within the pools. There is a disparity between the resistivity of the water measured in the pool return line and that of the water in the pool center, a disparity which is quite large in the case of EH1. We suspected that irregular ﬂow and increased potential organics between the pool wall and the liner, a semi-isolated circulation zone that might allow suﬃcient microbial growth, could cause this disparity. Subtle diﬀerences in conditions at the pool walls could also account for the large diﬀerences between the diﬀerent halls, as shown in Fig. 7. To test the suspicion that the ultimate drop in resistivity came from microbe-laden pool walls, we shut down physics data taking in EH2 temporarily, and installed a ball valve to (cid:13) shut oﬀ water coming from the zone in-between the Tyvek R lining and the pool wall. The 16 ##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##

text: Ag

text: s)

text: Fe

text: Zn

text: Zn

text: t

text: ni

text: u

text: y

text: ar

text: Fe

text: Cu

text: tr

text: Ag

text: Co

text: Ni

text: bi

text: r

text: A

text: Ca

text: Cr

text: y

text: −1

text: t

text: Al

text: Co

text: si

text: Mg

text: Cr

text: n

text: e

text: Ca

text: t

text: n

text: I

text: −2

text: X−Ray

text: Energy

text: (keV)

text: Figure

text: XRF

text: analysis

text: of

text: sludge

text: sample

text: taken

text: March

text: at

text: Rensselaer

text: Polytechnic

text: Institute

text: via

text: gas

text: chromatography

text: with

text: methanization

text: and

text: ﬂame

text: ionization

text: detection

text: (FID).

text: Unfortunately,

text: these

text: samples

text: failed

text: to

text: provide

text: any

text: conclusive

text: results.

text: Measured

text: Dissolved

text: Inorganic

text: Carbon

text: (DIC)

text: had

text: signiﬁcant

text: variation

text: within

text: samples

text: from

text: the

text: same

text: sample

text: port,

text: and

text: no

text: conclusions

text: or

text: trends

text: could

text: be

text: deduced

text: from

text: the

text: data

text: over

text: all.

text: This

text: is

text: most

text: likely

text: the

text: result

text: of

text: a

text: combination

text: of

text: low

text: concentrations

text: and

text: imperfect

text: sampling

text: technique.

text: Indeed,

text: a

text: theoretical

text: calculation

text: of

text: the

text: resistivity

text: attributed

text: solely

text: to

text: aqueous

text: CO

text: at

text: suggests

text: that

text: concentrations

text: as

text: low

text: as

text: ppb

text: (by

text: weight)

text: would

text: be

text: enough

text: to

text: aﬀect

text: resistivity

text: at

text: our

text: observed

text: levels.

text: Furthermore,

text: this

text: method

text: does

text: not

text: distinguish

text: between

text: CO

text: from

text: atmospheric

text: dissolution

text: and

text: microbial

text: metabolism.

text: Both

text: sources

text: may

text: have

text: contributed

text: to

text: the

text: concentrations

text: necessary

text: to

text: lower

text: the

text: resistivity

text: of

text: the

text: pools.

text: Table

text: summarizes

text: data

text: gathered

text: from

text: additional

text: sensors

text: (detailed

text: in

text: section

text: g)

text: that

text: pro-

text: vides

text: insight

text: into

text: the

text: resistivity

text: drop

text: that

text: seems

text: to

text: occur

text: within

text: the

text: pools.

text: There

text: is

text: a

text: disparity

text: between

text: the

text: resistivity

text: of

text: the

text: water

text: measured

text: in

text: the

text: pool

text: return

text: line

text: and

text: that

text: of

text: the

text: water

text: in

text: the

text: pool

text: center,

text: a

text: disparity

text: which

text: is

text: quite

text: large

text: in

text: the

text: case

text: of

text: We

text: suspected

text: that

text: irregular

text: ﬂow

text: and

text: increased

text: potential

text: organics

text: between

text: the

text: pool

text: wall

text: and

text: the

text: liner,

text: a

text: semi-isolated

text: circulation

text: zone

text: that

text: might

text: allow

text: suﬃcient

text: microbial

text: growth,

text: could

text: cause

text: this

text: disparity.

text: Subtle

text: diﬀerences

text: in

text: conditions

text: at

text: the

text: pool

text: walls

text: could

text: also

text: account

text: for

text: the

text: large

text: diﬀerences

text: between

text: the

text: diﬀerent

text: halls,

text: as

text: shown

text: in

text: Fig.

text: To

text: test

text: the

text: suspicion

text: that

text: the

text: ultimate

text: drop

text: in

text: resistivity

text: came

text: from

text: microbe-laden

text: pool

text: walls,

text: we

text: shut

text: down

text: physics

text: data

text: taking

text: in

text: temporarily,

text: and

text: installed

text: a

text: ball

text: valve

text: to

text: shut

text: oﬀ

text: water

text: coming

text: from

text: the

text: zone

text: in-between

text: the

text: Tyvek

text: R

text: lining

text: and

text: the

text: pool

text: wall.

text: The

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text: 2 T.Q. BAO, K. FELLNER, S. ROSENBERGER While the asymmetric localisation of Lgl is essential for the asymmetric cell division of SOP cells, the subsequent biological machinery, where the asymmetric localisation of Lgl leads to the asymmetric localisation allocation of the adaptor protein Pon (Partner of Numb) and the cell-fate determinate Numb is currently not well enough understood to be considered here. In this paper, we shall present and study a volume-surface reaction-diﬀusion model system describing the evolution of Lgl in its non-phosphorylated and phos- phorylated conformations both in the cytoplasm (i.e. in the cell volume) and at the cortex (i.e. the surface/membrane of the cell). More precisely, we shall denote by L(t, x) and P(t, x) the cytoplasmic concen- trations of non-phosphorylated and phosphorylated Lgl within the bounded cell ⊂ R domain Ω n, while l(t, x) and p(t, x) denote the cortical concentrations of the non-phosphorylated and phosphorylated Lgl at the boundary Γ := ∂Ω, which is assumed suﬃciently smooth (e.g. C2+α with α > 0). The reaction kinetics between the species L, P, l and p are depicted in Figure 1 and summarise the following processes: i) a reversible reaction between L and P with rates α and β on the domain Ω, ii) a reversible exchange between L and l at the boundary Γ with rates λ and γ, iii) an irreversible phosphorylation of l-Lgl into p-Lgl at the boundary Γ with rate σ and iv) an irreversible release of p-Lgl from the boundary Γ into the domain Ω with rate ξ. We emphasise that these reaction/sorption processes jointly conserve the total mass of Lgl (see the conservation law (1.4) below). Moreover, the dynamics of Fig. 1 forms a so-called weakly-reversible or complex balance reaction network, for which the convergence towards a steady state and well as the structure of the steady states are signiﬁcantly more subtle than for detailed balance models, see the discussion of the numerical examples in Section or also [21, 26] and the references therein. β L P α λ γ ξ (cid:96) p σ Figure 1. The weakly-reversible reaction dynamics of L, P, l and p. In the following, we propose a continuum model of partial diﬀerential equations, which describe the reactions and the diﬀusion processes of these species both on the domain Ω and on its surface Γ. The choice of a continuum model is based on the biological observation that protein concentrations in SOP cells are rather large and that stochastic eﬀects in the concentrations can thus be neglected, see [6]. The aim of the model is to qualitatively study, mathematically analyse and numerically simulate the reaction-diﬀusion dynamics between phosphorylated and un-phosphorylated Lgl in the cytoplasm and on the cortex with particular emphasis on the eﬀect of surface diﬀusion, see Sections and . ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: T.Q.

text: BAO,

text: K.

text: FELLNER,

text: S.

text: ROSENBERGER

text: While

text: the

text: asymmetric

text: localisation

text: of

text: Lgl

text: is

text: essential

text: for

text: the

text: asymmetric

text: cell

text: division

text: of

text: SOP

text: cells,

text: the

text: subsequent

text: biological

text: machinery,

text: where

text: the

text: asymmetric

text: localisation

text: of

text: Lgl

text: leads

text: to

text: the

text: asymmetric

text: localisation

text: allocation

text: of

text: the

text: adaptor

text: protein

text: Pon

text: (Partner

text: of

text: Numb)

text: and

text: the

text: cell-fate

text: determinate

text: Numb

text: is

text: currently

text: not

text: well

text: enough

text: understood

text: to

text: be

text: considered

text: here.

text: In

text: this

text: paper,

text: we

text: shall

text: present

text: and

text: study

text: a

text: volume-surface

text: reaction-diﬀusion

text: model

text: system

text: describing

text: the

text: evolution

text: of

text: Lgl

text: in

text: its

text: non-phosphorylated

text: and

text: phos-

text: phorylated

text: conformations

text: both

text: in

text: the

text: cytoplasm

text: (i.e.

text: in

text: the

text: cell

text: volume)

text: and

text: at

text: the

text: cortex

text: (i.e.

text: the

text: surface/membrane

text: of

text: the

text: cell).

text: More

text: precisely,

text: we

text: shall

text: denote

text: by

text: L(t,

text: x)

text: and

text: P(t,

text: x)

text: the

text: cytoplasmic

text: concen-

text: trations

text: of

text: non-phosphorylated

text: and

text: phosphorylated

text: Lgl

text: within

text: the

text: bounded

text: cell

text: R

text: domain

text: Ω

text: n,

text: while

text: l(t,

text: x)

text: and

text: p(t,

text: x)

text: denote

text: the

text: cortical

text: concentrations

text: of

text: the

text: non-phosphorylated

text: and

text: phosphorylated

text: Lgl

text: at

text: the

text: boundary

text: Γ

text: ∂Ω,

text: which

text: is

text: assumed

text: suﬃciently

text: smooth

text: (e.g.

text: with

text: α

text: The

text: reaction

text: kinetics

text: between

text: the

text: species

text: L,

text: P,

text: l

text: and

text: p

text: are

text: depicted

text: in

text: Figure

text: and

text: summarise

text: the

text: following

text: processes:

text: i)

text: a

text: reversible

text: reaction

text: between

text: L

text: and

text: P

text: with

text: rates

text: α

text: and

text: β

text: on

text: the

text: domain

text: Ω,

text: ii)

text: a

text: reversible

text: exchange

text: between

text: L

text: and

text: l

text: at

text: the

text: boundary

text: Γ

text: with

text: rates

text: λ

text: and

text: γ,

text: iii)

text: an

text: irreversible

text: phosphorylation

text: of

text: l-Lgl

text: into

text: p-Lgl

text: at

text: the

text: boundary

text: Γ

text: with

text: rate

text: σ

text: and

text: iv)

text: an

text: irreversible

text: release

text: of

text: p-Lgl

text: from

text: the

text: boundary

text: Γ

text: into

text: the

text: domain

text: Ω

text: with

text: rate

text: ξ.

text: We

text: emphasise

text: that

text: these

text: reaction/sorption

text: processes

text: jointly

text: conserve

text: the

text: total

text: mass

text: of

text: Lgl

text: (see

text: the

text: conservation

text: law

text: below).

text: Moreover,

text: the

text: dynamics

text: of

text: Fig.

text: forms

text: a

text: so-called

text: weakly-reversible

text: or

text: complex

text: balance

text: reaction

text: network,

text: for

text: which

text: the

text: convergence

text: towards

text: a

text: steady

text: state

text: and

text: well

text: as

text: the

text: structure

text: of

text: the

text: steady

text: states

text: are

text: signiﬁcantly

text: more

text: subtle

text: than

text: for

text: detailed

text: balance

text: models,

text: see

text: the

text: discussion

text: of

text: the

text: numerical

text: examples

text: in

text: Section

text: or

text: also

text: and

text: the

text: references

text: therein.

text: β

text: L

text: P

text: α

text: λ

text: γ

text: ξ

text: p

text: σ

text: Figure

text: The

text: weakly-reversible

text: reaction

text: dynamics

text: of

text: L,

text: P,

text: l

text: and

text: p.

text: In

text: the

text: following,

text: we

text: propose

text: a

text: continuum

text: model

text: of

text: partial

text: diﬀerential

text: equations,

text: which

text: describe

text: the

text: reactions

text: and

text: the

text: diﬀusion

text: processes

text: of

text: these

text: species

text: both

text: on

text: the

text: domain

text: Ω

text: and

text: on

text: its

text: surface

text: Γ.

text: The

text: choice

text: of

text: a

text: continuum

text: model

text: is

text: based

text: on

text: the

text: biological

text: observation

text: that

text: protein

text: concentrations

text: in

text: SOP

text: cells

text: are

text: rather

text: large

text: and

text: that

text: stochastic

text: eﬀects

text: in

text: the

text: concentrations

text: can

text: thus

text: be

text: neglected,

text: see

text: The

text: aim

text: of

text: the

text: model

text: is

text: to

text: qualitatively

text: study,

text: mathematically

text: analyse

text: and

text: numerically

text: simulate

text: the

text: reaction-diﬀusion

text: dynamics

text: between

text: phosphorylated

text: and

text: un-phosphorylated

text: Lgl

text: in

text: the

text: cytoplasm

text: and

text: on

text: the

text: cortex

text: with

text: particular

text: emphasis

text: on

text: the

text: eﬀect

text: of

text: surface

text: diﬀusion,

text: see

text: Sections

text: and

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: QSSA AND NUMERICS FOR A VOLUME-SURFACE REACTION-DIFFUSION SYSTEM 13 (cid:90) and ≤ (cid:107) (cid:107) (cid:107) (cid:107) Wl(0) dsdS = (φ(T), l(0)) l(0) φ(T) Γ Γ Γ ΓT ≤ (cid:107) (cid:107) (cid:107)∇ (cid:107) (cid:107) (cid:107) l(0) (C φ(T) + φ(T) ) (3.19) Γ Ω √ Ω ≤ 1(cid:107)∇ (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) φ(T) 2 + C l(0) 2 + T l(0) W , 8 Ω Γ Γ L2(ΩT) (cid:90) and similarly, √ ≤ 1(cid:107)∇ (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) Wp(0) dsdS φ(T) 2 + C p(0) 2 + T p(0) W . (3.20) 8 Ω Γ2 Γ2 L2(ΩT) Γ2,T (cid:90) Hence, by the non-negativity of L, P, l, p, we obtain from (3.16)–(3.20) √ ≤ (cid:107) (cid:107) WZ dsdx C T W + C. (3.21) L2(ΩT) ΩT ≤ { } It follows then from W max d , d Z and (3.21) that L P (cid:107) (cid:107) ≤ W C . (3.22) L2(ΩT) T Therefore, by the non-negativity of L and P and by keeping in mind that W = d L + d P, we conclude that L P (cid:107) (cid:107) (cid:107) (cid:107) ≤ L + P C . (3.23) L2(ΩT) L2(ΩT) T (cid:90) (cid:90) Next, by testing (3.1) with L, we estimate 1 d (cid:107) (cid:107) (cid:107)∇ (cid:107) − (cid:107) (cid:107) − (cid:107) (cid:107) L 2 + d L 2 = β L 2 + α LP dx λ L 2 + γ Ll dS 2 dt Ω L Ω Ω Γ Ω Γ (3.24) ≤ (cid:107) (cid:107) (cid:107) (cid:107) − λ(cid:107) (cid:107) C P 2 + C l 2 L 2. Ω Γ 2 Γ (cid:90) On the other hand, we get from (3.3) 1 d (cid:107) (cid:107) − (cid:107) (cid:107) − (cid:107) (cid:107) ≤ (cid:107) (cid:107) λ(cid:107) (cid:107) l 2 = γ l 2 σ l 2 + λ Ll dS C l 2 + L 2. (3.25) 2 dt Γ Γ Γ2 Γ 2 Γ Γ Summing (3.24) and (3.25) yields d (cid:107) (cid:107) (cid:107) (cid:107) (cid:107)∇ (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) ( L 2 + l 2) + 2d L 2 C P 2 + C l 2. (3.26) dt Ω Γ L Ω Ω Γ (cid:107) (cid:107) (cid:107) (cid:107) ≤ Therefore, by L + P C , we conclude that L2(ΩT) L2(ΩT) T (cid:107)∇ (cid:107) ≤ (cid:107) (cid:107) ≤ ≤ L C and l C eCT C . (3.27) L2(ΩT) T L2(ΓT) T T (cid:3) This ﬁnishes the proof. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: QSSA

text: AND

text: NUMERICS

text: FOR

text: A

text: VOLUME-SURFACE

text: REACTION-DIFFUSION

text: SYSTEM

label: equation

text: and

text: dsdS

text: (φ(T),

text: φ(T)

text: Γ

text: Γ

text: Γ

text: ΓT

text: (C

text: φ(T)

text: φ(T)

text: Γ

text: Ω

text: Ω

text: φ(T)

text: C

text: T

text: W

text: Ω

text: Γ

text: Γ

text: and

text: similarly,

text: dsdS

text: φ(T)

text: C

text: T

text: W

text: Ω

text: Γ2

text: Γ2

text: Hence,

text: by

text: the

text: non-negativity

text: of

text: L,

text: P,

text: l,

text: p,

text: we

text: obtain

text: from

text: WZ

text: dsdx

text: C

text: T

text: W

text: C.

text: ΩT

text: It

text: follows

text: then

text: from

text: W

text: max

text: d

text: d

text: Z

text: and

text: that

text: L

text: P

text: W

text: C

text: T

text: Therefore,

text: by

text: the

text: non-negativity

text: of

text: L

text: and

text: P

text: and

text: by

text: keeping

text: in

text: mind

text: that

text: W

text: d

text: L

text: d

text: P,

text: we

text: conclude

text: that

text: L

text: P

text: L

text: P

text: C

text: T

text: Next,

text: by

text: testing

text: with

text: L,

text: we

text: estimate

text: d

text: L

text: d

text: L

text: β

text: L

text: α

text: LP

text: dx

text: λ

text: L

text: γ

text: Ll

text: dS

text: dt

text: Ω

text: L

text: Ω

text: Ω

text: Γ

text: Ω

text: Γ

text: C

text: P

text: C

text: l

text: L

text: Ω

text: Γ

text: Γ

text: On

text: the

text: other

text: hand,

text: we

text: get

text: from

text: d

text: l

text: γ

text: l

text: σ

text: l

text: λ

text: Ll

text: dS

text: C

text: l

text: L

text: dt

text: Γ

text: Γ

text: Γ2

text: Γ

text: Γ

text: Γ

text: Summing

text: and

text: yields

text: d

text: L

text: l

text: L

text: C

text: P

text: C

text: l

text: dt

text: Ω

text: Γ

text: L

text: Ω

text: Ω

text: Γ

text: Therefore,

text: by

text: L

text: P

text: C

text: we

text: conclude

text: that

text: T

text: L

text: C

text: and

text: l

text: C

text: eCT

text: C

text: T

text: T

text: T

text: This

text: ﬁnishes

text: the

text: proof.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: QSSA AND NUMERICS FOR A VOLUME-SURFACE REACTION-DIFFUSION SYSTEM 19 ≡ ≡ ≡ ≡ L (x, y) 0.8, P (x, y) 0.6, l (x, y) 0.3, p (x, y) 0.4. (4.3) 0 0 0 0 The value of ξ will be chosen diﬀerently during the discussion of the numerical examples. Also the surface diﬀusion rates d and d will be speciﬁed later. l p (a) l-Lgl with surface diﬀusion (b) l-Lgl without surface diﬀusion (c) p-Lgl with surface diﬀusion (d) p-Lgl without surface diﬀusion Figure 3. Comparison of the numerical stationary states of l and p on a unit-circular cell (i.e. the blue circle centred at the origin) with surface diﬀusion rates d = 0.02, d = 0.04 (Figs. 3a and l p 3c) and without surface diﬀusion d = d = 0 (Figs. 3b and 3d) l p for the parameters (4.1), (4.2) and initial data (4.3): The l-Lgl concentration strongly decreases towards the active boundary part Γ located within the negative quadrant in the front (Figs. 3a and 2 3b). In Fig. 3a, the decrease of the l-Lgl is clearly smoothed by the surface diﬀusion. In Fig. 3b the decay proﬁle is steep yet still smoothed by an indirect surface diﬀusive eﬀect caused by volume diﬀusion and reversible reaction. This is conﬁrmed by Figs. 3c and 3d plotting the subsequent concentrations of phosphorylated p-Lgl. Fig 3d shows in particular the spikes of p-Lgl produced near ∂Γ due to the indirect surface diﬀusion eﬀect. 2 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: QSSA

text: AND

text: NUMERICS

text: FOR

text: A

text: VOLUME-SURFACE

text: REACTION-DIFFUSION

text: SYSTEM

text: L

text: (x,

text: y)

text: P

text: (x,

text: y)

text: l

text: (x,

text: y)

text: p

text: (x,

text: y)

text: The

text: value

text: of

text: ξ

text: will

text: be

text: chosen

text: diﬀerently

text: during

text: the

text: discussion

text: of

text: the

text: numerical

text: examples.

text: Also

text: the

text: surface

text: diﬀusion

text: rates

text: d

text: and

text: d

text: will

text: be

text: speciﬁed

text: later.

text: l

text: p

text: (a)

text: l-Lgl

label: caption

text: with

text: surface

text: diﬀusion

text: (b)

text: l-Lgl

text: without

text: surface

text: diﬀusion

text: (c)

text: p-Lgl

text: with

text: surface

text: diﬀusion

text: (d)

text: p-Lgl

text: without

text: surface

text: diﬀusion

text: Figure

text: Comparison

text: of

text: the

text: numerical

text: stationary

text: states

text: of

text: l

text: and

text: p

text: on

text: a

text: unit-circular

text: cell

text: (i.e.

text: the

text: blue

text: circle

text: centred

text: at

text: the

text: origin)

text: with

text: surface

text: diﬀusion

text: rates

text: d

text: d

text: (Figs.

text: and

text: l

text: p

text: and

text: without

text: surface

text: diﬀusion

text: d

text: d

text: (Figs.

text: and

text: l

text: p

text: for

text: the

text: parameters

text: and

text: initial

text: data

text: The

text: l-Lgl

text: concentration

text: strongly

text: decreases

text: towards

text: the

text: active

text: boundary

text: part

text: Γ

text: located

text: within

text: the

text: negative

text: quadrant

text: in

text: the

text: front

text: (Figs.

text: and

text: In

text: Fig.

text: the

text: decrease

text: of

text: the

text: l-Lgl

text: is

text: clearly

text: smoothed

text: by

text: the

text: surface

text: diﬀusion.

text: In

text: Fig.

text: the

text: decay

text: proﬁle

text: is

text: steep

text: yet

text: still

text: smoothed

text: by

text: an

text: indirect

text: surface

text: diﬀusive

text: eﬀect

text: caused

text: by

text: volume

text: diﬀusion

text: and

text: reversible

text: reaction.

text: This

text: is

text: conﬁrmed

text: by

text: Figs.

text: and

text: plotting

text: the

text: subsequent

text: concentrations

text: of

text: phosphorylated

text: p-Lgl.

text: Fig

text: shows

text: in

text: particular

text: the

text: spikes

text: of

text: p-Lgl

text: produced

text: near

text: ∂Γ

text: due

text: to

text: the

text: indirect

text: surface

text: diﬀusion

text: eﬀect.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##