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Multiple percolation transitions in a configuration model of network of networks

Ginestra BianconiSchool of Mathematical Sciences, Queen Mary University of London, London, E1 4NS, United Kingdom

Sergey N. DorogovtsevDepartamento de Fısica da Universidade de Aveiro, I3N, 3810-193, Aveiro, Portugal

A. F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia

Recently much attention has been paid to the study of the robustness of interdependent andmultiplex networks and, in particular, networks of networks. The robustness of interdependentnetworks can be evaluated by the size of a mutually connected component when a fraction of nodeshave been removed from these networks. Here we characterize the emergence of the mutuallyconnected component in a network of networks in which every node of a network (layer) α isconnected with qα randomly chosen replicas in some other networks and is interdependent of thesenodes with probability r. We find that when the superdegrees qα of different layers in the networkof networks are distributed heterogeneously, multiple percolation phase transition can occur, anddepending on the value of r these transition are continuous or discontinuous.

PACS numbers: 89.75.Fb, 64.60.aq, 05.70.Fh, 64.60.ah

I. INTRODUCTION

The complexity of a large variety of systems, from in-frastructures to the cell, is rooted in a network of in-teractions between their constituents [1–3]. Quantifyingthe robustness of complex networks is one of the mainchallenges of network of networks with implications infields as different as biology or policy making and riskassessment.In the last fifteen years it has been shown [4, 5] that

the structure of a single network is strictly related to itsrobustness. But only recently [6, 7], attention has beendrawn toward a previously neglected aspects of complexsystems, namely the interactions between several com-plex networks.Rarely single networks are isolated, while it is usu-

ally the case that several networks are interacting andinterdependent on each other. For example, in infras-tructures, the banking systems are interdependent withthe Internet and the electric power-grid, public trans-port, such as subway, is dependent on the power-grid,which relies on its turn on the water supply system tocool the power-plants, etc. In the cell the situation ismuch similar: all cellular networks, such as the metabolicnetworks, the protein-protein interaction networks, thesignaling networks, and the gene transcription networksare all dependent on each other, and the cell is only aliveif all these networks are functional. These are examplesof network of networks, i.e., networks formed by severalinteracting networks.A special class of network of networks are multiplex

networks [7–10], which are multilayer structures in whicheach layer is formed by the same set of nodes N intercon-nected by different kinds of links for different layers. Inother words, these are graphs with all nodes of one kindand with links of different colors. Multiplex networks areattracting great interest as they represent a large vari-ety of systems such as social networks where people can

be linked by different types of relationships (friendship,family tie, collaboration, citations, etc.) or, for example,in transportation networks, where different places can belinked by different types of transportation (train, flightconnections, flight connection of different airline compa-nies, etc.). Multiplex network datasets [8–10] are start-ing to be analysed, several modelling framework for thesenetworks have been proposed [11–13] and the characteri-zation of a large variety of dynamical processes is gettinga momentum [14–18].

A major progress in understanding the robustness ofmultilayer interdependent networks has been made in aseries of seminal papers [6, 19–22], where it has beenproposed that a natural measure for evaluating the ro-bustness of these structures to random failure is the sizeof a mutually connected giant component. The mutu-ally connected giant component is the component thatremains after breakdowns propagate back and forth be-tween different interdependent networks (layers) gener-ating a cascade of failure events. A node is in the mu-tually connected component of a multilayer network ifall the nodes on which it depends are also in the mutu-ally connected network and if at least one neighbor nodein its own network (layer) belongs to the mutually con-nected component [21–23]. Clearly, the giant mutuallyconnected component naturally generalizes the giant con-nected component (percolation cluster) in a single net-work. The robustness properties of multiplex networkshave been right now well understood [6, 19–23], includ-ing effects of degree correlations, the overlap of the linksor antagonistic effects in this novel type of percolationproblem [24–26]. As the fraction of 1 − p of removednodes—“igniters”—increases, multiplex networks are af-fected by cascading failures, until they reach a point forp = pc where the network abruptly collapses, and thesize of the mutually connected component shows a dis-continuous transition [6, 19–22]. In this case if a smallfraction 1−r of nodes in the multiplex are not interdepen-

2

dent, then the transition can change from discontinuousto continuous [20].Several works have considered afterwards a more gen-

eral (than multiplex) case of a network of networks inwhich the layers are formed by the same number of nodes,where each node has a replica node in each other networkand might depend only on its replica nodes. Despite themajor interest on the topic [27–32], only recently the fol-lowing key circumstance become clear [33]. When the in-terdependencies between the nodes are such that if a net-work depends on another network, all its nodes dependon the replica nodes of the other network, it turns outthat the mutually connected component coincides withthat in the corresponding fully connected network of net-works, which is actually a particular case of a multiplexnetwork.Here we show, nevertheless, that the situation changes

if the interdependence links are distributed between lay-ers more randomly, i.e., if we remove the above constraintthat all nodes in each layer are dependent on their repli-cas in the same set of other layers. We consider the situa-tion in which a superdegree qα is assigned to each layer α,so that each node of the network depends only on other qαreplica nodes, but now these replica nodes are chosen ran-domly and independently from different layers. We callthis network of network models the configuration modelof network of networks. We consider specifically the casein which all the layers have the same internal degree dis-tribution P (k) and where the superdegree distribution isP (q). We show for this network of networks that if r = 1,the layers with increasing superdegree q have a percola-tion transition at different values of p, where p is thefraction of not damaged nodes in the network. In otherwords, as p increases, in layers with higher and higherq, giant clusters of the mutual component emerge pro-gressively, or, one can also say, the mutual componentexpands to layers with higher q, see Fig. 2. All thesetransition are abrupt but also they are characterized bya singularity of the order parameter. Finally we showthat for r < 1, each of these transitions involving thelayers with qα = q can become continuous for r < rq.

II. PERCOLATION IN NETWORK OFNETWORKS: THE MESSAGE PASSING

APPROACH

A network of networks is formed by M networks (lay-ers), α = 1, 2 . . . ,M , each formed by N nodes, i =1, 2 . . . , N , where N is infinite. In fact, in this work wealso set M → ∞. Every node (i, α) is connected witha number of nodes (j, α) in within the same layer andwith a number of its “replica nodes” (i, β) in other lay-ers. We consider the situation in which each layer α isinterdependent on qα other networks, so that each nodei in layer α has exactly qα interlinks, which connect ionly to its replica nodes in qα of other layers. We call qαthe superdegree of layer α. Interlinks connecting replica

nodes within different sets, say, interlink (iα − iβ) andinterlink (jγ − jδ) are assumed to be independent (un-correlated). For example, if a node (i, α) of layer α isinterdependent on node (i, β) of layer β, then althoughanother node (j, α) of layer α may in principle occur in-terdependent on node (j, β), in general, it depends on qαreplica nodes (j, γ) sitting in any qα layers. We definethe network of networks with a super-adjacency matrixof elements aiα,jβ = 1 if there is a link between node(i, α) and node (j, β) and zero otherwise. In these net-works we have always aiα,jβ = 0 if both i 6= j and α 6= β.For each node i and all its replicas, we introduce a “localsupernetwork”, whose nodes are the layers and the linksare interdependencies within this set of replicas. Thislocal supernetwork is determined by the adjacency ma-trix Ai

α,β = ((aiα,iβ)) parametrized by the node i. Thisnetwork may consist of a number of connected compo-nents. Connected components from different local su-pernetworks are connected with each other through linkswithin individual layers. In this work we explore thiscomplicated system of interconnected components, whichis necessary to describe the emergence of the giant mu-tually connected component and its expansion (percola-tion) over different layers.

We define the mutually connected component as thefollowing. Each node (i, α) is in the mutually connectedcomponent if it has at least one neighbor (j, α) whichbelongs to the mutually connected component and if allthe linked nodes (i, β) in the interdependent networksare also in the mutually connected component. In theseproblems, the giant (i.e., containing a finite fraction ofnodes) mutually connected component is single. It im-mediately follows from this definition that, remarkably,within each connected component of any local supernet-work, all its replica nodes either together belong to themutually connected component or not. In our considera-tions, we will essentially exploit this strong consequence.

Given a network of networks it is easy to construct amessage passing algorithm [21, 23, 34, 35] determiningif node (i, α) is in the mutually connected component.Let us denote by σiα→jα = 1, 0 the message in withina layer, from node (i, α) to node (j, α) and indicating(σiα→jα = 1) if node (i, α) is in the mutually connectedcomponent when we remove the link (i, j) in network α.Furthermore, let us denote by S′

iα→iβ = 0, 1 the mes-

sage between the “replicas” (i, α) and (i, β) of node i inlayers α and β. The message S′

iα→iβ = 1 indicates if

the node (i, α) is in the mutually connected componentwhen we remove the link between node (i, α) and node(i, β). In addition to that we assume that the node (i, α)can be damaged and permanently removed from the net-work. This removal will launch an avalanche of failures(removals of nodes) spreading over the layers. Note that,of course, the node removal retains its interdependencelinks. We indicate with siα = 0 a node that is dam-aged, otherwise we have siα = 0. The message passing

3

equations for these messages are given by

σiα→jα = siα∏

β∈Ni(α)

S′iβ→iα

×

1−∏

ℓ∈Nα(i)\j

(1− σℓα→iα)

,

S′iα→iβ = siα

∏

γ∈Ni(α)\β

S′iγ→iα

×

1−∏

ℓ∈Nα(i)

(1− σℓα→iα)

, (1)

where Nα(i) indicates the set of nodes (ℓ, α) which areneighbors of node i in network α, and Ni(α) indicates thelayers β such that the nodes (i, β) are interdependent onthe node (i, α). Finally Siα indicates if a node (i, α) isin the mutually interdependent network (Siα = 1, 0) andthis indicator function can be expressed in terms of themessages as

Siα = si,α∏

β∈Ni(α)

S′iβ→iα

×

1−∏

ℓ∈Nα(i)

(1− σℓα→iα)

. (2)

The solution of the message passing equations is givenby the following closed expression,

σiα→jα =∏

γ∈C(i,α)\α

siγ

1−∏

ℓ∈Nγ(i)

(1 − σℓγ→iγ)

×siα

1−∏

ℓ∈Nα(i)\j

(1− σℓα→iα)

, (3)

where C(i, α) is the connected cluster of the “local su-pernetwork” of node i, i.e., the network between layersdetermined by the adjacency matrix Ai

α,β = ((aiα,iβ))parametrized by the node i. Finally Siα is given by

Siα = siα∏

β∈C(i,α)

1−∏

ℓ∈Nβ(i)

(1− σℓβ→iβ)

. (4)

For detailed derivation and explanation of this solutionsee our work [33] and Appendix.

III. PERCOLATION IN THE CONFIGURATIONMODEL OF NETWORK OF NETWORKS

We assume here that each network (layer) α is gener-ated from a configuration model with the same degree

distribution Pα(k) = P (k), and that each node (i, α) isconnected to qα other “replica” nodes (i, β) chosen uni-formly randomly. Moreover we assume that the degreesequence in each layer is {kαi } and that the degrees ofthe replicas of node i are uncorrelated. This implies thatwe are considering a network of networks ensemble, suchthat every network of networks with a super-adjacencymatrix a has a probability P (a) given by

P (a) =

M∏

α=1

N∏

i=1

δ

kαi ,

N∑

j=1

aiα,jα

×δ

qα,

M∑

β=1

aiα,iβ

, (5)

where δ(a, b) indicates the Kronecker delta. Moreover weassume that nodes (i, α) are removed with probability1 − p, i.e., we consider the following expression for theprobability P ({siα}) of the variables siα

P ({siα}) =M∏

α=1

N∏

i=1

psiα(1− p)1−siα . (6)

In order to quantify the expected size of the mutuallyconnected component in this ensemble, we can averagethe messages over this ensemble of the network of net-works. The message passage equations for this problemare given by Eqs. (3) . Therefore the equations for theaverage message in within a layer are given in terms ofthe parameter p = 〈siα〉 and the generating functionsGk

0(z), Gk1(z), G

q0(z), G

q1(z) given by

Gk0(z) =

∑

k

P (k)zk, Gk1(z) =

∑

k

kP (k)

〈k〉 zk−1,

Gq0(z) =

∑

q

P (q)zq, Gq1(z) =

∑

q

qP (q)

〈q〉 zq−1. (7)

In particular, if we indicate by σq the average messagesin within a layer α of degree qα = q we obtain

σq = p∑

s

P (s|q)

∑

q′

q′P (q′)

〈q〉 p[1−G0(1− σq′ )]

s−1

×[1−G1(1− σq)], (8)

where P (s|q) indicates the probability that a node i inlayer α with qα = q is in a connected component C(i, α) ofthe local supernetwork of cardinality (number of nodes)|C(i, α)| = s. Similarly, the probability that a node iin a layer α with superdegree qα = q is in the mutuallyconnected component Sq = 〈Siα〉 is given by

Sq = p∑

s

P (s|q)

∑

q′

q′P (q′)

〈q〉 p[1−G0(1 − σq′)]

s−1

×[1−G0(1− σq)]. (9)

4

Equations (8) and (9) are valid for any network of net-works ensemble described by Eqs. (5)–(6). In the follow-ing we study in particular the limiting case in which thenumber of layers M → ∞, and the local supernetwork issparse. In order to find a solution for Eqs. (8), we defineB as

B =

∑

q′

q′P (q′)

〈q〉 p[1−G0(1 − σq′)]

. (10)

Inserting this expression in Eq. (8) we get

σq = p

[

∑

s

P (s|q)Bs−1

]

[1−G1(1 − σq)]. (11)

From the definition of B we see that B ≤ 1 and thatB = 1 only if both p = 1 and σq = 1 ∀q. This implies thatin all the cases in which the layers are not all formed by asingle giant component or in which p < 1, we have B < 1.Moreover if B < 1, we can neglect in Eq. (8) the contri-bution coming from the giant component of the local su-pernetworks in the large M limit. Therefore in Eq. (11)we can replace P (s|q) with the probability Pf (s|q) thata node of degree q belongs to a finite component of size sin the supernetwork. Note that in our model, the statis-tics of all local supernetworks coincide. Let us considerthe quantity P (s) =

∑

q[qP (q)/〈q〉]Pf (s|q). Since P (s)only depends on the distribution of finite components,we have

P (s) =∑

q

qP (q)

〈q〉∑

s1,s2,...sq

q∏

ℓ=1

P (sℓ)δ

(

q∑

ℓ=1

sℓ, s− 1

)

. (12)

Therefore P (s) has the generating function H(z) =∑

s P (s)zs that satisfies the equation

H(z) = zGq1(H(z)). (13)

Moreover, since

Pf (s|q) =∑

s1,s2,...sq

q∏

ℓ=1

P (sℓ) δ

(

q∑

ℓ=1

sℓ, s− 1

)

, (14)

we find∑

s

Pf (s|q)Bs−1 = [H(B)]q. (15)

Therefore Eqs. (8) become

σq = p[H(B)]q[1−Gk1(1− σq)],

H(z) = zGq1(H(z)),

B =

∑

q′

q′P (q′)

〈q〉 p[1−Gk0(1 − σq′)]

. (16)

Putting H(B) = Σ, we can express Eqs. (8) and Eqs. (9)in the following simplified way,

σq = p(Σ)q[1−Gk1(1 − σq)],

Sq = p(Σ)q[1−Gk0(1 − σq)],

Σ =

∑

q′

q′P (q′)

〈q〉 p[1−Gk0(1− σq′ )]

×∑

q′

q′P (q′)

〈q〉 (Σ)q′−1. (17)

Therefore, given a configuration model of a network ofnetworks, the parameter Σ determines both σq and Sq

for any value of the superdegree q. For this reason wecan call Σ the order parameter for the entire network ofnetworks. Let us consider for simplicity the case in whicheach layer is formed by a Poisson network with averagedegree 〈k〉 = c. In this situation the previous equationsbecome

σq = Sq = p(Σ)q(1 − e−cσq ),

Σ =

∑

q′

q′P (q′)

〈q〉 p(1− e−cσq′ )

∑

q′

q′P (q′)

〈q〉 (Σ)q′−1. (18)

In particular, if the local supernetwork is regular, i.e.,P (q) = δ(q,m) we have σ = σm =

√Σ satisfying,

σ = pσq/2(1− e−cσ). (19)

This special random regular network of interdependentPoisson networks was recently considered in Ref. [31].In the general case of an arbitrary P (q) distribution theproblem defined in Eqs. (18) continues to have a singleorder parameter given by Σ. In fact the first equation inEqs. (18) has a solution expressed in terms of the princi-pal value of the Lambert function W (x), which is givenby

σq =1

c

[

pc(Σ)q +W(

−pc(Σ)qe−pc(Σ)q)]

. (20)

Inserting this solution back into the equation for Σ inEqs. (18) we find

5

Σ = Gq1(Σ)

[

p+∑

q

qP (q)

〈q〉1

c(Σ)−qW

(

−pc(Σ)qe−pc(Σ)q)

]

. (21)

This equation can be written as F (Σ, p) = 0. By imposing both F (Σ, p) = 0 and dF (Σ, p)/dΣ = 0 we can findthe set of critical points p = pc in which discontinuities (jumps) of the function Σ(p) take place. The equationdF (Σ, p)/dΣ = 0 reads as

1

p=∑

q

qP (q)

〈q〉 (1− e−cσq )∑

q

q(q − 1)P (q)

〈q〉 Σq−2 + cpGq1(Σ)

∑

q<qmax

q2P (q)

〈q〉 e−cσqΣq−1(1− e−cσq )

1− pcΣqe−cσq, (22)

with qmax = [− log(pc)/ logΣ]. Here [...] stands for theinteger part. Analysing Eq. (21) for the order parameterwe can show that, in addition to a jump, the order pa-rameter Σ has a singularity at each p = pc,Σ = Σc whereboth F (Σc, pc) = 0 and dF (Σ, p)/dΣ|Σ=Σc,p=pc

= 0. Forevery topology of the network of network, we have

Σ− Σc ∝ (p− pc)1/2, (23)

similarly to multiplex networks [22].In the special situation in which the minimal superde-

gree m of the local supernetworks is greater or equal to 2,i.e., m ≥ 2, each local supernetwork only has a giant con-nected component and does not have finite components.Therefore in this case we found that in the limit M → ∞the only viable solution for the order parameter is Σ = 0for any p and c and any superdegree distribution P (q).In another special case in which m = 0 there are layersthat are not interacting (q = 0) with other layers. Theselayers can be treated separately without losing the gen-erality of the treatment. Therefore the only non-trivialcase is the case of m = 1, which we explore in detail.In Fig. 1 we show the order parameter Σ vs. p for a con-

figuration model with a Poisson P (k) distribution with〈k〉 = c = 20 and a scale-free distribution P (q) ≃ q−γ

for q ∈ [m,Q] with γ = 2.8 and γ = 4.5, minimal su-perdegree m = 1 and maximal superdegree Q = 103.The supernetwork with γ = 2.8 is above the ordinarypercolation phase transition in the supernetwork (i.e.,it has a giant component) while the supernetwork withγ = 4.5 is below the ordinary percolation phase tran-sition (i.e., it consists only of finite components). No-tice that if the supernetwork has no giant component,then Σ(p = 1) = 1, otherwise, Σ(p = 1) < 1. The or-der parameter displays a series of discontinuous jumpscorresponding to the transitions in which layers with in-creasing values of q start to percolate. In other words,in layers with higher and higher superdegree q, giantclusters of the mutually connected component emergeprogressively. In order to see this, we observe thatσq = Sq = 0 for all values of Σ such that cpΣq < 1(note that for power-law superdegree distributions theminimal degree is always m ≥ 1). In Fig. 1 we plot themaximal superdegree of the percolating layers plus 1 as

qmax = [− log(cp)/ logΣ], where in this expression [. . .]indicates the integer part. From Fig. 1 it is clear thatthe discontinuities in the curve Σ = Σ(p) correspond tothe percolation transitions of layers of increasing superde-gree qmax. For γ = 2.8 the first, second and third transi-tions occur at (p1,Σ1) = (0.644285, 0.138357), (p2,Σ2) =(0.814055, 0.266496), and (p3,Σ3) = (0.927368, 0.37778).One can see that for γ = 4.5 the activations of lay-ers of increasing superdegree qmax become much morerapid. For these parameters the first, transitions oc-cur at (p1,Σ1) = (0.390686, 0.226456). For the case ofγ = 2.8, which we discussed above, we plot the observ-ables σq = Sq (fraction of nodes in a layer of superdegreeq, belonging to the mutual component) vs. p for dif-ferent values of q, see Fig. 2. This figure demonstrateshow giant clusters of the mutually connected componentemerge progressively in layers with higher and higher qas we increase the control parameter p. Note that, asis natural, each discontinuous emergence of a fractionof the mutual component nodes in layers of superdegreeq is accompanied by discontinuities of the dependenciesσq′(p) = Sq′(p) for all smaller superdegrees q′ < q.As a second example of the configuration model, we

have taken a network of networks ensemble with a Pois-son P (k) distribution characterized by 〈k〉 = c and aPoisson distribution P (q) = (c2)

qe−c2/q!, where we takeq ∈ [1, Q], excluding the layers with q = 0 that are notinteracting, and using the superdegree cutoff Q = 103

for performing the numerical calculations. In Fig. 3we have chosen c = 50 and c2 = 0.8, 1.2 as exam-ples of the supernetwork below (c2 = 0.8) and above(c2 = 1.2) the ordinary percolation phase transition inthe Poisson supernetwork. In order to see at whichvalue of p the layers with superdegree qmax become per-colating, in Fig. 3 we also plot the maximal superde-gree of the percolating layers qmax = [− log(cp)/ logΣ]vs. p. For c2 = 0.8 the first transitions are (p1,Σ1) =(0.478781, 0.07489530), (p2,Σ2) = (0.619669, 0.232843),and (p3,Σ3) = (0.667896, 0.338005). For c2 = 1.2 thefirst transitions are (p1,Σ1) = (0.714137, 0.0502185),(p2,Σ2) = (0.915646, 0.204011), and (p3,Σ3) =(0.918237, 0.421259). In Fig. 4 we plot the phase diagramcontaining a phase in which Σ = 0 (white region) and aphase in which Σ > 0 (shaded region) for p = 1. In par-

6

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

p

S

Γ=2 8

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

p

S

Γ=4 5

0 0.2 0.4 0.6 0.8 10

1

2

3

p

qm

ax

Γ=2 8

0 0.2 0.4 0.6 0.8 10

20

40

60

p

qm

ax

Γ=4 5

FIG. 1: Plot of the order parameter Σ vs. p and of the maximal superdegree of percolating layers qmax vs. p for a configurationmodel with a Poisson P (k) distribution with average 〈k〉 = c = 20 and a scale-free P (q) distribution with γ = 2.8 and γ = 4.5,and with minimal degree m = 1 and maximal superdegree Q = 103.

ticular, the phase diagram (γ, c) is plotted for a scale-freesupernetwork with power-law exponent γ and superde-grees q ∈ [m,Q] with m = 1 and Q = 103 and the phasediagram (c2, c) is plotted for a Poisson supernetwork withaverage superdegree 〈q〉 = c2 and the minimal and max-imal superdegrees given respectively by m = 1, Q = 103.As is natural, Figs. 1 and 3 correspond to points withinthe shaded areas in Fig. 4.

IV. PERCOLATION IN THE CONFIGURATIONMODEL OF NETWORK OF NETWORKS WITH

PARTIAL INTERDEPENDENCE

In this section we consider an interesting variation ofthe percolation problem in the configuration model ofthe network of networks. In particular we assume thatthe superdegree adjacency matrix a has probability P (a)given by Eq. (5), while the probability of P ({siα}) isgiven by Eq. (6). In addition to that we assume thatan interlink between replicas (i, α) and (i, β) is damagedand permanently removed with probability 1− r. In thiscase the nodes in a layer α with superdegree qα = q canbe interdependent on n ∈ [0, q] other randomly chosen

layers. In this case the message passing equations aregiven by

σiα→jα =∏

γ∈C(i,α)\α

siγ

1−∏

ℓ∈Nγ(i)

(1− σℓγ→iγ)

×siα

1−∏

ℓ∈Nα(i)\j

(1− σℓα→iα)

,

where C(i, α) is the connected cluster of the local su-pernetwork of node i when we consider only interdepen-dencies, i.e., only the interlinks that are not damaged.We can therefore indicate by σq the average messages inwithin a layer α of degree qα = q, obtaining

σq = p∑

s

q∑

n=0

(

q

n

)

rn(1− r)q−nP (s|n)Bs−1

×[1−Gk1(1 − σq)]. (24)

Here, P (s|n) indicates the probability that a node i inlayer α with qα = q and n interdependent layers is in aconnected component C(i, α) of the local supernetwork of

7

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

p

σq

q=3

q=2

q=1

FIG. 2: Plot of σq = Sq (fraction of nodes in a layer of su-perdegree q, belonging to the mutual component) vs. p fordifferent values q = 1, 2, 3 in the configuration model of thenetwork of networks having a Poisson P (k) distribution withaverage 〈k〉 = c = 20 and a scale-free P (q) distribution withγ = 2.8, minimal degree m = 1 and maximal superdegreeQ = 103. For each value of q = 1, 2, 3, σq emerges discon-tinuously, with a jump, which becomes smaller and smallerwith increasing q. The emergence of σ2 is accompanied by adiscontinuity of σ1(p). The emergence of σ3 is accompaniedby discontinuities of σ1(p) and σ2(p).

interdependent layers of cardinality |C(i, α)| = s. More-over, B is defined as

B =

∑

q′

q′P (q′)

〈q〉 p[1−Gk0(1− σq′ )]

. (25)

Similarly, the probability that a node i in a layer α withsuperdegree qα = q is in the mutually connected compo-nent Sq = 〈Si,α〉 is given by

Sq = p∑

s

q∑

n=0

(

q

n

)

rn(1− r)q−nP (s|n)Bs−1

×[1−Gk0(1 − σq)]. (26)

If the layers are not all formed by a single giant com-ponent or if p < 1 we have B < 1, and therefore wecan neglect in Eqs. (24)–(26) the contribution comingfrom the giant component of the local supernetworks inthe large M limit. Therefore in Eq. (24) we can re-place P (s|q) with the probability Pf (s|q) that a nodeof degree q belongs to a finite component of size s inthe supernetwork. Let us consider the quantity P (s) =∑

q[qP (q)/〈q〉]∑qn=0

(

qn

)

rq(1− r)q−nPf (s|n). Since P (s)only depends on the distribution of finite components wehave

P (s) =∑

q

qP (q)

〈q〉

q∑

n=0

(

q

n

)

rq(1− r)q−n

×∑

s1,s2,...sn

n∏

ℓ=1

P (sℓ)δ

(

q∑

ℓ=1

sℓ, s− 1

)

. (27)

Therefore P (s) has a generating function H(z) =∑

s P (s)zs that satisfies the equation

H(z) = zGq1(rH(z) + 1− r). (28)

Moreover, since

Pf (s|n) =∑

s1,s2,...sn

n∏

ℓ=1

P (sℓ)δ

(

n∑

ℓ=1

sℓ, s− 1

)

. (29)

we find

∑

s

q∑

n=0

(

q

n

)

rq(1 − r)1−nPf (s|n)Bs−1

= [rH(B) + 1− r]q. (30)

Therefore Eqs. (24), (28), and (25) have the form

σq = p[rH(B) + (1− r)]q [1−Gk1(1− σq)],

H(z) = zGq1(rH(z) + 1− r),

B =

∑

q′

q′P (q′)

〈q〉 p[1−Gk0(1− σq′ )]

. (31)

Putting H(B) = Σ we find that Eqs. (8) and Eqs. (9)become

σq = p(rΣ + 1− r)q [1−Gk1(1− σq)],

Sq = p(rΣ + 1− r)q [1−Gk0(1− σq)],

Σ =

∑

q′

q′P (q′)

〈q〉 p[1−Gk0(1− σq′ )]

,

×∑

q′

q′P (q′)

〈q〉 (rΣ + 1− r)q′−1. (32)

Let us consider for simplicity the case in which eachlayer is formed by a Poisson network with average degree〈k〉 = c. Then the previous equations become

σq = Sq = p(rΣ + 1− r)q(1− e−cσq ),

Σ =

∑

q′

q′P (q′)

〈q〉 p[1− e−cσq′ ]

×∑

q′

q′P (q′)

〈q〉 (rΣ + 1− r)q′−1. (33)

In the case in which the local supernetwork is regular, i.e.P (q) = δ(q,m), we have Σ[rΣ + 1− r]σm = σ satisfyingthe following equation

σ = p

[

1

2

(

1− r +√

(1− r)2 + 4rσ)

]q

(1− e−cσ). (34)

8

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

p

S

c2=0.8

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

p

S

c2=1.2

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

p

qm

ax

c2=0.8

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

p

qm

ax

c2=1.2

FIG. 3: Plot of the order parameter Σ vs. p and of the maximal superdegree of percolating layers qmax vs. p for a configurationmodel with a Poisson P (k) distribution with 〈k〉 = c = 50 and a Poisson P (q) distribution c2 = 0.8, 1.2 and q ∈ [1, Q] withmaximal superdegree Q = 103.

2.5 3 3.5 4 4.5 50

10

20

30

40

50

Γ

c

0 0.5 1 1.5 20

20406080

100120

c2

c

FIG. 4: Phase diagram of a power-law supernetwork (left panel) and a Poissonian supernetwork (right panel) at p = 1. Theshaded region shows the phase with Σ > 0, and in the white region, Σ = 0. The power-law super network has superdegreedistribution P (q) ∝ q−γ with q ∈ [m,Q], the Poisson supernetwork has superdegree distribution P (q) ∝ cq

2/q! with q ∈ [m,Q].

In both cases we have taken m = 1 and Q = 103.

For an arbitrary distribution P (q), the problem definedin Eqs. (18) still has a single order parameter Σ. As inSection III, the first equation in Eqs. (18) has solution

expressed in terms of the principal value of the Lambertfunction W (x), namely,

9

σq =1

c

[

pc(rΣ + 1− r)q +W(

−pc(rΣ + 1− r)qe−pc(rΣ+1−r)q)]

. (35)

Inserting this solution back in the equation for Σ in Eqs. (18) we find

Σ = Gq1(rΣ + 1− r)

[

p+∑

q

qP (q)

〈q〉1

c(rΣ + 1− r)−qW

(

−pc(rΣ + 1− r)qe−pc(rΣ+1−r)q)

]

. (36)

Similarly to Section III, we write this equation as F2(Σ, p, r) = 0. By imposing F2(Σ, p, r) = 0 and dF2(Σ, p, r)/dΣ = 0we can find the set of critical points p = pc at which the function Σ(p) turns out to be discontinuous. The equationdF2(Σ, p, r)/dΣ = 0 reads as

1

p=∑

q

qP (q)

〈q〉 (1−e−cσq )∑

q

q(q−1)P (q)

〈q〉 [rΣ+1−r]q−2+cpGq1(rΣ+1−r)

∑

q<qmax

q2P (q)

〈q〉 e−cσq[rΣ+1−r]q−1(1−e−cσq )

1− pc[rΣ+1−r]qe−cσq. (37)

where qmax = [− log(pc)/ log(rΣ + 1 − r)]. Note that incontrast to the case of r = 1, for r 6= 1 all values of theminimal superdegree m ≥ 0 provide non-trivial networkof networks. Indeed, for r < 1, even ifm ≥ 2, local super-networks have finite connected components. In additionto the abrupt phase transitions, this model displays alsoa set of continuous phase transition for different q, wherethe order parameter σq acquires a non-zero value. Thesetransitions occur at

p = pc =1

c(1− r + rΣ)q. (38)

These transitions are only stable for r below some specialvalue rq, and at r = rq they become discontinuous. Thevalue rq can be obtained by solving simultaneously thefollowing set of equations

1

c(1− r + rΣ)q= p,

F2(Σ, p, r) = 0,

dF2(Σ, p, r)

dΣ= 0. (39)

V. CONCLUSIONS

In this paper we have characterized the robustnessproperties of the configuration model of network of net-works by evaluating the size of the mutually connectedcomponent when a fraction 1 − p of nodes have beendamaged and removed. The configuration model of net-work of networks is an ensemble of multilayer networksin which each layer is formed by a network of N nodes,and where each node might depend only on its “replicanodes” on the other layers. We assign to each layer α asuperdegree qα indicating the number of interdependentnodes of each individual node (i, α) of the layer α andtake these qα “replica nodes” in qα uniformly randomly

chosen layers, independently for each node (i, α). Wehave shown that percolation in this ensemble of networksof networks demonstrate surprising features. Specifically,for low values of p, only the layers with low enough valueof the superdegree are percolating, and as we raise thevalue of p several discontinuous transition can occur inwhich layers of increasing value of superdegree q beginpercolate.

Here we observe a sharp contrast to ordinary percola-tion in which nodes of high degree belong to the giantconnected component (percolation cluster) with higherprobability. This principal difference is explained by thedefinition of the mutual component according to which anode is in a mutual component only if all the nodes inter-dependent with this node belong to the mutual compo-nent. This condition makes more difficult the entranceinto the mutual component for layers with a high su-perdegree. The non-trivial point here is that a layerof each given degree enters the mutual component notsmoothly but through a discontinuous transition.

The networks of networks which we considered inthis paper differ from those we studied in our previouswork [33] in one key aspect. In the present work, in-terdependence links of different nodes of a layer are notlead to the same other layers as in Ref. [33], but they aredistributed over the other layers essentially more ran-domly, independently for different nodes of a layer. Wehave found that this additional randomness dramaticallychange results and leads to new effects.

One should note that interlinking of only “replicanodes” is actually a great, very convenient simplificationwhich has enabled us to solve the problem analytically.Moreover, we have first assumed that all nodes in thesame layer have equal superdegree (number of interde-pendencies), which is a strong constraint. On the nextstep, we however have removed this restriction by intro-ducing the probability 1−r that an interdependence linkis removed. We found that when a fraction 1 − r > 0 ofinterdependent links are removed, each of these specific

10

transitions can change from discontinuous to continuous.In summary, we have found novel percolation phenom-

ena in a more general model of a network of networksthan the network models which were considered previ-ously. The multiple transitions, accompanying the ex-pansion of mutual component over the layers of such anetwork of networks, are in dramatic contrast to ordi-nary percolation and to more simple interdependent andmultiplex networks, e.g., for a pair of interdependent net-works. We suggest that our findings should be valid evenfor more general networks of networks.

Acknowledgments

This work was partially supported by the FCT projectPTDC/MAT/114515/2009 and the FET proactive IPproject MULTIPLEX number 317532.

Appendix A: Derivation of Eq. (3)

In Ref. [33] the percolation transition in a networkof network in which all the local adjacency matricesare the same, i.e., Ai

αβ = Aαβ ∀i, was considered,implying that the local supernetwork is the same forevery node i. In this setting, the percolation propertiesare determined by the message passing Eqs. (1) withNi(α) = Nα indicating the set of layers β which areneighbors of layer α in any local supernetwork. Forthis network of networks, it was shown in Ref. [33] thatEqs. (1) can be written as

S′iα→iβ =

∏

ξ∈C(α)

siξ

1−∏

ℓ∈Nξ(i)

(1 − σℓξ→iξ)

,

σiα→jα =∏

γ∈C(α)\α

siγ

1−∏

ℓ∈Nγ(i)

(1− σℓγ→iγ)

×siα

1−∏

ℓ∈Nα(i)\j

(1 − σℓα→iα)

, (A1)

where C(α) indicates the connected component of thesupernetwork to which layer α belongs. Moreover inRef. [33] it has been shown that Siα is given by

Siα = siα∏

β∈C(α)

1−∏

ℓ∈Nβ(i)

(1− σℓβ→iβ)

. (A2)

Now we observe that all the steps performed in Ref. [33]to obtain Eqs. (A1) and (A2) are in fact only operationsacting in the local supernetwork of node i. It follows im-mediately that for the network of networks coming fromthe configuration models, the same equations should bevalid, where we replace C(α) with the connected compo-nent of the local supernetwork C(i, α) of node i passingthrough layer α. Therefore for the configuration model ofa network of networks the solution to the message passingEqs. (1) and (2) is given by Eqs. (3) and Eqs. (4).

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