Large deviation theory of percolation on multiplex networks
LLarge deviation theory of percolation oninterdependent multiplex networks
Ginestra Bianconi
Alan Turing Institute, London, United KingdomSchool of Mathematical Sciences, Queen Mary University of London, London, UnitedKingdomE-mail: [email protected]
Abstract.
Recently increasing attention has been addressed to the fluctuationsobserved in percolation defined in single and multiplex networks. These fluctuations areextremely important to characterize the robustness of real finite networks but cannotbe captured by the traditionally adopted mean-field theory of percolation. Here wepropose a theoretical framework and a message passing algorithm that is able to fullycapture the large deviation of percolation in interdependent multiplex networks witha locally tree-like structure. This framework is here applied to study the robustness ofsingle instance multiplex networks and compared to the results obtained using extensivesimulations of the initial damage. For simplicity the method is here developed forinterdependent multiplex networks without link overlap, however it can be generalizedto treat multiplex networks with link overlap.
1. Introduction
Percolation is a fundamental critical phenomena [1, 2] defined in complex networks asit sheads light on the robustness of networks when a fraction f = 1 − p of nodesis randomly removed. In the last ten years important new insights on percolationtheory have been gained by considering percolation processes defined on multiplexnetworks [3–16]. Multiplex networks [6, 17–20] describe generalized network structuresformed by several interacting networks (layers). Examples of multiplex networks includeglobal infrastructures, interacting financial networks and biological networks in the cellor in the brain.Multiplex networks are often characterized by interdependencies [3] existingbetween nodes of different layers implying that the failure of one node in onelayer necessarily causes the failure of the interdependent nodes in the other layersindependently of the rest of the network. Interestingly interdependent multiplexnetworks are much more fragile than single networks [3–6]. In particular the percolationthreshold p = p c of multiplex networks is larger than the percolation threshold of theirsingle layers taken in isolation. Moreover if the layers are not formed by identical a r X i v : . [ c ond - m a t . d i s - nn ] J a n arge deviation theory of percolation on multiplex networks f = 1 − p ). However this approach is not ableto capture the fluctuations that can be observed in the size of the MCGC for differentrealizations of the initial damage.Recently, there has been a surge of interest in characterizing the risk of dramaticfailure of events as a consequence of a random damage [23–29]. In fact real networks aretypically finite and often sufficiently far from the thermodynamic limit. Therefore theoutcome of a initial damage configuration drawn from a given distribution can have largefluctuations not predicted by the mean-field approach to percolation. Characterizingthese fluctuations is of fundamental importance for applications as the average responseto perturbation captured by the mean-field theory of percolation can be dramaticallymisleading [28, 29]. Moreover characterizing the node sets that are responsible forsafeguarding the cohesiveness of a network [28] or for more efficiently disrupting anetwork (optimal percolation) [30–33] have a number of applications. These range fromthe design principles for robust infrastructures to the control of epidemic processes onnetworks. From the theoretical perspective characterizing these fluctuations is veryinteresting as well as it sheds new light on the nature of the percolation transition[23–27].These phenomena are naturally addressed using the theory of large deviations[34, 35]. On single networks the characterization of the large deviation properties ofrandom graphs has been first tackled using the properties of the Ising model [36] and arge deviation theory of percolation on multiplex networks − . This method has subsequentlybeen also used for numerically evaluating the distribution of the diameter [40] and ofthe largest biconnected component of networks [41].Recently [24] a large deviation theory of percolation has been proposed to capturethe fluctuations observed in percolation of single finite networks. This approach isbased on message-passing algorithm and specifically on Belief Propagation [42] and canbe applied to single instances of networks provided that they are locally tree-like. Thiswork reveals that when one considers aggravating initial damage configurations, thepercolation transition can become discontinuous also in percolation defined on singlenetworks.Here we generalize the large deviation theory of percolation to multiplex networks.For simplicity we consider only multiplex networks without link overlap. However itis possible to generalize the method to multiplex networks with link overlap. We showthat the percolation transition remains discontinuous for both aggravating and bufferingconfiguration of the initial damage. Interestingly we can measure the fluctuations in thesize of the MCGC and we observe that these fluctuations can remain very significant upto the percolation transition in the typical scenario. Finally we use the large deviationtheory to theoretically predict the convex envelop of the rate function I ( R ) of thedistribution π ( R ) the size R of the MCGC finding very good agreement with extensivenumerical simulations.The paper is structured as follows. In Sec. 2 we present the mean-field theory ofpercolation of interdependent duplex networks without link overlap. In Sec. 3 we presentthe large deviation theory of percolation. In Sec. 4 we show how the Belief Propagationapproach can be used to fully characterize the large deviation theory of percolation. InSec. 5 we compare the numerical results obtained with the Belief Propagation approachwith extensive numerical simulations. Finally in Sec. 6 we give the conclusions.
2. The mean-field theory of percolation in interdependent multiplexnetworks
We consider an interdependent duplex network (cid:126)G = ( G [1] , G [2] ) formed by two networks G [ α ] = ( V, E [ α ] ) (with α = 1 ,
2) among the same set of N nodes V = { i | i ∈ { , , . . . , N }} arge deviation theory of percolation on multiplex networks α is defined by the adjacency matrix a [ α ] . For each node i weconsider two replica nodes ( i,
1) and ( i,
2) indicating the identity of node i in layer 1and in layer 2 respectively. Each pair of replica nodes is connected by an interlink that has a different valence with respect to the links present in each layer. In factin interdependent multiplex networks interlinks indicate interdependencies whose rolein determining the robustness properties of the duplex network is explained in thefollowing. Duplex networks can be classified depending on the presence or the absenceof link overlap. The total overlap between the two layers of a duplex network is givenby the number of pairs of nodes that are connected in both layers, i.e. O = (cid:88) i 3. Large deviation theory of percolation While in infinite networks the percolation process is self-averaging, i.e. the typicalbehaviour characterizes a set of measure one of random instances of the initial damage,in finite networks deviations from the typical behaviour can be observed. It is thereforenecessary to establish the large deviation theory of percolation. Here our aim is togeneralize the framework proposed in Ref. [24] for characterizing the large deviation insingle networks in order to treat the large deviation of percolation in interdependentmultiplex networks. The large deviation theory of percolation of multiplex networksaims at characterizing the probability distribution π ( R ) of size R of the MCGC whenthe initial damage configuration x is chosen with probability ˜ P ( x ), i.e. π ( R ) = (cid:88) x ˜ P ( x ) δ ( R , R ) , (20)where with δ ( c, d ) we indicate the Kronecker delta. For large network sizes N (cid:29) p the probability π ( R ) has a scaling with the network size N determinedby the rate function I ( R ) ≥ 0. In particular we observe the scaling [34] π ( R ) ∼ e − NI ( R ) . (21)This expression implies that for N → ∞ the percolation is self-averaging and π ( R ) isnon-zero only for s R = ˆ R for which I ( R ) takes its minimum value I ( ˆ R ).Therefore inthe infinite network limit all realizations of the initial damage yield almost surely thesame size of the MCGC. However the rate function I ( R ) captures the fluctuations in thesize of the MCGC that can be observed in finite multiplex networks. In order to find I ( R ) we adopt a canonical approach and we introduce the partition function Z = Z ( ω )given by Z = (cid:88) x ˜ P ( x ) e − ω R . (22)Using the definition of π ( R ) given by Eq. (20) it can be easily shown that Z is thegenerating function of π ( R ) as Z can be written as Z = (cid:88) R π ( R ) e − ωR . (23)The corresponding free-energy F and the free energy density f can be calculated as ωF = ωN f = − log( Z ) . (24) arge deviation theory of percolation on multiplex networks I ( R ) [34] can be expressed in termsof the free energy density as ωf ( ω ) and we have ωf ( ω ) = inf R (cid:20) I ( R ) + ω RN (cid:21) . (25)Additionally as long ωf ( ω ) is differentiable, the Legendre-Fenchel transform ˆ I ( R ) of ωf ( ω ) given byˆ I ( R ) = sup ω (cid:20) ωf ( ω ) − ω RN (cid:21) . (26)fully determines the rate function I ( R ), i.e. I ( R ) = ˆ I ( R ).However when I ( R ) is non-convex ωf ( ω ) is not differentiable and the Legendre-Fenchel transform of ωf ( ω ) given by ˆ I ( R ) only provides the convex envelop of the ratefunction I ( R ) [34].Alternatevely it is possible to proceed as proposed in Ref. [39–41] and directlyextract the distribution π ( R ) from the biased distribution ˜ π ( R ) given by˜ π ( R ) = 1 Z (cid:88) x ˜ P ( x ) e − ω R δ ( R , R ) = π ( R ) e − ωR Z (27)getting π ( R ) = Ze ωR ˜ π ( R ) . (28)Therefore the knowledge of the biased distribution ˜ π ( R ) which can be sampled withthe Markov Chain Monte Carlo method is in principle sufficient to reconstruct the fulldistribution π ( R ). However this approach is entirely numerical.In the following we will consider the first approach and we will evaluate the partitionfunction Z using the Belief Propagation algorithm. In order to calculate the partition function Z we follow the approach proposed inRef. [24] and we consider a Gibbs measure P ( σ ) over the set σ of all messages. Theprobability distribution P ( σ ) weights the configuration of the messages σ accordingto the probability of the corresponding initial damage configurations. Moreover weintroduce a Lagrangian multiplier ω conjugated to the size of the MCGC R that isable to tune the relative weight of the configurations of the messages corresponding todifferent sizes of the MCGC. Therefore the Gibbs measure P ( σ ) is given by P ( σ ) = 1 Z (cid:88) x e − ω R ˜ P ( x ) χ ( σ , x ) , (29)where the function χ ( σ , x ) enforces the message passing Eqs. (3) , i.e. χ ( σ , x ) = (cid:89) α =1 , N (cid:89) i =1 (cid:89) j ∈ N α ( i ) δ σ αi → j , x i − (cid:89) (cid:96) ∈ N α ( i ) \ j (1 − σ α(cid:96) → i ) − (cid:89) (cid:96) ∈ N β ( i ) (1 − σ β(cid:96) → i ) . arge deviation theory of percolation on multiplex networks Z in Eq. (29) clearly reduces to Z defined in Eq. (23). Infact we have Z = (cid:88) σ (cid:88) x e − ω R ˜ P ( x ) χ ( x , σ ) = (cid:88) R π ( R ) e − ωR . (30)Therefore characterizing the partition function Z and the free energy F (given by Eq.(24)) corresponding to the Gibbs measure defined in Eq. (29) allows us to directlycalculate the Legendre-Fenchel transform of the rate function I ( R ).We note that the Gibbs measure P ( σ ) can be interpreted as a canonical ensemble,where ω plays the role of the inverse temperature and R plays the role of the energy.Since R is the sum of the node variable ρ i and ρ i can only take two values ( ρ i = 0-the node does not belong to the MCGC or ρ i = 1 the node does belong to the MCGC)the Gibbs measure can be interpreted as a a statistical mechanics problem of a twolevel system. It follows that in this case we can investigate the properties of the Gibbsmeasure for values of ω that can be also negative.For ω < 0, the Gibbs measure weights more the buffering configurations of theinitial damage resulting in a MCGC larger than the typical one. On the contrary for ω > aggravating configurations of the initialdamage resulting in a MCGC smaller than the typical one. For ω = 0 we recover thetypical scenario.The Gibbs measure P ( σ ) given by Eq. (29) can also be expressed as P ( σ ) = 1 Z N (cid:89) i =1 ψ i ( σ i , ω ) , (31)where the set of constraints ψ i ( σ i , ω ) for i = 1 , , . . . , N defined over all the messages σ i starting or ending to node i read ψ i ( σ i ) = (1 − p ) (cid:89) α =1 , (cid:89) j ∈ N α ( i ) δ ( σ αi → j , pe − ω ˜ ρ i (cid:89) α =1 , (cid:89) j ∈ N α ( i ) δ σ αi → j , − (cid:89) (cid:96) ∈ N α ( i ) \ j (1 − σ α(cid:96) → i ) − (cid:89) (cid:96) ∈ N β ( i ) \ j (1 − σ β(cid:96) → i ) . Here δ ( m, n ) indicates the Kronecker delta and ˜ ρ i is given by˜ ρ i = (cid:89) α =1 , − (cid:89) j ∈ N α ( i ) (1 − σ αj → i ) . (32)Finally, using Eq. (31) it can be easily shown that the partition function Z can be alsowritten as Z = (cid:88) σ N (cid:89) i =1 ψ i ( σ i , ω ) . (33)In the following sections we will characterize the large deviation properties ofpercolation on multiplex networks by calculating the Gibbs measure using BeliefPropagation (BP). arge deviation theory of percolation on multiplex networks 4. Belief Propagation approach On a locally tree-like duplex network without link overlap the Gibbs distribution P ( σ )can be expressed explicitly using the Belief Propagation (BP) algorithm [42] by findingthe messages ˆ P αi → j ( σ αi → j , σ αj → i ) that each node i sends to the generic neighbour node j in layer α . These messages satisfy the following recursive BP equations (see AppendixA for their explicit expression)ˆ P αi → j ( σ αi → j , σ αj → i ) = 1 D [ α ] i → j (cid:88) σ i ψ i ( σ i ) (cid:89) (cid:96) ∈ N α ( i ) \ j ˆ P α(cid:96) → i ( σ α(cid:96) → i , σ αi → (cid:96) ) (cid:89) (cid:96) ∈ N β ( i ) ˆ P β(cid:96) → i ( σ β(cid:96) → i , σ βi → (cid:96) ) , where α (cid:54) = β and where D [ α ] i → j are normalization constants enforcing the normalizationcondition (cid:88) σ αi → j =0 , (cid:88) σ αj → i =0 , ˆ P αi → j ( σ αi → j , σ αj → i ) = 1 . (34)In the Bethe approximation, valid on locally tree-like networks the probabilitydistribution P ( σ ) is given by [42] P ( σ ) = N (cid:89) i =1 P i ( σ i ) (cid:32) (cid:89) α =1 , (cid:89) α P αij ( σ αi → j , σ αj → i ) (cid:33) − (35)where P i ( σ i ) and P αij ( σ αi → j , σ αj → i ) indicate the marginal distribution of nodes and linksand are given by [42] P αij ( σ αi → j , σ αj → i ) = 1 C αij ˆ P αi → j ( σ αi → j , σ αj → i ) ˆ P αj → i ( σ αj → i , σ αi → j ) , P i ( σ i ) = 1 C i ψ i ( σ i ) (cid:89) α =1 , (cid:89) j ∈ N α ( i ) ˆ P αj → i ( σ j → i , σ i → j ) , (36)with C i and C [ α ] ij indicating normalization constants (see Appendix B for their explicitexpression). The free energy F given by Eq. (24) can be found by minimizing the Gibbs free energy F Gibbs given by ωF Gibbs = (cid:88) σ P ( σ ) ln (cid:18) P ( σ ) ψ ( σ ) (cid:19) , (37)where ψ ( σ ) indicates the set of constraints ψ ( σ ) = N (cid:89) i =1 ψ i ( σ i ) . (38) arge deviation theory of percolation on multiplex networks F Gibbs is minimal whencalculated over the probability distribution P ( σ ) given by Eq. (31) and that itsminimum value is ωF Gibbs = ωF = − ln Z. (39)On a locally tree-like duplex network the Gibbs measure P ( σ ) reduces to Eq. (35), andit can be easily see that the free energy can be expressed as ωF = (cid:88) α =1 , (cid:88) α log (cid:0) C αij (cid:1) − N (cid:88) i =1 log( C i ) . (40)Given the explicit expression of C [ α ] ij and of C i in terms of the messages (see AppendixB), the free energy F can be calculated easily when the BP equations have been solved. The energy and the specific heat corresponding to the Gibbs measure P ( σ ) have alsoa very important interpretation in terms of the underlying percolation process. Theenergy is given by the the average size of the MCGC R . In fact we have R = (cid:88) σ R P ( σ ) = − ∂ ln Z∂ω . (41)The mean-field percolation transition correspond to the phase transition from a non-percolating phase with R = 0 to a percolating phase R > p = p c and ω = 0.However the present approach allows to consider the full line of critical points ( p (cid:63) , ω (cid:63) )at which the transition is observed when the large deviations are considered.Following a statistical mechanics definition, we can also define the specific heat C as Cω = − ∂R∂ω . (42)The specific heat has the immediate interpretation in terms of the variance in the sizeof the MCGC, i.e. Cω = (cid:32)(cid:88) σ R P ( σ ) (cid:33) − (cid:32)(cid:88) σ R P ( σ ) (cid:33) . Both R and C/ω can be derived from the message passing algorithm. Indeed we have R = (cid:88) i r i , (43) Cω = N (cid:88) i =1 r i (1 − r i ) (44)where r i = (cid:88) σ ρ i P ( σ ) (45) arge deviation theory of percolation on multiplex networks i is in the giant component is given by r i = z i C i , (46)where the explicit expression of z i and C i is given in Appendix B. The quantity C/ω canbe also interpreted as the expected fraction of nodes that given two random realizationsof the initial damage are found in the MCGC in one realization but not in the other. Thisquantity generalizes the measure proposed in Ref. [23] to characterize the fluctuationsin percolation in single networks. It is instructive to see that the proposed large deviation theory of percolation reduces tothe mean-field theory of percolation in the typical scenario obtained by putting ω = 0.In this case we obtain that the BP equations have solution withˆ P αi → j (0 , 0) = ˆ P αi → j (0 , , ˆ P αi → j (1 , 1) = ˆ P αi → j (1 , , (47)and the BP equations for ω = 0 reduces to Eqs.(12) when we putˆ σ αi → j = ˆ P αi → j (1 , 1) + ˆ P αi → j (1 , . (48)Similarly it is easy to show that r i reduces to ˆ ρ i given by Eq. (14). 5. Numerical results Here we consider the results obtained by running the BP algorithm over a Poissonduplex network with average degree z = 6 and network size N = 100 (see Figure 1).The typical scenario observed for ω = 0 gives a discontinuous transition of the averagesize R of the MCGC at p = p c = 0 . . . . with R = R c = 0 . . . . for theconsidered duplex network as predicted by the mean-field theory of percolation. Thefluctuations of the size of the MCGC measured by C/ω for ω → p = p c . Therefore as we approach the critical percolation threshold from above (i.e.for p → p + c ) we observe significant fluctuations in the size of the MCGC. However thesefluctuations are maximal only for larger values of p .Interestingly the BP results allow us to investigate also how the average size of theMCGC R and its fluctuations C/ω change if we deviate from the typical scenario, i.e.for ω (cid:54) = 0. For ω < p , for ω > p . In both cases the transition remains discontinuous in the investigated rangeof values of ω . Additionally we note that for buffering configurations, as ω decreases thediscontinuous jump in the size of the MCGC R appears to reach a constant value.This suggests that the discontinuity might be preserved even beyond the observedrange of values of ω . The fluctuations in the size of the MCGC C/ω observed at arge deviation theory of percolation on multiplex networks C/ω also in the typical scenario ω = 0 while in single networks C/ω is continuous at thetransition point. Therefore the observed discontinuity of C/ω at p = p c and ω = 0 isa purely multiplex network phenomenon not observed in percolation of single networks.In fact the presence of significant fluctuations of the percolation order parameter at thepercolation transition can be only observed if the order parameter has a discontinuousjump and in this case only for p → p + c when R → R c > 0. Finally we note thatin the multiplex case the discontinuity of C/ω extend also for negative value of ω corresponding to buffering configurations of the initial damage while in single networkswe observe a continuous behaviour of C/ω .In order to test the validity of the BP algorithm, we have compared the rate function I ( R ) measured starting from 10 initial damage configurations performed on the sameduplex network instance on which the BP algorithm is run, with ˆ I ( R ) provided by theBP algorithm finding very good results (see Figure 3). We notice that while for largevalues of p the rate function I ( R ) is convex and therefore I ( R ) = ˆ I ( R ), as we approachthe percolation transition the rate function I ( R ) becomes non-convex and ˆ I ( R ) onlyprovides the convex envelop of I ( R ). 6. Conclusions In this paper we have characterized the large deviation of percolation of interdependentmultiplex networks without link overlap using a Belief Propagation algorithm. Inthe typical scenario, well captured by the mean-field theory of percolation weobserve a discontinuous percolation transition. Our analysis reveals that when wedepart from the typical scenario the percolation transition can occur for values of p significantly distant from the percolation threshold in the typical scenario whileremaining discontinuous both for buffering configurations of the random damage andaggravating ones. Interestingly our study show a significant difference of interdependentpercolation with respect to percolation in single layers. In fact when we considerpercolation of isolated networks the fluctuations in the size of the giant componentgo to zero at the critical point. However in interdependent percolation we can observesignificant fluctuations of the size of the MCGC when we approach the critical point fromabove, i.e. for p → p + c . Finally we have compared the rate function I ( R ) of observinga MCGC of size R with the predicted ˆ I ( R ) obtained using the proposed canonical BPalgorithm. As observed in the single layer scenario the rate function I ( R ) becomesbimodal for small values of p , and therefore the proposed canonical BP algorithm canonly capture its convex envelop, i.e. we have ˆ I ( R ) (cid:54) = I ( R ). arge deviation theory of percolation on multiplex networks p R / N =-0.4=-0.3=-0.2=-0.1=0.0=0.1=0.2=0.3=0.4 p C / ( N ) Figure 1. The average fraction of nodes of the MCGC R/N and the normalizedfluctuations C/ ( N ω ) of the size of the MCGC are plotted as a function of p fordifferent values of ω . The considered multiplex network is a duplex Poisson networkwith average degree z = 6 and total number of nodes N = 100. p R / N =-0.4=-0.3=-0.2=-0.1=0.0=0.1=0.2=0.3=0.4 p C / ( N ) Figure 2. The average fraction of nodes of the giant component R/N and thenormalized fluctuations C/ ( N ω ) of the size of the giant component are plotted asa function of p for different values of ω . The considered network is a single Poissonnetwork with average degree z = 4 and total number of nodes N = 100 with percolationthreshold p c = 1 / arge deviation theory of percolation on multiplex networks R -202468101214 I ( R ) Figure 3. The rate function I ( R ) measured starting from 10 random realizations ofthe initial damage (symbols) and compared to its convex envelop obtained using theBP algorithm (solid lines). The different color and symbols correspond to differentvalues of p : p = 0 . p = 0 . p = 0 . p = 0 . p = 0 . p = 0 . z = 6 and total number of nodes N = 100. The proposed method can be extended in different ways by considering percolationand directed percolation in multiplex network with link overlap and by developing amicrocanonical approach in which instead of introducing the Lagrangian multiplier ω fixing the size of the MCGC in average, we consider the corresponding hard constraint. Acknowledgements We thank Alexander Hartmann, Francesco Coghi, Giorgio Parisi, Riccardo Zecchinaand Robert Ziff for interesting discussions. References [1] Dorogovtsev S N, Goltsev A V and Mendes J F F 2008 Rev. Mod. Phys. Eur. Phys. Jour. Spe. Top. Nature EPL (EurophysicsLetters) Phys. Rev. Lett. Multilayer Networks: Structure and Function (Oxford:Oxford University Press)[7] Parshani R, Buldyrev S V and Havlin S 2010 Phys. Rev. Lett. Nature Physics arge deviation theory of percolation on multiplex networks [9] Min B, Do Y, S, Lee KM and Goh, K I 2014 Phys. Rev. E Phys. Rev. E Phys. Rev. E, Phys. Rev. E Chaos, Solitons & Fractals Sci. Rep. Phys. Rev. E Phys. Rep. Jour. Comp.Net. Eur. Phys. Jour. B Nature Physics Phys. Rev. E Phys. Rev. E, Phys. Rev. E Phys. Rev. E, Phys. Rev. E , Phys. Rev. E Nature Proc. Nat. Aca. Sci. Nature Comm. Phys. Rep. Large deviations (Vol. 14) (American Mathematical Society)[36] Biskup M, Chayes L and Smith S A 2007 Random Structures & Algorithms Jour. Stat. Phys. Jour. Phys. A Eur. Phys. Jour. B Phys. Rev. E Information, physics, and computation (Oxford:OxfordUniversity Press) Appendix A. Explicit Belief Propagation equations In this appendix we provide the explicit expression of the Belief Propagation equations(34). When degree k [ α ] i > k [ β ] i > D [ α ] i → j ˆ P αi → j (0 , 0) = (1 − p ) (cid:89) (cid:96) ∈ N α ( i ) \ j (cid:104) ˆ P α(cid:96) → i (0 , 0) + ˆ P α(cid:96) → i (1 , (cid:105) (cid:89) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (0 , 0) + ˆ P β(cid:96) → i (1 , (cid:105) arge deviation theory of percolation on multiplex networks p (cid:89) (cid:96) ∈ N α ( i ) \ j (cid:104) ˆ P α(cid:96) → i (0 , 0) + ˆ P α(cid:96) → i (1 , (cid:105) (cid:89) (cid:96) ∈ N β ( i ) ˆ P β(cid:96) → i (0 , + p (cid:89) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (0 , 0) + ˆ P β(cid:96) → i (1 , (cid:105) − (cid:89) (cid:96) ∈ N β ( i ) ˆ P β(cid:96) → i (0 , (cid:89) (cid:96) ∈ N α ( i ) \ j ˆ P α(cid:96) → i (0 , , D [ α ] i → j ˆ P αi → j (0 , 1) = (1 − p ) (cid:89) (cid:96) ∈ N α ( i ) \ j (cid:104) ˆ P α(cid:96) → i (0 , 0) + ˆ P α(cid:96) → i (1 , (cid:105) (cid:89) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (0 , 0) + ˆ P β(cid:96) → i (1 , (cid:105) + p (cid:89) (cid:96) ∈ N α ( i ) \ j (cid:104) ˆ P α(cid:96) → i (0 , 0) + ˆ P α(cid:96) → i (1 , (cid:105) (cid:89) (cid:96) ∈ N β ( i ) ˆ P β(cid:96) → i (0 , + pe − ω (cid:89) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (0 , 1) + ˆ P β(cid:96) → i (1 , (cid:105) − (cid:89) (cid:96) ∈ N β ( i ) ˆ P β(cid:96) → i (0 , (cid:88) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (1 , − ˆ P β(cid:96) → i (1 , (cid:105) (cid:89) (cid:96) (cid:48) ∈ N β ( i ) \ (cid:96) ˆ P β(cid:96) (cid:48) → i (0 , (cid:89) (cid:96) ∈ N α ( i ) \ j ˆ P α(cid:96) → i (0 , , D [ α ] i → j ˆ P αi → j (1 , 0) = pe − ω (cid:89) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (0 , 1) + ˆ P β(cid:96) → i (1 , (cid:105) − (cid:89) (cid:96) ∈ N β ( i ) ˆ P β(cid:96) → i (0 , (cid:88) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (1 , − ˆ P β(cid:96) → i (1 , (cid:105) (cid:89) (cid:96) (cid:48) ∈ N β ( i ) \ (cid:96) ˆ P β(cid:96) (cid:48) → i (0 , × (cid:89) (cid:96) ∈ N α ( i ) \ j (cid:104) ˆ P α(cid:96) → i (0 , 1) + ˆ P α(cid:96) → i (1 , (cid:105) − (cid:89) (cid:96) ∈ N α ( i ) \ j ˆ P α(cid:96) → i (0 , (cid:88) (cid:96) ∈ N α ( i ) \ j (cid:104) ˆ P α(cid:96) → i (1 , − ˆ P α(cid:96) → i (1 , (cid:105) (cid:89) (cid:96) (cid:48) ∈ N α ( i ) \ j,(cid:96) ˆ P α(cid:96) (cid:48) → i (0 , D [ α ] i → j ˆ P αi → j (1 , 1) = pe − ω (cid:89) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (0 , 1) + ˆ P β(cid:96) → i (1 , (cid:105) − (cid:89) (cid:96) ∈ N β ( i ) ˆ P β(cid:96) → i (0 , (cid:88) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (1 , − ˆ P β(cid:96) → i (1 , (cid:105) (cid:89) (cid:96) (cid:48) ∈ N β ( i ) \ (cid:96) ˆ P β(cid:96) (cid:48) → i (0 , × (cid:89) (cid:96) ∈ N α ( i ) \ j (cid:104) ˆ P α(cid:96) → i (0 , 1) + ˆ P α(cid:96) → i (1 , (cid:105) − (cid:89) (cid:96) ∈ N α ( i ) \ j ˆ P α(cid:96) → i (0 , . (A.1) arge deviation theory of percolation on multiplex networks D [ α ] i → j are normalization constants fixed by the conditions expressed in Eq. (34).When degree k [ α ] i = 1 and degree k [ β ] i > D [ α ] i → j ˆ P αi → j (0 , 0) = (cid:89) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (0 , 0) + ˆ P β(cid:96) → i (1 , (cid:105) , D [ α ] i → j ˆ P αi → j (0 , 1) = (1 − p ) (cid:89) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (0 , 0) + ˆ P β(cid:96) → i (1 , (cid:105) + p (cid:89) (cid:96) ∈ N β ( i ) ˆ P β(cid:96) → i (0 , + pe − ω (cid:89) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (0 , 1) + ˆ P β(cid:96) → i (1 , (cid:105) − (cid:89) (cid:96) ∈ N β ( i ) ˆ P β(cid:96) → i (0 , (cid:88) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (1 , − ˆ P β(cid:96) → i (1 , (cid:105) (cid:89) (cid:96) (cid:48) ∈ N β ( i ) \ (cid:96) ˆ P β(cid:96) (cid:48) → i (0 , , ˆ P αi → j (1 , 0) = 0 , ˆ P αi → j (1 , 1) = 0 . (A.2)When for arbitrary degree k [ α ] i ≥ k [ β ] i = 0 these equations readˆ P αi → j (0 , 0) = 12 , ˆ P αi → j (0 , 1) = 12 , ˆ P αi → j (1 , 0) = 0 , ˆ P αi → j (1 , 1) = 0 . (A.3) Appendix B. Explicit expression of C αij , C i and z i In this appendix we give the explict expression of C αij , C i and z i in terms of the messagesthat solve the BP equations. In particular by normalizing the marginals in Eqs. (36)we obtain C αij = ˆ P αi → j (0 , 0) ˆ P αj → i (0 , 0) + ˆ P αi → j (0 , 1) ˆ P αj → i (1 , 0) + ˆ P αi → j (1 , 0) ˆ P αj → i (0 , 1) + ˆ P αi → j (1 , 1) ˆ P αj → i (1 , , C i = (1 − p ) (cid:89) (cid:96) ∈ N α ( i ) (cid:104) ˆ P α(cid:96) → i (0 , 0) + ˆ P α(cid:96) → i (1 , (cid:105) (cid:89) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (0 , 0) + ˆ P β(cid:96) → i (1 , (cid:105) + p (cid:89) (cid:96) ∈ N α ( i ) (cid:104) ˆ P α(cid:96) → i (0 , 0) + ˆ P α(cid:96) → i (1 , (cid:105) (cid:89) (cid:96) ∈ N β ( i ) ˆ P β(cid:96) → i (0 , + p (cid:89) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (0 , 0) + ˆ P β(cid:96) → i (1 , (cid:105) − (cid:89) (cid:96) ∈ N β ( i ) ˆ P β(cid:96) → i (0 , (cid:89) (cid:96) ∈ N α ( i ) ˆ P α(cid:96) → i (0 , + z i ,z i = pe − ω (cid:89) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (0 , 1) + ˆ P β(cid:96) → i (1 , (cid:105) − (cid:89) (cid:96) ∈ N β ( i ) ˆ P β(cid:96) → i (0 , arge deviation theory of percolation on multiplex networks (cid:88) (cid:96) ∈ N β ( i ) (cid:104) ˆ P β(cid:96) → i (1 , − ˆ P β(cid:96) → i (1 , (cid:105) (cid:89) (cid:96) (cid:48) ∈ N β ( i ) \ (cid:96) ˆ P β(cid:96) (cid:48) → i (0 , × (cid:89) (cid:96) ∈ N α ( i ) \ j (cid:104) ˆ P α(cid:96) → i (0 , 1) + ˆ P α(cid:96) → i (1 , (cid:105) − (cid:89) (cid:96) ∈ N α ( i ) \ j ˆ P α(cid:96) → i (0 , (cid:88) (cid:96) ∈ N α ( i ) (cid:104) ˆ P α(cid:96) → i (1 , − ˆ P α(cid:96) → i (1 , (cid:105) (cid:89) (cid:96) (cid:48) ∈ N α ( i ) ,(cid:96) ˆ P α(cid:96) (cid:48) → i (0 ,