depth penetration and scope extension of failures...

Research ArticleDepth Penetration and Scope Extension of Failures in theCascading of Multilayer Networks

Wen-Jun Jiang Run-Ran Liu and Chun-Xiao Jia

Research Center for Complexity Sciences Hangzhou Normal University Hangzhou Zhejiang 311121 China

Correspondence should be addressed to Run-Ran Liu runranliu163com

Received 16 December 2019 Revised 7 March 2020 Accepted 17 March 2020 Published 25 April 2020

Guest Editor Tomas Veloz

Copyright copy 2020Wen-Jun Jiang et alis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Real-world complex systems always interact with each other which causes these systems to collapse in an avalanche or cascadingmanner in the case of random failures or malicious attacks e robustness of multilayer networks has attracted great interestwhere the modeling and theoretical studies of which always rely on the concept of multilayer networks and percolation methodsA straightforward and tacit assumption is that the interdependence across network layers is strong which means that a node willfail entirely with the removal of all links if one of its interdependent nodes in other network layers fails However thisoversimplification cannot describe the general form of interactions across the network layers in a real-world multilayer system Inthis paper we reveal the nature of the avalanche disintegration of general multilayer networks with arbitrary interdependencystrength across network layers Specifically we identify that the avalanche process of the whole system can essentially bedecomposed into twomicroscopic cascading dynamics in terms of the propagation direction of the failures depth penetration andscope extension In the process of depth penetration the failures propagate from layer to layer where the greater the number offailed nodes is the greater is the destructive power that will emerge in an interdependency group In the process of scopeextension failures propagate with the removal of connections in each network layer Under the synergy of the two processes wefind that the percolation transition of the system can be discontinuous or continuous with changes in the interdependencystrength across network layers which means that a sudden system-wide collapse can be avoided by controlling the interde-pendency strength across network layers Our work not only reveals the microscopic mechanism of global collapse in multilayerinfrastructure systems but also provides stimulating ideas on intervention programs and approaches for cascade failures

1 Introduction

Many real-world complex systems both natural [1] and manmade [2ndash5] can be described as multilayer or interdepen-dent networks given the existence of different levels of in-terdependence across network layers Recent theoreticalstudies on networks with two or more layers show that whenthe nodes in each network are interdependent on the nodesin other networks even small initial failures can propagateback and forth and lead to the abrupt collapse of the wholesystem [5ndash10] In this sense multilayer networks are morefragile than single-layer networks in resisting the propa-gation of initial failures [6] In recent years we have wit-nessed considerable progress in the study of multilayernetworks with the aid of percolation theory [11ndash13] It hasbeen found that multilayer networks are not as fragile as in

theoretical studies under certain specific conditions such asthose given link overlap [14 15] geometric correlations[16 17] correlated community structures [18] interlayerdegree correlations [19 20] intralayer degree correlations[21] asymmetry in interdependence [22] and autonomousnodes [23ndash25] being able to facilitate the viability of nodesand alleviate the suddenness of the collapse in an interde-pendent system In addition some features of real inter-dependent systems such as spatial constraints [26ndash30]clustering [31 32] and degree distribution [33 34] alsoenhance the robustness and mitigate cascading failures ofinterdependent networks

A key question in the modeling of multilayer networks ishow to describe the interdependencies across networklayers A straightforward method employed in most pre-vious models of cascading failures in multilayer networks is

HindawiComplexityVolume 2020 Article ID 3578736 11 pageshttpsdoiorg10115520203578736

assuming that the layer interdependence is ldquostrongrdquo where afailure node can cause all of its interdependent neighbors tofail completely [6 23 35ndash40] is assumption has alreadybeen extended extensively to the study of cascading dy-namics in networks under different conditions such as in-terdependency groups in single-layer networks [41ndash43] andk-core percolation [44] weak percolation [45] and re-dundant percolation [46] in multilayer networks Never-theless this assumption is somewhat simplistic and cannotcover the case where nodes are weakly interdependent Forinstance in a civil transportation system the flow of pas-sengers from city to city depends on a number of trans-portation modes such as coaches trains airplanes andferries When any mode becomes unavailable the totalfailure of the other three modes seems impossible eg whena local train station is shut down passenger flow into the citymay be decreased some passengers destined for this city maycancel their trips and the transferring passengers wouldswitch to other cities to reach their destinations e re-duction of passenger flow can cause some routes in othermodes to not operate properly and carriers experience fi-nancial or other losses for instance airlines may cancelflights if passenger numbers are below expectations Spe-cifically the interdependence across network layers can beldquoweakrdquo and the failure of a node cannot destroy all of itsdependency nodes in other network layers with certainty[47 48] Under these circumstances the failure of a trainstation can cause one or more of its interdependent nodes inother network layers to suffer damage or even failure egthe failure of a local coach station which can further lead tofailures in more modes and deteriorate the connectivity ofthe city By this token there may exist a cascading processunderlying a group of interdependent nodes across networklayers which means that the microscopic mechanism ofglobal collapse inmultilayer networks could include not onlythe propagation of failures from node to node inside acertain network layer but also the cascading process offailures across network layers In this paper we regard thepropagation of failures inside a network layer as ldquoinner-layercascadingrdquo and the propagation of failures in a dependencygroup across network layers as ldquocross-layer cascadingrdquo

Previous networks that have considered the ldquostrongrdquointerdependence ignore the microscopic process of ldquocross-layer cascadingrdquo as the failure of one node will destroy all itsinterdependent neighbors In this paper we aim to modelthe cascading dynamics in multilayer networks within amore general situation by using the assumption of ldquoweakrdquointerdependence [47] where the strength of interdepen-dence can be tuned by introducing a tolerance parameter αUsing a comprehensive theoretical study and numericalsimulations we find that the cascading dynamic in multi-layer networks is essentially the synergistic result of ldquocross-layer cascadingrdquo and ldquoinner-layer cascadingrdquo In particularwe find that the system can undergo different types ofpercolation with changing tolerance parameters α Specifi-cally the system percolates as an abrupt (first-order) per-colation transition for small values of α With increasing αexceeding a critical value αc the system percolates in acontinuous (second-order) manner However for scale-free

networks the phenomenon of double phase transitionsoccurs for some moderate parameter values of α where thenetworks in the system first percolate in a continuous(second-order) manner and then experience a first-orderphase transition in an abrupt manner at another phasetransition point

2 Model

We consider a multilayer network consisting of M layers ofnetworks where each network layer has N nodes We labelthe network layers with Latin letters A B C and so on andthe nodes in each network layer are labeled with Arabicnumbers 1 2 N erefore each node in a certainnetwork layer can be identified as a pair of coordinates (xX) with x denoting the node label and X denoting the layerlabel e nodes across network layers with the same Arabicnumber are interdependent on each other ie they arereplica nodes As shown in Figure 1 the nodes (1 A) (1 B)(1 C) and (1 D) are interdependent nodes in network layersA B C and D respectively When one of these nodes failsthe other three nodes will suffer damages Similarly theother nodes can be also divided into several interdependentgroups by their Arabic number labels e nodes in the samenetwork layer X are linked by a set of connectivity links andthe connectivity degree of nodes follows a degree distri-bution pX

k In this paper we consider the case where the M

network layers within the multilayer system have an iden-tical degree distribution for simplicity

e cascading in the multilayer networks is triggered byrandomly selecting a fraction 1 minus p of nodes and theirreplicas as initially failed nodes For each network layer allthe links of failed nodes are removed simultaneously whichbreaks the network layer into a number of connectedcomponents [49] e nodes in the isolated components aretreated as failed ones and the nodes in the giant componentare functional Due to the interdependency among thereplica nodes across network layers a failed node will cause acertain level of damage to it replicas where the damagedegree is controlled by the tolerance parameter α Specifi-cally when one node in a network layer fails each con-nectivity link of its viable interdependency replicas in othernetwork layers will be removed with a probability 1 minus αAlong with the removal of the connectivity links the net-work layers will be further fragmented and thus lead tomore failures After a number of iterations of link removalcaused by node failures and node isolations resulting fromnetwork fragmentation the system can reach a steady stateIn this iterative process the failures can propagate from layerto layer through the interdependencies among replica nodesand the failures can also propagate from node to node alongwith the removal of connectivity links in a certain networklayer simultaneously Here we regard the propagation offailures inside a network layer as ldquoscope extensionrdquo and thepropagation of failures in an interdependency group acrossnetwork layers as ldquodepth penetrationrdquo Especially it shouldbe noticed that there is a microscopic cascading in theprocess of depth penetration a failed node can lead morenode failures in its replicas and thus lead more and more

2 Complexity

node failures in its replicas Schematic illustration of cas-cading process of a four-layer system is shown in Figure 1 Inthis paper we use the relative size SX of the giant componentin each layer of network X to measure the robustness of thenetwork

3 Theory

We use the method of probability generation functions[50 51] to obtain the theoretical solution of the model andthe generating function GX

0 (x) 1113936kpXk xk is employed to

generate the degree distribution pXk of layer X Similarly the

generating function GX1 (x) 1113936kpX

k kxkminus 1langkrangX is used togenerate the excess degree distribution of a node reached byfollowing a random link where langkrang equiv 1113936kpX

k k represents theaverage degree of the network layer X In particular wedefine RX as the probability that a randomly chosen link innetwork X leads to its giant component in the steady state ofthe system For the case where the M network layers in thesystem have a same degree distribution pX

k pk Accord-ingly GX

0 (x) equiv G0(x) GX1 (x) equiv G1(x) RX equiv R and

langkrangX equiv langkrang Assuming that each network of M networklayers is tree-like we aim to obtain the probability SX that arandom node is in the giant component of layer X Because

Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(a)

Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(b)

Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(c)Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(d)

Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(e)

Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(f)

Figure 1 Schematic diagram of the cascading process in a four-layer network e nodes in different layers with the same Arabic numberare replica nodes and they are connected by dotted lines e funtional links are marked in solid lines and the failed links are marked indashed lines e functional nodes are marked in green the failed nodes are marked in red and the yellow nodes are still viable after beingdamaged At stage (A) the replicas with the Arabic number 1 are removed from all layers At stage (B) the node (2 D) becomes isolatedfrom the giant component in layerD and fails which leads to the link removal of its replicas in layersA B andC At stage (C) the node (3C)becomes isolated from the giant component in layer C and fails which leads to the link removal of its replicas in layers A B and DSimultaneously node (2 A) becomes isolated and fails due to the removal of link 23 in layer A which further leads to the link removal of itsreplicas in layers B and C At stage (D) the node (4 A) becomes isolated from the giant component in layer A and fails which leads to thelink removal of its replicas in layers B C and D Simultaneously node (2 A) becomes isolated and fails due to the removal of link 23 in layerB At stage (E) the node (4 C) becomes isolated from the giant component and fails which leads to the link removal of its replicas in layers B

and D At stage (F) the node (4 D) becomes isolated and fails and the system reaches the final steady state (a) Stage A (b) Stage B (c) StageC (d) Stage D (e) Stage E (f ) Stage F

Complexity 3

each layer has the same degree distribution we haveSA SB SC middot middot middot equiv S Following a randomly chosen link inthe layer X we arrive at a node (x X) of degree k with t

failed replicas erefore each link of node (x X) can bepreserved with a probability αt Considering that the degreek follows the probability distribution kpklangkrang the proba-bility that a random link can lead to the giant componentfollows αt[1 minus kpklangkrang(1 minus αtR)kminus 1] which can be simpli-fied as αt[1 minus G1(1 minus αtR)] in terms of the generatingfunction G1(x) If the number t of failed replicas for a givennode follows a probability distribution f(t) we can obtainthe self-consistent equation for R by summing over allpossible t

R p 1113944Mminus 1

t0αt 1 minus G1 1 minus αt

R1113872 11138731113960 1113961f(t) equiv h(R) (1)

Similarly we can obtain the probability S that a randomnode is in the giant component

S p 1113944Mminus 1

t01 minus G0 1 minus αt

R1113872 11138731113960 1113961f(t) (2)

e solution process of equations (1) and (2) utilizes theprobability distribution function f(t) which can be ob-tained by using the probability R Considering that there aret failed replicas for a random node (x X) in the layer X atsteady state the viable probability of each replica is1 minus G0(1 minus αtR) and the remaining M minus t minus 1 replicas are allviable with probability [1 minus G0(1 minus αtR)]Mminus tminus 1 Because thefailures can propagate from layer to layer through the in-terdependencies among replica nodes we assume that thereare s failed replicas caused by the link removal of other nodesin the corresponding network layers and there are t minus s

failed nodes induced by the s failed replicas e probabilityof s failed replicas existing caused by isolation is Gs

0(1 minus R)After that the probability of t minus s additional replicas failingis [G0(1 minus αsR) minus G0(1 minus R)]tminus s erefore f(t) satisfies

f(t) M minus 1

t1113888 1113889 1 minus G0 1 minus αt

R1113872 11138731113960 1113961Mminus tminus 1

1113944

t

s0

t

s1113888 1113889G

s0

middot (1 minus R) G0 1 minus αsR( 1113857 minus G0(1 minus R)1113858 1113859

tminus s

(3)

For a given degree distribution pk we can obtain thefinal size S of the giant component in a certain layer bysolving equations (1) and (2) simultaneously

When α⟶ 1 the interdependence across networklayers is weakest and the system percolates in a second-order manner as in single-layer networks [52 53] Whenα⟶ 0 the interdependence across network layers is thestrongest and the system percolates in a first-order manner[10] erefore the manner of percolation transitions can bedetermined by the value of α and the critical value αc isdefined as the switch point of the percolation transition froma second-order to a first-order For the second-order per-colation transition the probability R tends to zero when p isclose to the second-order percolation point pII

c We can use

the Taylor expansion of equation (1) for R equiv ϵ⟶ 0 andp⟶ pII

c

h(ϵ) hprime(0)ϵ +12hPrime(0)ϵ2 + O ϵ31113872 1113873 ϵ (4)

Ignoring the high-order terms of ϵ we have hprime(0) 1when p⟶ pII

c We thus have the condition for the sec-ond-order percolation transitions

pIIc α

2Mminus 2G1prime (1) 1 (5)

and the second-order percolation point

pIIc

1α2Mminus 2G1prime(1)

(6)

When α 1 or M 1 the second-order percolationtransition point pII

c 1G1prime(1) which is coincident with theresult for the ordinary percolation in a single-layer network[53] In addition we can also find the second-order per-colation point pII

c increases with the increase of M whichmeans the more network layers a system has the morefragile it becomes

At the first-order phase transition point pIc the proba-

bility R jumps from Rc to zero or a nontrivial value and thecurve of y h(R) minus R is tangent with the straight line y 0

dh(R)

dR

1113868111386811138681113868111386811138681113868RRcppIc

1 (7)

In this paper we resort to the numerical method to solveequations (1) and (7) for the first-order percolation tran-sition point pI

cWhen α αc the conditions for the first- and second-

order percolation transitions are satisfied simultaneouslyie pI

c pIIc for Rc⟶ 0 at the percolation transition point

At this time equation (4) reduces to12hPrime(0)ϵ2 + O ϵ31113872 1113873 0 (8)

erefore we know that hPrime(0) 0 when p⟶ pc andα αc We can have

α3G1Prime(1) + 2 α2 minus 11113872 1113873(M minus 1)G1prime (1)G0prime (1) 0 (9)

By the solution of equation (9) we can obtain the switchpoint αc of first-order and second-order percolation tran-sitions Formultilayer random networks the degree of nodesfollows Poisson distribution and equation (9) can be sim-plified as

α3c +(2M minus 2)α2c minus (2M minus 2) 0 (10)

erefore the value of αc is only relevant to M For scale-free networks equation (9) can be written asα3clangk(k minus 1)(k minus 2)rang

langkranglangk(k minus 1)rang+(2M minus 2)α2c minus (2M minus 2) 0 (11)

where the brackets langmiddotrang denote the average over the degreedistribution erefore αc depends not only on M but alsoon the degree distribution pk for multilayer scale-freenetworks

4 Complexity

Figure 2 shows the function curves y h(R) minus R fordifferent values of p from which we can validate the ex-istence of first- and second-order percolation transitionsFigure 2(a) shows the graphical solutions of equation (1) forαlt αc from which we can find that the solution of R is givenby the tangent point when p pI

c indicating a discontin-uous percolation transition From Figure 2(c) we can findthat the nontrivial solution of R emerges at the point p pII

c at which the function curve y h(R) minus R is tangent with theR axis at R 0 indicating a continuous percolation tran-sition Interestingly we also found that the system un-dergoes double phase transitions for some moderate valuesof α if the degree distribution pk of the systems is scale-freeas shown in Figure 2(b) In this case the system first per-colates in a second-order manner and then experiences afirst-order phase transition with increased p and conditions(5) and (7) should be satisfied at the phase transition pointspII

c and pIc successively If condition (5) cannot be satisfied

the double phase transition reduces to a single first-orderpercolation transition and if condition (7) cannot be sat-isfied the double phase transition reduces to a single second-order percolation transition Using these conditions we canlocate the boundary between double phase transitions andsingle first-order phase transitions and the boundary be-tween double phase transitions and single second-orderpercolation transitions

4 Results

41 Synthetic Network In this paper we take two specialmultilayer networks with M 3 and M 4 layers as ex-amples to illustrate the characteristics of percolation dy-namics Figures 3(a) and 3(b) show the size S of the giantcomponent as functions of p for different values of α inthree-layer random networks and four-layer randomnetworks with langkrang 4 respectively For both three-layerand four-layer random networks we find that the systemcan percolate in a discontinuous manner for α 08 or in acontinuous manner for a larger value of α 095 In ad-dition the percolation transition point of three-layer

networks is less than that of four-layer networks whichmeans that three-layer networks are always more robustthan four-layer networks Simultaneously we find that thecritical point αc separating the types of percolation tran-sitions depends on the number of layers in the systemSimilar results can also be found for three-layer randomnetworks and four-layer random networks with langkrang 5 InFigure 3 the theoretical predictions for the size S of thegiant component as functions of p and percolationtransition points pI

c(pIIc ) are also provided from which we

can find that they agree with the simulation results verywell

e results for multilayer scale-free networks are alsoprovided in Figure 4 For each network layer of the systemthe degree of the nodes follows a truncated power-lawdistribution with an average langkrang and the degree distributionis pk sim kminus c(kmin le kle kmax) where kmin and kmax are theminimum and maximum of the degree respectively and c

denotes the power law exponent Similarly we can also findthat the system can percolate in a discontinuous manner fora small value of α or in a continuous manner for a large valueof α and the three-layer networks are more robust than four-layer networks for both langkrang 4 and langkrang 5 Interestinglywe can also find that the four-layer networks can undergo adouble phase transition for α 065 Specifically the systemfirst percolates as a continuous phase transition and thenundergoes a discontinuous phase transition with increasingp With increasing α the discontinuous phase transitiondisappears and the system reduces to a single continuousphase transition With decreasing α the continuous phasetransition disappears and the system reduces to a singlediscontinuous phase transition

Figures 5(a) and 5(b) show the percolation transitionpoints pc as functions of α for three- and four-layer randomnetworks with different average degrees respectively Forboth three- and four-layer random networks the manners ofpercolation transition are classified as discontinuous andcontinuous by a critical value of αc and the critical value ofαc only depends on the number of network layers M in asystem which coincides the theoretical result provided by

01

00

ndash01

y

p = 09

p = 08149

p = 07

00 03 06 09R

(a)

y

p = 08

p = 06

p = 07174p = 06928

005

00

ndash00500 04 08

R

(b)

p = 03

p = 01

p = 01993

y

005

00

ndash00500 005 01

R

(c)

Figure 2e graphical solutions of equation (1) for different values of p and α as marked by the black dots (a)e result for α 05 (b) theresult for α 065 and (c) the result for α 09 For each panel the degree distribution of networks follows a truncated power-lawdistribution pk sim kminus c(kmin le kle kmax) where kmin 2 kmax 141 and c 23

Complexity 5

S

02 03 04 05 06

pcII asymp 0307 pc

II asymp 0381 pcI asymp 0460 pc

I asymp 0503

p

α = 095

α = 0

9 α = 0

85 α = 0

8

06

04

02

00

(a)

pcII asymp 0340 pc

I asymp 0452 pcI asymp 0534 pc

I asymp 0596

025 035 045 055 065p

α = 095 α = 09 α =

085 α =

08

S

06

04

02

00

(b)

pcII asymp 0246 pc


I asymp 0403

02 03 04 05 06p

α = 095

α = 0

9 α = 0

85 α = 0

8

S

06

04

02

00

(c)

pcII asymp 0272 pc


I asymp 0477

025 035 045 055 065p

α = 095

α = 09 α =

085 α =

08

S

06

04

02

00

(d)

Figure 3 Simulation results for percolation transitions on three-layer and four-layer random networks (a) (b) e results for three-layerand four-layer random networks with langkrang 4 respectively (c) (d) e results for three-layer random networks and four-layer randomnetworks with langkrang 5 respectively e results were obtained by averaging over 100 independent realizations and the network size wasN 105 e solid lines behind the symbols denote the theoretical predictions that were obtained by equations (1) and (2) e verticaldashed lines denote the first- and second-order percolation transition points predicted by equations (5) and (7) respectively

02 04 06 08

S

09

06

03

0

pcII asymp 0460 pc


I asymp 0768

α = 07

α = 0

65α =

06

α =

05

p

(a)

04 06 08 10

pcII asymp 0421 pc

I asymp 0768

pcI asymp 0835 pc

I asymp 0868

α = 08

α =

05α

= 0

7

α =

06

S

09

06

03

0

p

(b)

Figure 4 Continued

6 Complexity

pcII asymp 0218 pc

II asymp 0292 pcII asymp 0403 pc

I asymp 0653

α = 07

α = 065α = 06

α = 0

5

02 04 06 08p

S09

06

03

0

(c)

pcII asymp 0444 pc

I asymp 0693

pcI asymp 0764 pc

I asymp 0815

α = 0

7 α =

065

04 06 08 10p

S

09

06

03

0

α =

06

α =

05

(d)

Figure 4 Simulation results for percolation transitions on three- and four-layer scale-free networks (a) (b)e results for three- and four-layer scale-free networks with langkrang 4 respectively (c) (d) e results for three- and four-layer scale-free networks with langkrang 5 re-spectively When the average degree langkrang 4 the parameter settings for the power-law distribution are kmin 2 kmax 63 and c 25When the average degree langkrang 5 the parameter settings are kmin 2 kmax 141 and c 23 e simulation results were obtained byaveraging over 100 independent realizations and the network size is N 105 e solid lines behind the symbols denote the theoreticalpredictions that were obtained by equations (1) and (2) e vertical dashed lines denote the first- and second-order percolation transitionpoints predicted by equations (5) and (7) respectively

10080604020000

02

04

06

08

10

p c

α

ltkgt = 4ltkgt = 5ltkgt = 6

(a)

00

02

04

06

08

10

p c

100806040200α


(b)

Figure 5 Continued

Complexity 7

equation (10) Figures 5(c) and 5(d) show the percolationtransition points pc as functions of α for three- and four-layer scale-free networks with different average degreesrespectively fromwhich we can also find that themanners ofpercolation transition are classified as discontinuous andcontinuous by a critical value of αc however the specificvalue of αc depends on the parameter settings of the degreedistributions ese results imply that the robustness of amultilayer network increases with the increase of the average

degree of network layers and decreases with the increase ofthe number M of network layers For multilayer randomnetworks the collapse manner is irrelevant to the averagedegree of the networks and abrupt breakdown cannot beavoided by the increase of the average degree For multilayerscale-free networks the collapse manner is relevant to thenumber M of network layers and the parameters of degreedistribution ig minimum degree maximum degree andpower law exponent

00

02

04

06

08

10p c

100806040200α


(c)

00

02

04

06

08

10

p c

100806040200α


(d)

Figure 5 (Color online) (a)e percolation transition point pc ((pIc) or (pII

c )) versus α for three-layer random networks where the averagedegree langkrang is 4 5 and 6 from top to bottom respectively (b) Corresponding results for four-layer random networks (c)e correspondingresults for three-layer scale-free networks of average degree langkrang 4 5 and 6 (corresponding to a power-law exponent of degree distributionminus 26 minus 23 and minus 21 from top to bottom respectively) with minimum degree 2 (d) Corresponding results for four-layer scale-free networks

00 02 04 06p

04

03

02

01

SA

00

α = 00α = 03

α = 06α = 09

(a)

04

03

02

01

SB

0000 02 04 06

p

α = 00α = 03

α = 06α = 09

(b)

04

03

02

01

SC

0000 02 04 06

p

α = 00α = 03

α = 06α = 09

(c)

Figure 6 Percolation inmultilayer empirical networkse system consists of the three layers of networks A B and Cwhich represent threemajor carriers American Airlines (AA) Delta Air Lines (DL) and United Airlines (UA) respectively (andashc) e sizes of the giantcomponents of the network layers A B and C as functions of p for different values of α respectively e data points are the result ofaveraging over 1000 statistical realizations (a) AA (b) DL (c) UA

8 Complexity

42 Empirical Networks To address the percolation processin empirical multilayer networks we consider a three-layersystem constituting the three major carriers in the UnitedStates Delta Air Lines (DL) American Airlines (AA) andUnited Airlines (UA) In each layer of a network nodesrepresent airports and links between two airports areconnected in the layers if there is at least one flight operatedby a given carrier We construct the multilayer system usingthe dataset from OpenFlights (httpsopenflightsorgdatahtml) For civil flights of the three major carriers there are intotal N 310 nodes (functioning airports) Some of thenodes do not appear as connected in all the layers leading toa difference in the relative sizes of the giant componentsFigures 6(a)ndash6(c)) show the sizes S of the giant component asfunctions of p for AA DL and UA respectivelyWe can findthat a large value of α always leads to a larger size S of thegiant component for three layers which means that therobustness of the system can be improved greatly byrestricting the interdependence across network layers withincreasing α

5 Conclusions and Discussion

e interdependence of real multilayer networks is generallyweak in layer-to-layer interactions where the failure of onenode usually does not result in failures of interdependentnodes across all network layers In this paper we haveexamined the percolation process and the robustness of amultilayer network when the interdependence of nodesacross networks is weak We reveal that the avalancheprocess of the whole system can be essentially decomposedinto two microscopic cascading dynamics in terms of thepropagation direction of failures depth penetration andscope extension Specifically the former describes thepropagation of failures across network layers and thus isregarded as ldquocross-layer cascadingrdquo while the latter de-scribes the propagation of failures inside a network layerand thus is regarded as ldquoinner-layer cascadingrdquo With thecoaction of the two cascading dynamics a multilayer net-work can disintegrate via first- or second-order percolationtransitions in the case of initial failures where the inter-dependence across network layers plays important roles indetermining the percolation behaviors of the system Whenthe interdependence of network layers is weak the failures ofnodes can neither penetrate into deep network layers norcause great destructiveness to their interdependent replicaswhich inhibits the spread of failure and makes the systempercolate via a second-order percolation transition Whenthe interdependence of network layers is strong the failuresof nodes can penetrate into deep network layers in a cas-cading manner and spread with a broad scope throughvarious network layers which thus makes the systemcollapse abruptly ese results prove that the process ofldquocross-layer cascadingrdquo dominates over the process of ldquoin-ner-layer cascadingrdquo and plays a crucial role in determiningthe robustness of a multilayer system

e present work essentially reveals the complexity ofcascading failures and previous works ignoring the weakinterdependence may underestimate the complexity

Specifically the cascading dynamics that occur across net-work layers cannot be produced in a strong layer-to-layerinterdependence of multilayer networks erefore ourmethod insights for the intervention of cascading failures ina multilayer network and evidences the idea that imposingrestrictions on ldquocross-layer cascadingrdquo can restrain thespread of failures more effectively Our work not only offersa new understanding of the cascading failure dynamics ofmultilayer networks but also implies that the strength ofinterdependence can be used for enhancing the robustnessof multilayer structured infrastructure systems at is tosay the penetration depth of cascading failures can besignificantly reduced and thus the scale of cascading failuresis also decreased by imposing restrictions on the strength ofinterdependence if the strength of interdependence is ad-justable For some other systems with fixed or nonadjustableinterdependence across network layers our finding is alsomeaningful for the robustness assessment In this case wecan evaluate the robustness of such system by the parameterssuch as average degrees number of network layers strengthof interdependence and so on

It has been found that overlapping links [14 15] of realmultilayer networks are able to relieve the destruction ofcascading failures and improve robustness of the systemserefore it is meaningful to study the percolation onmultilayer networks with overlapping links in terms oftunable interdependency strength across networks in futureworks Specifically for a system with M network layer anytwo networks of it may have overlapping links and thedegree of overlap may change with the varying of combi-nation of network layers We believe that the study of thisissue will yield richer results

Data Availability

e data of empirical networks used in this paper weredownloaded from the website OpenFlights (httpsopenflightsorgdatahtml) which contains basic data formultiple airports and flights worldwide and we extract thedata of civil flights of United States to built a three-layernetworke data of simulation used to support the findingsof this study are available from the corresponding authorupon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China under grant no 61773148

References

[1] D F Klosik A Grimbs S Bornholdt and M-T Htt ldquoeinterdependent network of gene regulation and metabolism isrobust where it needs to berdquo Nature Communications vol 8no 1 p 534 2017

Complexity 9

[2] M Ouyang ldquoReview on modeling and simulation of inter-dependent critical infrastructure systemsrdquo Reliability Engi-neering amp System Safety vol 121 pp 43ndash60 2014

[3] S M Rinaldi J P Peerenboom and T K Kelly ldquoIdentifyingunderstanding and analyzing critical infrastructure interde-pendenciesrdquo IEEE Control Systems vol 21 no 6 pp 11ndash252001

[4] F Radicchi ldquoPercolation in real interdependent networksrdquoNature Physics vol 11 no 7 pp 597ndash602 2015

[5] J Wu J Zhong Z Chen and B Chen ldquoOptimal couplingpatterns in interconnected communication networksrdquo IEEETransactions on Circuits and Systems II Express Briefs vol 65no 8 pp 1109ndash1113 2018

[6] S V Buldyrev R Parshani G Paul H E Stanley andS Havlin ldquoCatastrophic cascade of failures in interdependentnetworksrdquo Nature vol 464 no 7291 pp 1025ndash1028 2010

[7] M Kivela A Arenas M Barthelemy J P GleesonY Moreno and M A Porter ldquoMultilayer networksrdquo Journalof Complex Networks vol 2 no 3 pp 203ndash271 2014

[8] J Gao S V Buldyrev S Havlin and H E Stanley ldquoRo-bustness of a network of networksrdquo Physical Review Lettersvol 107 no 19 Article ID 195701 2011

[9] J Gao S V Buldyrev H E Stanley and S Havlin ldquoNetworksformed from interdependent networksrdquo Nature Physicsvol 8 no 1 pp 40ndash48 2012

[10] G J Baxter S N Dorogovtsev A V Goltsev andJ F F Mendes ldquoAvalanche collapse of interdependent net-worksrdquo Physical Review Letters vol 109 no 24 Article ID248701 2012

[11] H Kesten Percolation3eory for Mathematicians BirkhauserPress Boston MA USA 1982

[12] D Stauffer and A Aharony Introduction to Percolation3eory Tailor amp Francis Press London UK 1992

[13] S-W Son G Bizhani C Christensen P Grassberger andM Paczuski ldquoPercolation theory on interdependent networksbased on epidemic spreadingrdquo EPL (Europhysics Letters)vol 97 no 1 p 16006 2012

[14] D Cellai E Lopez J Zhou J P Gleeson and G BianconildquoPercolation in multiplex networks with overlaprdquo PhysicalReview Letters vol 88 Article ID 052811 2013

[15] Y Hu D Zhou R Zhang Z Han C Rozenblat andS Havlin ldquoPercolation of interdependent networks withintersimilarityrdquo Physical Review E vol 88 no 5 Article ID052805 2013

[16] K-K Kleineberg M Boguntildea M Angeles Serrano andF Papadopoulos ldquoHidden geometric correlations in realmultiplexnetworksrdquo Nature Physics vol 12 no 11pp 1076ndash1081 2016

[17] K-K Kleineberg L Buzna F Papadopoulos M Boguntildea andM A Serrano ldquoGeometric correlations mitigate the extremevulnerability of multiplex networks against targeted attacksrdquoPhysical Review Letters vol 118 no 21 Article ID 2183012017

[18] A Faqeeh S Osat and F Radicchi ldquoCharacterizing theanalogy between hyperbolic embedding and communitystructure of complex networksrdquo Physical Review Lettersvol 121 no 9 Article ID 098301 2018

[19] B Min S D Yi K-M Lee and K-I Goh ldquoNetwork ro-bustness of multiplex networks with interlayer degree cor-relationsrdquo Physical Review E vol 89 no 4 Article ID 0428112014

[20] R Parshani C Rozenblat D Ietri C Ducruet and S HavlinldquoInter-similarity between coupled networksrdquo EPL (Euro-physics Letters) vol 92 no 6 p 68002 2011

[21] L D Ducruet P A Macri H E Stanley and L A BraunsteinldquoTriple point in correlated interdependent networksrdquo PhysicalReview E vol 88 no 5 Article ID 050803 2013

[22] R-R Liu C-X Jia and Y-C Lai ldquoAsymmetry in interde-pendence makes a multilayer system more robust againstcascading failuresrdquo Physical Review E vol 100 no 5 ArticleID 052306 2019

[23] R Parshani S V Buldyrev and S Havlin ldquoInterdependentnetworks reducing the coupling strength leads to a changefrom a first to second order percolation transitionrdquo PhysicalReview Letters vol 105 no 4 Article ID 048701 2010

[24] C M Schneider N Yazdani N A M Arajo S Havlin andH J Herrmann ldquoTowards designing robust coupled net-worksrdquo Scientific Reports vol 3 no 1 p 1969 2013

[25] L D Valdez P A Macri and L A Braunstein ldquoA triple pointinduced by targeted autonomization on interdependent scale-free networksrdquo Journal of Physics A Mathematical and3eoretical vol 47 no 5 Article ID 055002 2014

[26] Y Berezin A Bashan M M Danziger D Li and S HavlinldquoLocalized attacks on spatially embedded networks with de-pendenciesrdquo Scientific Reports vol 5 no 1 p 8934 2015

[27] A Bashan Y Berezin S V Buldyrev and S Havlin ldquoeextreme vulnerability of interdependent spatially embeddednetworksrdquo Nature Physics vol 9 no 10 pp 667ndash672 2013

[28] M M Danziger A Bashan Y Berezin and S HavlinldquoPercolation and cascade dynamics of spatial networks withpartial dependencyrdquo Journal of Complex Networks vol 2no 4 pp 460ndash474 2014

[29] L M Shekhtman Y Berezin M M Danziger and S HavlinldquoRobustness of a network formed of spatially embeddednetworksrdquo Physical Review E vol 90 no 1 Article ID 0128092014

[30] K Gong J-J Wu Y Liu Q Li R-R Liu and M Tang ldquoeeffective healing strategy against localized attacks on inter-dependent spatially embedded networksrdquo Complexityvol 2019 Article ID 7912857 10 pages 2019

[31] S Shao X Huang H E Stanley and S Havlin ldquoRobustness ofa partially interdependent network formed of clustered net-worksrdquo Physical Review E vol 89 Article ID 032812 2014

[32] X Huang S Shao H Wang S V Buldyrev H EugeneStanley and S Havlin ldquoe robustness of interdependentclustered networksrdquo EPL (Europhysics Letters) vol 101 no 1p 18002 2013

[33] T Emmerich A Bunde and S Havlin ldquoStructural andfunctional properties of spatially embedded scale-free net-worksrdquo Physical Review E vol 89 no 6 Article ID 0628062014

[34] X Yuan S Shao H E Stanley and S Havlin ldquoHow breadthof degree distribution influences network robustness com-paring localized and random attacksrdquo Physical Review Evol 92 no 3 Article ID 032122 2015

[35] X Yuan Y Hu H E Stanley and S Havlin ldquoEradicatingcatastrophic collapse in interdependent networks via rein-forced nodesrdquo Proceedings of the National Academy of Sci-ences vol 114 no 13 pp 3311ndash3315 2017

[36] L B Shaw and I B Schwartz ldquoEnhanced vaccine control ofepidemics in adaptive networksrdquo Physical Review E vol 81no 4 Article ID 046120 2010

[37] D Zhao L Wang S Li Z Wang L Wang and B GaoldquoImmunization of epidemics in multiplex networksrdquo PLoSOne vol 9 no 11 Article ID e112018 2014

[38] J C Nacher and T Akutsu ldquoStructurally robust control ofcomplex networksrdquo Physical Review E vol 91 no 1 ArticleID 012826 2015

10 Complexity

[39] Z Wang D Zhou and Y Hu ldquoGroup percolation in in-terdependent networksrdquo Physical Review E vol 97 no 3Article ID 032306 2018

[40] R-R Liu C-X Jia and Y-C Lai ldquoRemote control of cas-cading dynamics on complex multilayer networksrdquo NewJournal of Physics vol 21 Article ID 045002 2019

[41] A Bashan and S Havlin ldquoe combined effect of connectivityand dependency links on percolation of networksrdquo Journal ofStatistical Physics vol 145 no 3 pp 686ndash695 2011

[42] A Bashan R Parshani and S Havlin ldquoPercolation in net-works composed of connectivity and dependency linksrdquoPhysical Review E vol 83 no 5 Article ID 051127 2011

[43] M Li R-R Liu C-X Jia and B-H Wang ldquoCritical effects ofoverlapping of connectivity and dependence links on per-colation of networksrdquo New Journal of Physics vol 15 no 9Article ID 093013 2013

[44] N Azimi-Tafreshi J Gomez-Gardentildees and S N Dorogovtsevldquok-core percolation on multiplex networksrdquo Physical Review Evol 90 no 3 Article ID 032816 2014

[45] G J Baxter S N Dorogovtsev J F F Mendes and D CellaildquoWeak percolation onmultiplex networksrdquo Physical Review Evol 89 no 4 Article ID 042801 2014

[46] F Radicchi and G Bianconi ldquoRedundant interdependenciesboost the robustness of multiplex networksrdquo Physical ReviewX vol 7 no 1 Article ID 011013 2017

[47] R-R Liu D A Eisenberg T P Seager and Y-C Lai ldquoeldquoweakrdquo interdependence of infrastructure systems producesmixed percolation transitions in multilayer networksrdquo Sci-entific Reports vol 8 no 1 p 2111 2018

[48] R-R Liu M Li and C-X Jia ldquoCascading failures in couplednetworks the critical role of node-coupling strength acrossnetworksrdquo Scientific Reports vol 6 p 35352 2016

[49] M Molloy and B Reed ldquoA critical point for random graphswith a given degree sequencerdquo Random Structures amp Algo-rithms vol 6 no 2-3 pp 161ndash180 1995

[50] S-W Son G Bizhani C Christensen P Grassberger andM Paczuski ldquoPercolation theory on interdependent networksbased on epidemic spreadingrdquo EPL vol 97 p 16006 2011

[51] L Feng C P Monterola and Y Hu ldquoe simplified self-consistent probabilities method for percolation and its ap-plication to interdependent networksrdquo New Journal ofPhysics vol 17 Article ID 063025 2015

[52] R Cohen K Erez D ben-Avraham and S Havlin ldquoResil-ience of the internet to random breakdownsrdquo Physical ReviewLetters vol 85 no 21 pp 4626ndash4628 2000

[53] D S Callaway M E J Newman S H Strogatz andD J Watts ldquoNetwork robustness and fragility percolation onrandom graphsrdquo Physical Review Letters vol 85 no 25pp 5468ndash5471 2000

Complexity 11

assuming that the layer interdependence is ldquostrongrdquo where afailure node can cause all of its interdependent neighbors tofail completely [6 23 35ndash40] is assumption has alreadybeen extended extensively to the study of cascading dy-namics in networks under different conditions such as in-terdependency groups in single-layer networks [41ndash43] andk-core percolation [44] weak percolation [45] and re-dundant percolation [46] in multilayer networks Never-theless this assumption is somewhat simplistic and cannotcover the case where nodes are weakly interdependent Forinstance in a civil transportation system the flow of pas-sengers from city to city depends on a number of trans-portation modes such as coaches trains airplanes andferries When any mode becomes unavailable the totalfailure of the other three modes seems impossible eg whena local train station is shut down passenger flow into the citymay be decreased some passengers destined for this city maycancel their trips and the transferring passengers wouldswitch to other cities to reach their destinations e re-duction of passenger flow can cause some routes in othermodes to not operate properly and carriers experience fi-nancial or other losses for instance airlines may cancelflights if passenger numbers are below expectations Spe-cifically the interdependence across network layers can beldquoweakrdquo and the failure of a node cannot destroy all of itsdependency nodes in other network layers with certainty[47 48] Under these circumstances the failure of a trainstation can cause one or more of its interdependent nodes inother network layers to suffer damage or even failure egthe failure of a local coach station which can further lead tofailures in more modes and deteriorate the connectivity ofthe city By this token there may exist a cascading processunderlying a group of interdependent nodes across networklayers which means that the microscopic mechanism ofglobal collapse inmultilayer networks could include not onlythe propagation of failures from node to node inside acertain network layer but also the cascading process offailures across network layers In this paper we regard thepropagation of failures inside a network layer as ldquoinner-layercascadingrdquo and the propagation of failures in a dependencygroup across network layers as ldquocross-layer cascadingrdquo

Previous networks that have considered the ldquostrongrdquointerdependence ignore the microscopic process of ldquocross-layer cascadingrdquo as the failure of one node will destroy all itsinterdependent neighbors In this paper we aim to modelthe cascading dynamics in multilayer networks within amore general situation by using the assumption of ldquoweakrdquointerdependence [47] where the strength of interdepen-dence can be tuned by introducing a tolerance parameter αUsing a comprehensive theoretical study and numericalsimulations we find that the cascading dynamic in multi-layer networks is essentially the synergistic result of ldquocross-layer cascadingrdquo and ldquoinner-layer cascadingrdquo In particularwe find that the system can undergo different types ofpercolation with changing tolerance parameters α Specifi-cally the system percolates as an abrupt (first-order) per-colation transition for small values of α With increasing αexceeding a critical value αc the system percolates in acontinuous (second-order) manner However for scale-free

networks the phenomenon of double phase transitionsoccurs for some moderate parameter values of α where thenetworks in the system first percolate in a continuous(second-order) manner and then experience a first-orderphase transition in an abrupt manner at another phasetransition point

2 Model

We consider a multilayer network consisting of M layers ofnetworks where each network layer has N nodes We labelthe network layers with Latin letters A B C and so on andthe nodes in each network layer are labeled with Arabicnumbers 1 2 N erefore each node in a certainnetwork layer can be identified as a pair of coordinates (xX) with x denoting the node label and X denoting the layerlabel e nodes across network layers with the same Arabicnumber are interdependent on each other ie they arereplica nodes As shown in Figure 1 the nodes (1 A) (1 B)(1 C) and (1 D) are interdependent nodes in network layersA B C and D respectively When one of these nodes failsthe other three nodes will suffer damages Similarly theother nodes can be also divided into several interdependentgroups by their Arabic number labels e nodes in the samenetwork layer X are linked by a set of connectivity links andthe connectivity degree of nodes follows a degree distri-bution pX

k In this paper we consider the case where the M

network layers within the multilayer system have an iden-tical degree distribution for simplicity

e cascading in the multilayer networks is triggered byrandomly selecting a fraction 1 minus p of nodes and theirreplicas as initially failed nodes For each network layer allthe links of failed nodes are removed simultaneously whichbreaks the network layer into a number of connectedcomponents [49] e nodes in the isolated components aretreated as failed ones and the nodes in the giant componentare functional Due to the interdependency among thereplica nodes across network layers a failed node will cause acertain level of damage to it replicas where the damagedegree is controlled by the tolerance parameter α Specifi-cally when one node in a network layer fails each con-nectivity link of its viable interdependency replicas in othernetwork layers will be removed with a probability 1 minus αAlong with the removal of the connectivity links the net-work layers will be further fragmented and thus lead tomore failures After a number of iterations of link removalcaused by node failures and node isolations resulting fromnetwork fragmentation the system can reach a steady stateIn this iterative process the failures can propagate from layerto layer through the interdependencies among replica nodesand the failures can also propagate from node to node alongwith the removal of connectivity links in a certain networklayer simultaneously Here we regard the propagation offailures inside a network layer as ldquoscope extensionrdquo and thepropagation of failures in an interdependency group acrossnetwork layers as ldquodepth penetrationrdquo Especially it shouldbe noticed that there is a microscopic cascading in theprocess of depth penetration a failed node can lead morenode failures in its replicas and thus lead more and more

2 Complexity


3 Theory










Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(a)

Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(b)

Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(c)Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(d)

Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(e)

Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(f)



Complexity 3



R p 1113944Mminus 1


R1113872 11138731113960 1113961f(t) equiv h(R) (1)


S p 1113944Mminus 1


R1113872 11138731113960 1113961f(t) (2)




f(t) M minus 1

t1113888 1113889 1 minus G0 1 minus αt

R1113872 11138731113960 1113961Mminus tminus 1

1113944

t

s0

t

s1113888 1113889G

s0


tminus s

(3)



c We can use


c




pIIc α



pIIc


(6)






dh(R)

dR

1113868111386811138681113868111386811138681113868RRcppIc

1 (7)













4 Complexity






4 Results








01

00

ndash01

y

p = 09

p = 08149

p = 07

00 03 06 09R

(a)

y

p = 08

p = 06

p = 07174p = 06928

005

00

ndash00500 04 08

R

(b)

p = 03

p = 01

p = 01993

y

005

00

ndash00500 005 01

R

(c)


Complexity 5

S

02 03 04 05 06

pcII asymp 0307 pc


I asymp 0503

p

α = 095

α = 0

9 α = 0

85 α = 0

8

06

04

02

00

(a)

pcII asymp 0340 pc


I asymp 0596

025 035 045 055 065p

α = 095 α = 09 α =

085 α =

08

S

06

04

02

00

(b)

pcII asymp 0246 pc


I asymp 0403

02 03 04 05 06p

α = 095

α = 0

9 α = 0

85 α = 0

8

S

06

04

02

00

(c)

pcII asymp 0272 pc


I asymp 0477

025 035 045 055 065p

α = 095

α = 09 α =

085 α =

08

S

06

04

02

00

(d)


02 04 06 08

S

09

06

03

0

pcII asymp 0460 pc


I asymp 0768

α = 07

α = 0

65α =

06

α =

05

p

(a)

04 06 08 10

pcII asymp 0421 pc

I asymp 0768

pcI asymp 0835 pc

I asymp 0868

α = 08

α =

05α

= 0

7

α =

06

S

09

06

03

0

p

(b)

Figure 4 Continued

6 Complexity

pcII asymp 0218 pc


I asymp 0653

α = 07

α = 065α = 06

α = 0

5

02 04 06 08p

S09

06

03

0

(c)

pcII asymp 0444 pc

I asymp 0693

pcI asymp 0764 pc

I asymp 0815

α = 0

7 α =

065

04 06 08 10p

S

09

06

03

0

α =

06

α =

05

(d)


10080604020000

02

04

06

08

10

p c

α


(a)

00

02

04

06

08

10

p c

100806040200α


(b)

Figure 5 Continued

Complexity 7



00

02

04

06

08

10p c

100806040200α


(c)

00

02

04

06

08

10

p c

100806040200α


(d)



00 02 04 06p

04

03

02

01

SA

00

α = 00α = 03

α = 06α = 09

(a)

04

03

02

01

SB

0000 02 04 06

p

α = 00α = 03

α = 06α = 09

(b)

04

03

02

01

SC

0000 02 04 06

p

α = 00α = 03

α = 06α = 09

(c)


8 Complexity







Data Availability




Acknowledgments


References


Complexity 9






































10 Complexity
















Complexity 11


3 Theory










Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(a)

Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(b)

Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(c)Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(d)

Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(e)

Layer A

Layer B

Layer C

Layer D

1

1

1

1

2

2

2

2

3

3

3

3

4

4

5

5

5

5

4

4

(f)



Complexity 3



R p 1113944Mminus 1


R1113872 11138731113960 1113961f(t) equiv h(R) (1)


S p 1113944Mminus 1


R1113872 11138731113960 1113961f(t) (2)




f(t) M minus 1

t1113888 1113889 1 minus G0 1 minus αt

R1113872 11138731113960 1113961Mminus tminus 1

1113944

t

s0

t

s1113888 1113889G

s0


tminus s

(3)



c We can use


c




pIIc α



pIIc


(6)






dh(R)

dR

1113868111386811138681113868111386811138681113868RRcppIc

1 (7)













4 Complexity






4 Results








01

00

ndash01

y

p = 09

p = 08149

p = 07

00 03 06 09R

(a)

y

p = 08

p = 06

p = 07174p = 06928

005

00

ndash00500 04 08

R

(b)

p = 03

p = 01

p = 01993

y

005

00

ndash00500 005 01

R

(c)


Complexity 5

S

02 03 04 05 06

pcII asymp 0307 pc


I asymp 0503

p

α = 095

α = 0

9 α = 0

85 α = 0

8

06

04

02

00

(a)

pcII asymp 0340 pc


I asymp 0596

025 035 045 055 065p

α = 095 α = 09 α =

085 α =

08

S

06

04

02

00

(b)

pcII asymp 0246 pc


I asymp 0403

02 03 04 05 06p

α = 095

α = 0

9 α = 0

85 α = 0

8

S

06

04

02

00

(c)

pcII asymp 0272 pc


I asymp 0477

025 035 045 055 065p

α = 095

α = 09 α =

085 α =

08

S

06

04

02

00

(d)


02 04 06 08

S

09

06

03

0

pcII asymp 0460 pc


I asymp 0768

α = 07

α = 0

65α =

06

α =

05

p

(a)

04 06 08 10

pcII asymp 0421 pc

I asymp 0768

pcI asymp 0835 pc

I asymp 0868

α = 08

α =

05α

= 0

7

α =

06

S

09

06

03

0

p

(b)

Figure 4 Continued

6 Complexity

pcII asymp 0218 pc


I asymp 0653

α = 07

α = 065α = 06

α = 0

5

02 04 06 08p

S09

06

03

0

(c)

pcII asymp 0444 pc

I asymp 0693

pcI asymp 0764 pc

I asymp 0815

α = 0

7 α =

065

04 06 08 10p

S

09

06

03

0

α =

06

α =

05

(d)


10080604020000

02

04

06

08

10

p c

α


(a)

00

02

04

06

08

10

p c

100806040200α


(b)

Figure 5 Continued

Complexity 7



00

02

04

06

08

10p c

100806040200α


(c)

00

02

04

06

08

10

p c

100806040200α


(d)



00 02 04 06p

04

03

02

01

SA

00

α = 00α = 03

α = 06α = 09

(a)

04

03

02

01

SB

0000 02 04 06

p

α = 00α = 03

α = 06α = 09

(b)

04

03

02

01

SC

0000 02 04 06

p

α = 00α = 03

α = 06α = 09

(c)


8 Complexity







Data Availability




Acknowledgments


References


Complexity 9






































10 Complexity
















Complexity 11



R p 1113944Mminus 1


R1113872 11138731113960 1113961f(t) equiv h(R) (1)


S p 1113944Mminus 1


R1113872 11138731113960 1113961f(t) (2)




f(t) M minus 1

t1113888 1113889 1 minus G0 1 minus αt

R1113872 11138731113960 1113961Mminus tminus 1

1113944

t

s0

t

s1113888 1113889G

s0


tminus s

(3)



c We can use


c




pIIc α



pIIc


(6)






dh(R)

dR

1113868111386811138681113868111386811138681113868RRcppIc

1 (7)













4 Complexity






4 Results








01

00

ndash01

y

p = 09

p = 08149

p = 07

00 03 06 09R

(a)

y

p = 08

p = 06

p = 07174p = 06928

005

00

ndash00500 04 08

R

(b)

p = 03

p = 01

p = 01993

y

005

00

ndash00500 005 01

R

(c)


Complexity 5

S

02 03 04 05 06

pcII asymp 0307 pc


I asymp 0503

p

α = 095

α = 0

9 α = 0

85 α = 0

8

06

04

02

00

(a)

pcII asymp 0340 pc


I asymp 0596

025 035 045 055 065p

α = 095 α = 09 α =

085 α =

08

S

06

04

02

00

(b)

pcII asymp 0246 pc


I asymp 0403

02 03 04 05 06p

α = 095

α = 0

9 α = 0

85 α = 0

8

S

06

04

02

00

(c)

pcII asymp 0272 pc


I asymp 0477

025 035 045 055 065p

α = 095

α = 09 α =

085 α =

08

S

06

04

02

00

(d)


02 04 06 08

S

09

06

03

0

pcII asymp 0460 pc


I asymp 0768

α = 07

α = 0

65α =

06

α =

05

p

(a)

04 06 08 10

pcII asymp 0421 pc

I asymp 0768

pcI asymp 0835 pc

I asymp 0868

α = 08

α =

05α

= 0

7

α =

06

S

09

06

03

0

p

(b)

Figure 4 Continued

6 Complexity

pcII asymp 0218 pc


I asymp 0653

α = 07

α = 065α = 06

α = 0

5

02 04 06 08p

S09

06

03

0

(c)

pcII asymp 0444 pc

I asymp 0693

pcI asymp 0764 pc

I asymp 0815

α = 0

7 α =

065

04 06 08 10p

S

09

06

03

0

α =

06

α =

05

(d)


10080604020000

02

04

06

08

10

p c

α


(a)

00

02

04

06

08

10

p c

100806040200α


(b)

Figure 5 Continued

Complexity 7



00

02

04

06

08

10p c

100806040200α


(c)

00

02

04

06

08

10

p c

100806040200α


(d)



00 02 04 06p

04

03

02

01

SA

00

α = 00α = 03

α = 06α = 09

(a)

04

03

02

01

SB

0000 02 04 06

p

α = 00α = 03

α = 06α = 09

(b)

04

03

02

01

SC

0000 02 04 06

p

α = 00α = 03

α = 06α = 09

(c)


8 Complexity







Data Availability




Acknowledgments


References


Complexity 9






































10 Complexity
















Complexity 11






4 Results








01

00

ndash01

y

p = 09

p = 08149

p = 07

00 03 06 09R

(a)

y

p = 08

p = 06

p = 07174p = 06928

005

00

ndash00500 04 08

R

(b)

p = 03

p = 01

p = 01993

y

005

00

ndash00500 005 01

R

(c)


Complexity 5

S

02 03 04 05 06

pcII asymp 0307 pc


I asymp 0503

p

α = 095

α = 0

9 α = 0

85 α = 0

8

06

04

02

00

(a)

pcII asymp 0340 pc


I asymp 0596

025 035 045 055 065p

α = 095 α = 09 α =

085 α =

08

S

06

04

02

00

(b)

pcII asymp 0246 pc


I asymp 0403

02 03 04 05 06p

α = 095

α = 0

9 α = 0

85 α = 0

8

S

06

04

02

00

(c)

pcII asymp 0272 pc


I asymp 0477

025 035 045 055 065p

α = 095

α = 09 α =

085 α =

08

S

06

04

02

00

(d)


02 04 06 08

S

09

06

03

0

pcII asymp 0460 pc


I asymp 0768

α = 07

α = 0

65α =

06

α =

05

p

(a)

04 06 08 10

pcII asymp 0421 pc

I asymp 0768

pcI asymp 0835 pc

I asymp 0868

α = 08

α =

05α

= 0

7

α =

06

S

09

06

03

0

p

(b)

Figure 4 Continued

6 Complexity

pcII asymp 0218 pc


I asymp 0653

α = 07

α = 065α = 06

α = 0

5

02 04 06 08p

S09

06

03

0

(c)

pcII asymp 0444 pc

I asymp 0693

pcI asymp 0764 pc

I asymp 0815

α = 0

7 α =

065

04 06 08 10p

S

09

06

03

0

α =

06

α =

05

(d)


10080604020000

02

04

06

08

10

p c

α


(a)

00

02

04

06

08

10

p c

100806040200α


(b)

Figure 5 Continued

Complexity 7



00

02

04

06

08

10p c

100806040200α


(c)

00

02

04

06

08

10

p c

100806040200α


(d)



00 02 04 06p

04

03

02

01

SA

00

α = 00α = 03

α = 06α = 09

(a)

04

03

02

01

SB

0000 02 04 06

p

α = 00α = 03

α = 06α = 09

(b)

04

03

02

01

SC

0000 02 04 06

p

α = 00α = 03

α = 06α = 09

(c)


8 Complexity







Data Availability




Acknowledgments


References


Complexity 9






































10 Complexity
















Complexity 11

S

02 03 04 05 06

pcII asymp 0307 pc


I asymp 0503

p

α = 095

α = 0

9 α = 0

85 α = 0

8

06

04

02

00

(a)

pcII asymp 0340 pc


I asymp 0596

025 035 045 055 065p

α = 095 α = 09 α =

085 α =

08

S

06

04

02

00

(b)

pcII asymp 0246 pc


I asymp 0403

02 03 04 05 06p

α = 095

α = 0

9 α = 0

85 α = 0

8

S

06

04

02

00

(c)

pcII asymp 0272 pc


I asymp 0477

025 035 045 055 065p

α = 095

α = 09 α =

085 α =

08

S

06

04

02

00

(d)


02 04 06 08

S

09

06

03

0

pcII asymp 0460 pc


I asymp 0768

α = 07

α = 0

65α =

06

α =

05

p

(a)

04 06 08 10

pcII asymp 0421 pc

I asymp 0768

pcI asymp 0835 pc

I asymp 0868

α = 08

α =

05α

= 0

7

α =

06

S

09

06

03

0

p

(b)

Figure 4 Continued

6 Complexity

pcII asymp 0218 pc


I asymp 0653

α = 07

α = 065α = 06

α = 0

5

02 04 06 08p

S09

06

03

0

(c)

pcII asymp 0444 pc

I asymp 0693

pcI asymp 0764 pc

I asymp 0815

α = 0

7 α =

065

04 06 08 10p

S

09

06

03

0

α =

06

α =

05

(d)


10080604020000

02

04

06

08

10

p c

α


(a)

00

02

04

06

08

10

p c

100806040200α


(b)

Figure 5 Continued

Complexity 7



00

02

04

06

08

10p c

100806040200α


(c)

00

02

04

06

08

10

p c

100806040200α


(d)



00 02 04 06p

04

03

02

01

SA

00

α = 00α = 03

α = 06α = 09

(a)

04

03

02

01

SB

0000 02 04 06

p

α = 00α = 03

α = 06α = 09

(b)

04

03

02

01

SC

0000 02 04 06

p

α = 00α = 03

α = 06α = 09

(c)


8 Complexity







Data Availability




Acknowledgments


References


Complexity 9






































10 Complexity
















Complexity 11

pcII asymp 0218 pc


I asymp 0653

α = 07

α = 065α = 06

α = 0

5

02 04 06 08p

S09

06

03

0

(c)

pcII asymp 0444 pc

I asymp 0693

pcI asymp 0764 pc

I asymp 0815

α = 0

7 α =

065

04 06 08 10p

S

09

06

03

0

α =

06

α =

05

(d)


10080604020000

02

04

06

08

10

p c

α


(a)

00

02

04

06

08

10

p c

100806040200α


(b)

Figure 5 Continued

Complexity 7



00

02

04

06

08

10p c

100806040200α


(c)

00

02

04

06

08

10

p c

100806040200α


(d)



00 02 04 06p

04

03

02

01

SA

00

α = 00α = 03

α = 06α = 09

(a)

04

03

02

01

SB

0000 02 04 06

p

α = 00α = 03

α = 06α = 09

(b)

04

03

02

01

SC

0000 02 04 06

p

α = 00α = 03

α = 06α = 09

(c)


8 Complexity







Data Availability




Acknowledgments


References


Complexity 9






































10 Complexity
















Complexity 11



00

02

04

06

08

10p c

100806040200α


(c)

00

02

04

06

08

10

p c

100806040200α


(d)



00 02 04 06p

04

03

02

01

SA

00

α = 00α = 03

α = 06α = 09

(a)

04

03

02

01

SB

0000 02 04 06

p

α = 00α = 03

α = 06α = 09

(b)

04

03

02

01

SC

0000 02 04 06

p

α = 00α = 03

α = 06α = 09

(c)


8 Complexity







Data Availability




Acknowledgments


References


Complexity 9






































10 Complexity
















Complexity 11







Data Availability




Acknowledgments


References


Complexity 9






































10 Complexity
















Complexity 11






































10 Complexity
















Complexity 11
















Complexity 11

depth penetration and scope extension of failures...

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