Control of Multilayer Networks
CControl of Multilayer Networks
Giulia Menichetti, Luca Dall’Asta,
2, 3 and Ginestra Bianconi Department of Physics and Astronomy and INFN Sez. Bologna,Bologna University, Viale B. Pichat 6/2 40127 Bologna, Italy ∗ Department of Applied Science and Technology, DISAT,Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy Collegio Carlo Alberto, Via Real Collegio 30, 10024 Moncalieri, Italy † School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom ‡ The controllability of a network is a theoretical problem of relevance in a variety ofcontexts ranging from financial markets to the brain. Until now, network controllabilityhas been characterized only on isolated networks, while the vast majority of complexsystems are formed by multilayer networks. Here we build a theoretical frameworkfor the linear controllability of multilayer networks by mapping the problem into acombinatorial matching problem. We found that correlating the external signals inthe different layers can significantly reduce the multiplex network robustness to noderemoval, as it can be seen in conjunction with a hybrid phase transition occurringin interacting Poisson networks. Moreover we observe that multilayer networks canstabilize the fully controllable multiplex network configuration that can be stable alsowhen the full controllability of the single network is not stable.
Most of the real networks are not isolated but interact with each other forming multilayer structures [1, 2]. Forexample, banks are linked to each other by different types of contracts and relationships, gene regulation in the cellis mediated by the different types of interactions between different kinds of molecules, brain data are described bymultilayer brain networks. Studying the controllability properties of these networks is important for assessing therisk of a financial crash [3, 4], for drug discovery [5] and for characterizing brain dynamics [6–10]. Therefore thecontrollability of multilayer networks is a problem of fundamental importance for a large variety of applications.Recently, linear [11–19] and non-linear [20–30] approaches are providing new scenarios for the characterization ofthe controllability of single complex networks. In particular, in the seminal paper by Liu et al. [12] the structuralcontrollability of complex networks has been addressed by mapping this problem into a Maximum Matching Problemthat can be studied using statistical mechanics techniques [31–36]. Other works approach the related problem ofnetwork observability [37], or target control [38] which focuses on controlling just a subset of the nodes. Despite thesignificant interest in network controllability, all linear and non-linear approaches for the controllability of networksare still restricted to single networks while it has been recently found that the multiplexity of networks can haveprofound effects on the dynamical processes taking place on them [39–44]. For example, percolation processes thatusually present continuous phase transitions on single networks can become discontinuous on such structures [39–43]and are characterized by large avalanches of disruption events.Here, we consider the elegant framework of structural controllability [12] and investigate how the multilayer structureof networks can affect their controllability. We focus on multiplex networks, which are multilayer networks in whichthe same set of nodes are connected by different types of interactions. Multiplex network controllability is studiedunder the assumption that input nodes are the same in all network layers, thus mimicking the situation in whichinput nodes can send different signals in the different layers of the multiplex but the position of the external signalsin the layers is correlated.We show that controlling the dynamics of multiplex networks is more costly than controlling single layers takenin isolation. Moreover, the controllability of multiplex networks displays unexpected new phenomena. In fact thesenetworks can become extremely sensible to damage in conjunction with a discontinuous phase transition characterizedby a jump in the number of input points (driver nodes). A careful investigation of this phase transition reveals thatthis is a hybrid phase transition with a square root singularity, therefore in the same universality class of the emergenceof the mutually connected component in multiplex networks [1, 39, 41]. The number of driver nodes in the multiplexnetwork is in general higher than the number of driver nodes in the single layers taken in isolation. Neverthelessthe degree correlations between low-degree nodes in the different layers can affect the controllability of the multiplexnetwork and modulate the number of its driver nodes. Moreover, a fully controllable configuration can be stable in a ∗ Electronic address: [email protected] † corresponding author; Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ phy s i c s . s o c - ph ] J a n !" FIG. 1:
Control of a multiplex network . The controllability of a duplex network (multiplex with M = 2 layers) can bemapped to a Maximum Matching Problem in which the unmatched nodes (indicated with a white circle) are the driver nodesof the duplex network. Here we have indicated with red thick links the matched links and by black thin links the unmatchedlinks. multilayer network even if it is not stable in the isolated networks that form the multilayer structure. RESULTS
We consider multiplex networks [1] in which every node i = 1 , , . . . , N has a replica node ( i, α ) in each layer α andevery layer is formed by a directed network between the corresponding replica nodes. We assume that each replicanode ( i, α ) is characterized by a different dynamical variable x αi ∈ R and that each layer is characterized by a possiblydifferent dynamical process. We consider for simplicity a duplex, i.e a multiplex formed by two layers { A, B } whereeach layer α ∈ { A, B } is a directed network. The state of the network at time t is governed by a linear dynamicalsystem d X ( t ) dt = G X + K u , (1)in which the 2 N -dimensional vector X ( t ) describes the dynamical state of each replica node, i.e. X i = x Ai and X N + i = x Bi for i = 1 , , . . . , N . The matrix G is a 2 N × N (asymmetric) matrix and K is a 2 N × P matrix. Theyhave the following block structure G = (cid:18) g A g B (cid:19) , K = (cid:18) K A K B (cid:19) , (2)in which g α are the N × N matrices describing the directed weighted interactions within the layers and K α are the N × P α matrices describing the coupling between the nodes of each layer α and P α ≤ N external signals. Thelatter are represented by a vector u ( t ) of elements u γ and γ = 1 , . . . P = P A + P B . Here we consider the conceptof structural controllability [11, 12] that guarantees the controllability of a networks for any choice of the non-zerosentries of G and K , except for a variety of zero Lebesgue measure in the parameter space. Therefore each layer of theduplex networks can be structurally controlled by identifying a minimum number of driver nodes , that are controllednodes which do not share input vertices. If different replicas of the same node can be independently controlled, thenthe controllability properties of the multiplex network factorize and each layer can be studied as if was taken inisolation [11, 12, 16, 20]. Liu et al. [12] showed that in a single layer the minimum set of driver nodes can be foundby mapping the problem into a matching problem. In real multiplex networks however nodes are usually univocallydefined and share common properties across different layers, therefore we make the assumption that each node of theduplex network is either a driver node in each layer or it is not a driver node in any layer . The problem of findingthe driver nodes of the duplex network can be thus mapped into a maximum matching problem in which every nodehas at most one matched incoming link and at most one matched outgoing link, with the constraint that two replicanodes either have no matched incoming links on each layer or have one matched incoming link in each layer (seeFigure 1). This problem can be studied, using statistical mechanics techniques, such as the cavity method and theBelief Propagation (BP) algorithm. Following [12, 16], we consider the variables s αij = 1 , i, α ) to node ( j, α ) in layer α = A, B is matched or not. In order to have a matching ineach layer of the duplex the following constraints have always to be satisfied (cid:88) j ∈ ∂ α + i s αij ≤ , (cid:88) i ∈ ∂ α − j s αij ≤ . (3)where ∂ α + i is the set of replica nodes ( j, α ) in layer α that are reached by directed links from ( i, α ) and ∂ α − j is the setof replica nodes ( i, α ) in layer α that point to ( j, α ). In addition, we impose that the driver nodes in the two layers(the unmatched nodes) are replica nodes, i.e. (cid:88) i ∈ ∂ A − j s Aij = (cid:88) i ∈ ∂ B − j s Bij . (4)In this formalism, computing the maximum matching corresponds to minimize an energy function E = N D = N n D where N D is the number of unmatched replica nodes associated to each matching. The energy E for a given matching,can be expressed in terms of the variables s ij as E = (cid:88) α (cid:88) j − (cid:88) i ∈ ∂ α − j s αij . (5)In order to study this novel statistical mechanics problem, we derived the BP equations [33, 35, 36] (see Methods andSupplementary Material) valid in the locally tree-like approximation, as described for the case of a single networkproblem in [12, 16, 31, 32, 34]. DISCUSSION
The controllability of multiplex networks displays a rich phenomenology, coming from the interplay between thedynamical and the structural properties of multiplex networks. Here we characterize the controllability of multiplexnetworks with different degree distribution and with tunable level of structural correlations.
Phase transition in Poisson duplex networks–
We consider duplex networks in which the two layers are realizationsof uncorrelated directed random graphs characterized by Poisson distributions for in-degree and out-degree with sameaverage value c , i.e. (cid:104) k Ain (cid:105) = (cid:104) k Aout (cid:105) = (cid:104) k Bin (cid:105) = (cid:104) k Bout (cid:105) = c . In Figure 2A we report the average rescaled number of drivernodes n D as function of the average degree c computed from the solutions of Eqs.(8) on single instances and from thegraph ensemble analysis. A comparison with two independent layers with the same topological properties shows thatthe controllability of a duplex network is in general more demanding in terms of number of driver nodes than thecontrollability of independent single layers, in particular for low average degrees. In addition, a discontinuity in thenumber of driver nodes at c = c (cid:63) = 3 . . . . marks a change in the controllability properties of duplex networks thatis not observed in uncoupled networks. This is due to a structural change in the solution of the matching problem,in which a finite density of zero valued cavity fields emerges. A careful investigation (see Supplementary Material )reveals that this is a hybrid phase transition with a square root singularity, therefore in the same universality class ofthe emergence of the mutually connected component in multiplex networks [1, 39, 41].In correspondence to this phase transition the network responds non trivially to perturbations. This is observed byperforming a numerical calculation of the robustness of the networks. Following [12] we classify the nodes into three c n D c n c n o n r c ∗ c ∗ A BtheoryBP
FIG. 2:
Controllability of Poisson duplex networks with average degrees (cid:10) k A,in (cid:11) = (cid:10) k A,out (cid:11) = (cid:10) k B,in (cid:11) = (cid:10) k B,out (cid:11) = c . In panel A the fraction n D of driver nodes in a Poisson duplex network with (cid:10) k A,in (cid:11) = (cid:10) k A,out (cid:11) = (cid:10) k B,in (cid:11) = (cid:10) k B,out (cid:11) = c ,plotted as a function of the average degree c . The points indicate the average BP results obtained over 5 single realizations ofthe Poisson duplex networks with average degree c and N = 10 , the solid line is the theoretical expectation (the error bar,indicating the interval of one standard deviation from the mean, is always smaller or comparable to marker size). The dashedline represents twice the density of driver nodes for a single Poisson network with the same average degree.In panel B the densities n c , n r and n o respectively of critical redundant and ordinary nodes are shown as functions of c for thesame type of duplex networks with N = 10 , where each point is the average over 100 different instances.In both panels the dot-dashed vertical line indicates the phase transition average degree c ∗ = 3 . . . . . categories: critical nodes , redundant nodes and ordinary nodes . When a critical node is removed from the (multiplex)network, controllability is sustained at the cost of increasing the number of driver nodes. If the number of driver nodesdecreases or is unchanged, the removed nodes are classified as redundant and ordinary respectively. Figure 2B showsthat the fraction of critical nodes reaches a maximum at the transition, revealing an increased fragility of the duplexnetwork to random damage with respect to single layers. While an abrupt change in the number of driver nodes canresult from a small change in the network topology, it is important to stress that the non-monotonic behavior of thesequantities around the critical average degree value could be interpreted as a precursor of the discontinuity.In a duplex network formed by directed Poisson random graphs with different average degree in the two layers (i.e. (cid:104) k αin (cid:105) = (cid:104) k αout (cid:105) = c α ) a similar discontinuous phase transition is observed (see Figure 3). Nevertheless we checkedthat this discontinuous phase transition is not occurring for every multiplex network structure (see SupplementaryMaterial). Effect of degree correlations on the controllability of duplex networks –
We consider a model of duplex network inwhich the replica nodes of the directed random graphs in the two layers have correlated degrees. In particular, weconsider a case in which only the low in-degree nodes (nodes with in-degree equal to 0 , ,
2) are correlated (replicanodes in different layers have same degree with probability p ) and a case in which the in-degrees of the replica nodesare equal with probability p independently of their value (see Supplementary Material for details). The controllabilityof the network is affected by these correlations as shown in Figure 4. In fact, the number of driver nodes n D decreasesas the level of correlation increases. In duplex networks with Poisson degree distribution, low-degree correlationsmodify both the position of the hybrid transition and the size of the discontinuity. Once the replica nodes with lowin-degree are correlated, a further correlation of the remaining replica nodes does not substantially change the numberof driver nodes. This result confirms that structural controllability is essentially determined by the control of lowdegree nodes [16]. Stability of the fully controllable solution –
A fully controllable solution, in which a single driver node is necessary tocontrol the whole duplex network, exists if the minimum in-degree and the minimum out-degree are both greater than1 in both layers. This solution of the cavity equations gives the correct solution to the maximum matching problemdescribing the controllability of multilayer networks only if no instabilities take place. The stability conditions are thenfound by imposing that the Jacobian of the systems of equations derived by the cavity method has all its eigenvalues λ i of modulus less than one, i.e. | λ i | <
1. In random duplex networks with the same degree distribution in the twolayers, the fully controllable solution is stable (see Supplementary Material for the details of the derivation) if and c A c B FIG. 3:
Phase diagram of the controllability for a Poisson duplex networks with average degrees (cid:10) k A,in (cid:11) = (cid:10) k A,out (cid:11) = c A and (cid:10) k B,in (cid:11) = (cid:10) k B,out (cid:11) = c B . The color code indicates the density of driver nodes n D = E/N in the multiplexnetwork. (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) (cid:1)
BP p = (cid:2) BP ld p = (cid:3) BP ld p = (cid:4) BP td p = A c n D (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:4) (cid:1) BP p = (cid:2) BP ld p = (cid:3) BP ld p = (cid:4) BP td p = B (cid:1) n D FIG. 4:
The effect of the degree correlation between replica nodes in different layers on the controllabilityof multiplex networks.
Correlations between the low in-degrees (ld) and correlations between any in-degree node (td),parametrized by p , affect the fraction of driver nodes in the network n D , both in the case of Poisson networks with the same inand out average degree c across the two layers (Panel A) and in the case of scale-free networks with the same in and out degreedistribution across the layers, given by P ( k ) ∝ k − γ and minimum in/out degree 1 (Panel B). When p = 0 there is no degreecorrelation between replica nodes in different layers. The BP data are shown for networks with N = 10 , and are averaged 5times for panel A and 20 times for panel B. N D nu m b er o f i n s t a n ce s Layer ALayer BDuplex
FIG. 5:
The fully controllable solution can be stable for the multiplex network also if it is not stable for thesingle layers taken in isolation.
Histogram of the number of networks that out of 100 realizations have N D driver nodes.The results obtained for the control of a duplex networks and its two layers are compared. The duplex networks are formed bytwo scale-free networks with N = 10 and P A,in ( k ) = P B,in ∝ k − γ for k > P A,out ( k ) = P B,out ( k ) ∝ k − γ for k >
2, with γ = 2 .
3, the networks have minimum in-degree equal to 2 and minimum out-degree equal to 3 and P A,in (2) = P B,in (2) = 0 . only if P outα (2) < (cid:104) k α (cid:105) in (cid:104) k α (cid:105) out (cid:104) k α ( k α − (cid:105) in , (6)for α = A, B . On single networks it was instead recently found [16] that the fully controllable configuration is onlystable for P in (2) < (cid:104) k (cid:105) in (cid:104) k ( k − (cid:105) out , P out (2) < (cid:104) k (cid:105) in (cid:104) k ( k − (cid:105) in . (7)This implies that for multiplex networks with asymmetric in-degree and out-degree distributions it might occur thatthe fully controllable solution is stable in the multiplex network but unstable in the single networks taken in isolation(see Figure 5 for the characterization of the controllability of a similar type of multiplex networks). Therefore amultiplex structure can help to stabilize the fully controllable solution.In conclusion, within the framework of structural controllability, we have considered the controllability propertiesof multiplex networks in which the nodes are either driver nodes in all the layers or they are not driver nodes in anylayer. Our results show that controlling multiplex networks is more demanding, in terms of number of driver nodes,than controlling networks composed of a single layer. In random duplex networks with Poisson degree distribution, itis possible to observe a hybrid phase transition with a discontinuity in the number of driver nodes as a function of theaverage degree, that is phenomenologically similar to the emergence of mutually connected components. Close to thisphase transition the duplex network exhibits an increased fragility to random damage. The existence of correlationsbetween the degrees of replica nodes in different layers, in particular between low-degree nodes, has the effect ofreducing the number of driver nodes necessary to control duplex networks. Finally, multiplex structure of networkscan stabilize the fully controllable solution also if this solution is not stable in the single layers that form the multiplexnetwork. METHODS
The BP equations –
The BP equations of this problem are derived using the cavity method [33, 35, 36] as describedfor the case of a single network problem in [12, 16, 31, 32, 34]. The same approximation methods can be applied here,as long as the structure of the interconnected layers is locally tree-like both within the layers and across them. Underthe decorrelation (replica-symmetric) assumption, the cavity fields (or messages) { h αi → j } and { ˆ h αi → j } , defined on thedirected links between neighboring nodes ( i, α ) and ( j, α ) in the same layer α = A, B satisfy the zero-temperaturelimit of the BP equations, also known as Max-Sum equations, h αi → j = − max (cid:20) − , max k ∈ ∂ + i \ j ˆ h αk → i (cid:21) ˆ h Ai → j = − max (cid:34) max k ∈ ∂ A − i \ j h Ak → i , − max k ∈ ∂ B − j h Bk → i (cid:35) ˆ h Bi → j = − max (cid:34) max k ∈ ∂ B − i \ j h Bk → i , − max k ∈ ∂ A − j h Ak → i (cid:35) (8)in which the fields are defined to take values in the discrete set {− , , } and here and in the following we use theconvention that the maximum over a null set is equal to − E in Eq.(5) becomes E = − (cid:88) α N (cid:88) i =1 max (cid:20) − , max k ∈ ∂ α + i ˆ h αk → i (cid:21) + (cid:88) α (cid:88) ( i,j ) max (cid:104) , h αi → j + ˆ h αj → i (cid:105) − (cid:88) i =1 ,N max (cid:34) , max k ∈ ∂ A − i h Ak → i + max k ∈ ∂ B − i h Bk → i (cid:35) . (9) BP equations over ensemble of networks-
Let us consider the case of uncorrelated duplex networks in which thedegree of the same node in different layers are uncorrelated and there is no overlap of the links. In each layer α = A, B we consider a maximally random network with in-degree distribution P α,in ( k ) and out-degree distribution P α,out ( k ).At the ensemble level, each link of (the infinitely large) random network forming layer α has the same statisticalproperties, that we describe through distributions P α ( h α ) and ˆ P α (ˆ h α ) of cavity fields that are defined on the supportof Eqs. 8, i.e. P α ( h α ) = w α δ ( h α −
1) + w α δ ( h α + 1) + w α δ ( h α ) , ˆ P α (ˆ h α ) = ˆ w α δ (ˆ h α −
1) + ˆ w α δ (ˆ h α + 1) + ˆ w α δ (ˆ h α ) , (10)where α = A, B and where the probabilities w α , w α , w α are normalized w α + w α + w α = 1 as well as the probabilitiesˆ w α , ˆ w α , ˆ w α that satisfy the equation ˆ w α + ˆ w α + ˆ w α = 1. The cavity method at the network ensemble level is alsoknown as density evolution method [36].It is useful to introduce the generating functions G α,in/out ( z ) , and G α,in/out ( z ) of the multiplex network as G α,in/out ( z ) = (cid:80) k P α,in/out ( k ) z k , G α,in/out ( z ) = (cid:80) k k (cid:104) k α (cid:105) P α,in/out ( k ) z k − , with α = A, B . In this way, we canderive recursive equations for the probabilities { w αi } i =1 , , and { ˆ w αi } i =1 , , , that are the analogous of the BP equa-tions for an ensemble of uncorrelated duplex networks w α = G α,out ( ˆ w α ) ,w α = (cid:2) − G α,out (1 − ˆ w α ) (cid:3) , ˆ w A = G A,in ( w A ) (cid:104) − G B,in (1 − w B ) (cid:105) , ˆ w A = (cid:104) − G A,in (1 − w A ) + G A,in (1 − w A ) G B,in (cid:0) w B (cid:1)(cid:105) , ˆ w B = G B,in ( w B ) (cid:104) − G A,in (1 − w A ) (cid:105) , ˆ w B = (cid:104) − G B,in (1 − w B ) + G B,in (1 − w B ) G A,in (cid:0) w A (cid:1)(cid:105) , (11)with w α = 1 − w α − w α , and ˆ w α = 1 − ˆ w α − ˆ w α . The energy E and the entropy density s of the matching problemcan be also expressed in terms of the { w αi } i =1 , , and { ˆ w αi } i =1 , , (see Supplementary Material for details). Hybrid transition for Poisson duplex network–
Here we consider the case of two Poisson networks with the samein/out average degree. In other words, we consider the situation in which (cid:10) k A,in (cid:11) = (cid:10) k A,out (cid:11) = (cid:10) k B,in (cid:11) = (cid:10) k B,out (cid:11) = c .We notice that the BP equations can be rewritten to form a closed subsystem of equations for ˆ w and ˆ w (seeSupplementary Material for details),ˆ w = h ( ˆ w , ˆ w ) = e − ce − c ˆ w (cid:104) − e − ce − c (1 − ˆ w (cid:105) (12a)ˆ w = h ( ˆ w , ˆ w ) = (cid:104) − e − ce − c (1 − ˆ w + e − ce − c (1 − ˆ w e − ce − c ˆ w (cid:105) (12b)from the solution of which the remaining quantities can be determined.The value c (cid:63) of the average degree c at which the discontinuity in the number of driver nodes n D observed is foundby imposing that Eqs. (12) are satisfied together with the condition | J | = 0 , (13)with J indicating the Jacobian of the system of equations (12). Imposing that Eqs. (12) and condition (13) aresimultaneously satisfied, the solution c (cid:63) = 3 . . . . is found. For c < c (cid:63) we observe that w = ˆ w = 0. At c (cid:63) we observe a discontinuity in both w and ˆ w , but for c > c (cid:63) the functions h ( ˆ w , ˆ w ) and h ( ˆ w , ˆ w ) are analytic,and analyzing Eqs. (12 a ) − (12 b ) we obtain the behaviour of the order parameters w and ˆ w for c > c (cid:63) w − w (cid:63) ∝ ( c − c (cid:63) ) / ˆ w − ˆ w (cid:63) ∝ ( c − c (cid:63) ) / , (14)showing that the transition is hybrid. Acknowledgements
LD aknowledges the European Research Council for grant n. 267915. GM aknowledges the European ProjectMIMOmics.
Author Contributions
GM, LD and GB designed the study, developed the methodology, performed the analysis and wrote the manuscript.The authors declare no competing financial interests.
Supplemental Material ”Control of Multilayer Networks”A. Introduction
This Supplemental Material is structured as follows.In Sec. II we define the problem of structural controllability of multiplex networks, focusing on the case of a duplexnetwork. Moreover we define the driver nodes, as the set of nodes that, if stimulated by an external signal, can drivethe dynamical state of the network to any desired configuration.In Sec. III we map the problem of structural controllability of a duplex network to a Maximum Matching Problem,and we derive the Belief Propagation (BP) equations determining the driver nodes, and their zero-temperature limitknown as Max-Sum equations.In Sec IV we consider the controllability of uncorrelated duplex networks, characterizing the BP equations valid for thisproblem, the stability conditions for the solutions of the BP equations, and the entropy of the solutions. Moreover, weconsider duplex networks formed by two Poisson layers and we characterize their hybrid phase transition. Finally weconsider the controllability of duplex networks formed by layers with power law in-degree and out-degree distributions.In Sec. V we consider ensembles of duplex networks in which the in-degrees of replica nodes are correlated and wederive the BP equations assuming either that only the low in-degrees of replica nodes are correlated or that all thein-degrees of replica nodes are correlated.
B. The structural controllability of a multiplex network
We consider a multiplex network in which every node i = 1 , , . . . , N has a replica node in each layer and every layeris formed by a directed networks between the corresponding replica nodes [1]. We assume that each replica node canhave a different dynamical state and can send different signals in the different networks (each layers is characterizedby a different dynamical process). In this case the controllability of the multiplex network can be treated by controltheory methods used for the single layers taken in isolation [11, 12, 16, 20]. Nevertheless here we will consider anadditional constraint on the number of driver nodes. In fact we impose that corresponding replica nodes are eitherdriver nodes in all layers or they are not driver nodes in any layer.We consider for simplicity a duplex, i.e a multiplex formed by two layers where each layer is formed by a directednetwork. We call the two layers layer α = A, B . We consider a linear dynamical system determining the networkdynamics d X ( t ) dt = G X ( t ) + K u ( t ) , (15)in which the vector X ( t ) describes the dynamical state of each replica node in the duplex, and has 2 N elements. Thefirst set of N elements represents the dynamical state x Ai of node i in layer A (i.e. X i = x Ai for i = 1 , . . . , N ),while the elements X N + i represent the dynamical state of the node i in layer B, and are given by X N + i = x Bi for i = 1 , , . . . , N . The matrix G is a 2 N × N (asymmetric) matrix and the matrix K is a 2 N × M matrix. The matrices G and K have the following block structure G = (cid:18) g A g B (cid:19) , K = (cid:18) K A K B (cid:19) , (16)where g α with α = A, B are the N × N matrices describing the directed weighted interactions within each of thenetworks in the two layers and K α are the N × M α matrices describing the interaction between the nodes of thenetwork α and the M α ≤ N external signals for layer α . The external signals are indicated by the vector u ( t ) ofelements u γ and γ = 1 , . . . M = M A + M B .Given block structure of both matrix G and K described in Eq. (16), the problem of duplex network controllabilitydefined by Eq. (15), can be exactly recast into the problem of controllability of the single layers that form the duplexnetwork.Here we adopt the framework of structural controllability [11] aimed at characterizing if a given duplex network iscontrollable when the non-zero matrix elements of G and K given by Eq. (16) are free parameters. A duplex networksin which the linear dynamics described by the Eqs. (15) and (16) take place, is structurally controllable if both layers α = A, B are structurally controllable.Each layer α is structurally controllable if for any choice of the free parameters in g α and K α , except for a variety ofzero Lebesgue measure in the parameter space, the Kalman’s condition is fulfilled [11]. Since structural controllabilityonly distinguishes between zero and non-zero entries of the matrices g α and K α , a given directed network in layer α is0structurally controllable if it is possible to determine the input nodes (i.e. the position of the non-zero entries of thematrix K α ) in a way to control the dynamics described by any realization of the matrix g α with the same non-zeroelements, except for atypical realizations of zero measure. In practice, a single network can be structurally controlledby identifying a minimum number of driver nodes , that are controlled nodes which do not share input vertices in bothlayers. In their seminal paper [12], Liu and coworkers showed that on single networks this control theoretic problemcan be reduced to a well-known optimization problem: their Minimum Input Theorem states that the minimum setof driver nodes that guarantees the full structural controllability of a network is the set of unmatched nodes in amaximum matching of the same directed network. Their result for a single network remains valid for the duplexnetwork described by Eqs. (15) − (16). Therefore the structural controllability of duplex networks, in the absenceof further constraints can be mapped to a Maximum Matching problem defined on the single layers of the duplexnetworks. Here nevertheless, we consider a further constraint to be imposed on the driver nodes, which enforce anew type of dependence between the layers of the duplex. In particular we impose that the replica nodes ( i, α ) with α = A, B and a given index i , are either both driver nodes or neither is a driver node. This implies that these tworeplica nodes are either both linked to independent and external signals or none of them is connected to externalsignals. C. The Maximum Matching Problem for the Controllability of Duplex Networks
1. Mapping duplex controllability into a constrained Maximum Matching Problem
In order to build an algorithm able to find the driver nodes of a duplex network we consider the variables s αij = 1 , i, α ) to node ( j, α ) in layer α = A, B is matched or not. In thetwo layers of the duplex network we want to have a matching, i.e. the following constraints must always be satisfiedfor α = A, B , (cid:88) j ∈ ∂ α + i s αij ≤ , (17a) (cid:88) j ∈ ∂ α − i s αji ≤ , (17b)where here and in the following we indicate with ∂ α + the set of nodes j that are pointed by node i in layer α and with ∂ α − i the set of nodes j pointing to node i in layer α . In addition we impose that the driver nodes in the two networks arereplica nodes, i.e. in the matching problem either two replica nodes are both matched or both unmatched. Thereforethe variable s αij satisfy the following additional constraints (cid:88) i ∈ ∂ A − j s Aji = (cid:88) i ∈ ∂ B − j s Bji . (18)Finally we need to minimize the number of driver nodes in the multiplex network. Therefore we minimize the energy E of the problem given by E = (cid:88) α (cid:88) j − (cid:88) i ∈ ∂ α − j s αij = (cid:88) α (cid:88) i E αi , (19)with E αi = 1 − (cid:88) j ∈ ∂ α + i s αij . (20)The energy E is given by the number N D of driver replica nodes in the duplex network by E = N D = N n D . (21)
2. Derivation of the BP equations at finite inverse temperature β We consider here the Maximum Matching Problem defined in Sec C 1. The goal is to find the configuration of thevariables { s αij } associated to every directed edge i → j in layer α , such that the energy E given by the number of1driver replica nodes in the duplex network is minimized provided that the conditions given by Eqs. (17), (18) aresatisfied. Introducing as an auxiliary variable the “inverse temperature” β we cast this problem into a statisticalmechanics problem where our first aim is finding the distribution P ( { s ij } ), parametrized by the inverse temperature β , and given by P ( { s ij } ) = e − βE Z N (cid:89) i =1 (cid:89) α θ − (cid:88) j ∈ ∂ + i s αij θ − (cid:88) j ∈ ∂ − i s αji δ (cid:88) i ∈ ∂ A − j s Aij , (cid:88) i ∈ ∂ B − j s Bij , (22)where θ ( x ) = 1 for x ≥ θ ( x ) = 0 for x < δ ( x ) is the Kronecker delta, and where Z is the normalizationconstant, that corresponds to the partition function of the statistical mechanics problem. Subsequently, we plan toperform the limit β → ∞ in order to characterize the optimal (i.e. the maximum-sized) matching in the networksatisfying Eqs. (17), (18). The free-energy of the problem F ( β ) is defined as βF ( β ) = − ln Z, (23)and the energy E is therefore given by E = ∂ [ βF ( β )] ∂β . (24)The distribution P ( { s ij } ) on a locally tree-like network can be (approximately) estimated by the cavity method inthe replica symmetric assumption (i.e. by deriving Belief Propagation equations) [12, 16, 31–36]. In this respect, ineach layer α of the duplex network, we define two probability marginals on each directed link, one going in the samedirection of the link P αi → j ( s ij ) and one in the opposite direction ˆ P αi → j ( s ji ). The BP equations for these quantities are P αi → j ( s ij ) = 1 D αi → j (cid:88) { s αik }| k ∈ ∂ α + i \ j θ − (cid:88) k ∈ ∂ α + i s αik exp − β − (cid:88) k ∈ ∂ α + i s αik (cid:89) k ∈ ∂ α + i \ j ˆ P αk → i ( s αik ) , (25a)ˆ P Ai → j ( s Aji ) = 1ˆ D Ai → j (cid:88) { s Aki }\ s Aji ,k ∈ ∂ A − i θ − (cid:88) k ∈ ∂ A − i s Aki (cid:88) { s Bki }| k ∈ ∂ B − i θ − (cid:88) k ∈ ∂ B − i s Bki δ (cid:88) i ∈ ∂ A − j s Aij , (cid:88) i ∈ ∂ B − j s Bij × (cid:89) k ∈ ∂ A − i \ j P Ak → i ( s Aki ) (cid:89) k ∈ ∂ B − i P Bk → i ( s Bki ) , (25b)ˆ P Bi → j ( s Bji ) = 1ˆ D Bi → j (cid:88) { s Bki }| k ∈ ∂ B − \ j θ − (cid:88) k ∈ ∂ B − i s Bki (cid:88) { s Aki }| k ∈ ∂ A − i θ − (cid:88) k ∈ ∂ A − i s Aki δ (cid:88) i ∈ ∂ A − j s Aij , (cid:88) i ∈ ∂ B − j s Bij × (cid:89) k ∈ ∂ B − i \ j P Bk → i ( s Bki ) (cid:89) k ∈ ∂ A − i P Ak → i ( s Aki ) (25c)where D αi → j and ˆ D αi → j are normalization constants. The probability marginals { P αi → j ( s αij ), ˆ P αi → j ( s αji ) } can beparametrized by the cavity fields h αi → j and ˆ h αi → j defined by P αi → j ( s αij ) = exp [ βh αi → j s αij ] [ βh αi → j ] ˆ P αi → j ( s αji ) = exp [ β ˆ h αi → j s αji ] [ β ˆ h αi → j ] . (26)In terms of the cavity fields (or messages), Eqs. (25) reduce to the following set of finite temperature BP equations, h αi → j = − β log e − β + (cid:88) k ∈ ∂ α + i \ j e β ˆ h αk → i , (27a)ˆ h Ai → j = − β log (cid:80) k ∈ ∂ B − i e βh Bk → i + (cid:88) k ∈ ∂ A − i \ j e βh Ak → i , (27b)ˆ h Bi → j = − β log (cid:80) k ∈ ∂ A − i e βh Ak → j + (cid:88) k ∈ ∂ B − i \ j e βh Bk → i , (27c)2The free energy F and the energy E = ∂βF∂β of the model are given respectively by − βF = (cid:88) α N (cid:88) i =1 ln e − β + (cid:88) k ∈ ∂ α + i e β ˆ h αk → i + (cid:88) i =1 ,N ln (cid:88) k ∈ ∂ A − i e βh Ak → i (cid:88) k (cid:48) ∈ ∂ B − i e βh Bk (cid:48)→ i − (cid:88) α (cid:88) α ln (cid:16) e β ( h αi → j +ˆ h αj → i ) (cid:17) , (28)and by E = (cid:88) α N (cid:88) i =1 e − β − (cid:80) k ∈ ∂ α + i ˆ h αk → i e β ˆ h αk → i e − β + (cid:80) k ∈ ∂ α + i e β ˆ h αk → i − N (cid:88) i =1 (cid:80) k ∈ ∂ A − i h Ak → i e βh Ak → i (cid:80) k (cid:48) ∈ ∂ B − i e βh Bk (cid:48)→ i (cid:80) k ∈ ∂ A − i e βh Ak → i (cid:80) k (cid:48) ∈ ∂ B − i e βh Bk (cid:48)→ i − (cid:88) i (cid:80) k ∈ ∂ A − i e βh Ak → i (cid:80) k (cid:48) ∈ ∂ B − i h Bk (cid:48) → i e βh Bk (cid:48)→ i (cid:80) k ∈ ∂ A − i e βh Ak → i (cid:80) k (cid:48) ∈ ∂ B − i e βh Bk (cid:48)→ i + (cid:88) α (cid:88) α ( h αi → j + ˆ h αj → i ) e β ( h αi → j +ˆ h αj → i ) e β ( h αi → j +ˆ h αj → i ) . (29)
3. BP Equations for β → ∞ The BP equations in the limit β → ∞ are derived from the Eqs. (27). In the limit β → ∞ the solution is expressedin terms of the fields h αi → j or ˆ h αi → j sent from a node ( i, α ) to the linked node ( j, α ) in layer α = A, B . The cavityfields have a simple interpretation as messages between neighboring replica nodes [31]: h αi → j = ˆ h αi → j = 1 means “match me” , h αi → j = ˆ h αi → j = − “do not match me” , and h αi → j = ˆ h αi → j = 0 means “do what you want” . Thezero-temperature BP (or Max-Sum) equations determining the values of these fields in the limit β → ∞ are are givenby h αi → j = − max (cid:20) − , max k ∈ ∂ + i \ j ˆ h αk → i (cid:21) (30a)ˆ h Ai → j = − max (cid:34) max k ∈ ∂ A − i \ j h Ak → i , − max k ∈ ∂ B − j h Bk → i (cid:35) (30b)ˆ h Bi → j = − max (cid:34) max k ∈ ∂ B − i \ j h Bk → i , − max k ∈ ∂ A − j h Ak → i (cid:35) (30c)in which the fields are defined to take values in the discrete set { , , − } and we defined the maximum over a nullset equal to −
1. It follows that for k B,ini = 0 we have ˆ h Ai → j = − k A,ini = 0 we have ˆ h Bi → j = − E can also be expressed in terms of these fields and is given by E = − (cid:88) α N (cid:88) i =1 max (cid:20) − , max k ∈ ∂ α + i ˆ h αk → i (cid:21) + (cid:88) α (cid:88) max (cid:104) , h αi → j + ˆ h αj → i (cid:105) − (cid:88) i =1 ,N max (cid:34) , max k ∈ ∂ A − i h Ak → i + max k ∈ ∂ B − i h Bk → i (cid:35) , (31)where < i, j > indicates pair of nodes that are nearest neighbors in the network and where we take the maximumover a null set equal to -1.3 D. Controllability of uncorrelated multiplex networks with given in-degree and out-degree distribution
1. Cavity equations for an uncorrelated multiplex network ensemble
Let us consider the case of uncorrelated duplex networks in which the degree of the same node in different layersare uncorrelated and there is no overlap of the links. In each layer α = A, B we consider a maximally randomnetwork with in-degree distribution P α,in ( k ) and out-degree distribution P α,out ( k ). At the ensemble level, each linkof (the infinitely large) random network forming layer α has the same statistical properties, that we describe throughdistributions P α ( h α ) and ˆ P α (ˆ h α ) of cavity fields that are defined on the support of Eqs.30, i.e. P α ( h α ) = w α δ ( h α −
1) + w α δ ( h α + 1) + w α δ ( h α ) , ˆ P α (ˆ h α ) = ˆ w α δ (ˆ h α −
1) + ˆ w α δ (ˆ h α + 1) + ˆ w α δ (ˆ h α ) , (32)where α = A, B and where the probabilities w α , w α , w α are normalized w α + w α + w α = 1 as well as the probabilitiesˆ w α , ˆ w α , ˆ w α that satisfy the equation ˆ w α + ˆ w α + ˆ w α = 1. The cavity method at the network ensemble level is alsoknown as density evolution method [36].It is useful to introduce the generating functions G α,in/out ( z ) , and G α,in/out ( z ) of the multiplex network as G α,in ( z ) = (cid:88) k P α,in ( k ) z k ,G α,in ( z ) = (cid:88) k k (cid:104) k α (cid:105) P α,in ( k ) z k − ,G α,out ( z ) = (cid:88) k P α,out ( k ) z k ,G α,out ( z ) = (cid:88) k k (cid:104) k α (cid:105) P α,out ( k ) z k − , (33)with α = A, B . In this way, we can derive recursive equations for the probabilities { w αi } i =1 , , and { ˆ w αi } i =1 , , , thatare the analogous of Eqs. 30 for an ensemble of uncorrelated duplex networks w α = G α,out ( ˆ w α ) ,w α = (cid:2) − G α,out (1 − ˆ w α ) (cid:3) ,w α = 1 − w α − w α , ˆ w α = 1 − ˆ w α − ˆ w α , ˆ w A = G A,in ( w A ) (cid:104) − G B,in (1 − w B ) (cid:105) , ˆ w A = (cid:104) − G A,in (1 − w A ) + G A,in (1 − w A ) G B,in (cid:0) w B (cid:1)(cid:105) , ˆ w B = G B,in ( w B ) (cid:104) − G A,in (1 − w A ) (cid:105) , ˆ w B = (cid:104) − G B,in (1 − w B ) + G B,in (1 − w B ) G A,in (cid:0) w A (cid:1)(cid:105) . (34)The energy E of the matching problem can be also expressed in terms of the { w αi } i =1 , , and { ˆ w αi } i =1 , , giving E = (cid:88) α (cid:8) G α,out ( ˆ w α ) − (cid:2) − G α,out (1 − ˆ w α ) (cid:3)(cid:9) − (cid:110) [1 − G A,in (1 − w A )][1 − G B,in ( w B )]+[1 − G B,in (1 − w B )][1 − G A,in ( w A )] (cid:111) + (cid:88) α (cid:104) k α (cid:105) in [ ˆ w α (1 − w α ) + w α (1 − ˆ w α )] . (35)
2. Stability condition
The Eqs.34 might have multiple solutions. In order to evaluate the stability of these solutions, using a methodalready used in the context of single networks [16, 31] here we compute the Jacobian of the system of Eqs. (34) and4impose that all its eigenvalues have modulus less than one. We avoid to consider w α and ˆ w α because they influenceonly the number of null eigenvalues (4 eigenvalues upon 12). The 12 ×
12 Jacobian matrix becomes 8 × × J = (cid:18) H H , H , H , (cid:19) . with H = G A,out ( ˆ w A )0 0 G A,out (1 − ˆ w A ) 00 G A,in ( w A )(1 − G B,in (1 − w B )) 0 0 G A,in (1 − w A )(1 − G B,in ( w B )) 0 0 0 , (36) H , = G B,in ( w B ) (cid:104) k (cid:105) A,in G A,in (1 − w A ) 0 0 00 G B,in (1 − w B ) (cid:104) k (cid:105) A,in G A,in ( w A ) 0 0 , (37) H , = G A,in ( w A ) (cid:104) k (cid:105) B,in G B,in (1 − w B ) 0 0 00 G A,in (1 − w A ) (cid:104) k (cid:105) B,in G B,in ( w B ) 0 0 , (38)and H , = G B,out ( ˆ w B )0 0 G B,out (1 − ˆ w B ) 00 G B,in ( w B )(1 − G A,in (1 − w A )) 0 0 G B,in (1 − w B )(1 − G A,in ( w A )) 0 0 0 . (39)Here the generating functions G α,in/out and G α,in/out are given by Eqs. (33) and the generating functions G α,in ( x )and G α,out ( x ) are defined as G α,in ( z ) = (cid:88) k k ( k − (cid:104) k α (cid:105) in P inα ( k ) z k − G α,out ( z ) = (cid:88) k k ( k − (cid:104) k α (cid:105) out P outα ( k ) z k − . (40)Of particular interest is the characterization of the stability of the solution w α = ˆ w α = w α = ˆ w α = 0 and w α = ˆ w α = 1,corresponding to the full controllability of the network, a configuration with E = N D = 0. This solution emerges for P inα (1) = P outα (1) = 0 for α = A, B . Therefore if the minimum in-degree and the minimum out-degree are both greaterthan one, the analysis at the ensemble level is consistent with the full controllability of the network. Nevertheless thissolution might be not stable. By analyzing the Jacobian J for w α = ˆ w α = w α = ˆ w α = 0 and w α = ˆ w α = 1, we candetermine under which condition the full controllability solution is stable. The Jacobian matrix, in this case simplifysignificantly and is given by J = P outA (2) (cid:104) k A (cid:105) out (cid:104) k A ( k A − (cid:105) out (cid:104) k A (cid:105) out (cid:104) k A ( k A − (cid:105) in (cid:104) k A (cid:105) in P outB (2) (cid:104) k B (cid:105) out (cid:104) k B ( k B − (cid:105) out (cid:104) k B (cid:105) out
00 0 0 0 0 0 0 00 0 0 0 (cid:104) k B ( k B − (cid:105) in (cid:104) k B (cid:105) in (41)5Four eigenvalues of J are zero, the other four have degenerate modulus, therefore the stability conditions are2 (cid:10) k A ( k A − (cid:11) in (cid:104) k A (cid:105) in P outA (2) (cid:104) k A (cid:105) out < (cid:10) k B ( k B − (cid:11) in (cid:104) k B (cid:105) in P outB (2) (cid:104) k B (cid:105) out < . (42)When P inA ( k ) = P outA ( k ) = P inB ( k ) = P outB ( k ) = P ( k ) we have just one stability criterion solution and it reads P (2) < (cid:104) k (cid:105) (cid:104) k ( k − (cid:105) (43)We observe here that on the single layers α = A, B the full controllability solution, is instead only stable [16] for2 (cid:104) k α ( k α − (cid:105) in (cid:104) k α (cid:105) in P outα (2) (cid:104) k α (cid:105) out < (cid:104) k α ( k α − (cid:105) out (cid:104) k α (cid:105) out P inα (2) (cid:104) k α (cid:105) in < . (44)It follows that for duplex networks in which both layers have the same in-degree and out-degree distributions, i.e. P α,in ( k ) = P α,out ( k ) the stability of the full controllability solution on single layers is the same as the stability onthe duplex network. Nevertheless, for duplex networks formed by layers in which the in-degree distribution and theout-degree distribution are not the same there can be cases in which for the duplex network the fully controllablesolution is stable while for the single layers it is not stable (See main text for discussion of this phenomenon andsimulation results).
3. Entropy
In order to evaluate the number of maximum matchings, here we evaluate the entropy of the ground state solutionsin the case of uncorrelated layers. The entropy density is given by s = S /N and can be computed by expanding thefree energy at low temperatures f ( β → ∞ ) = e − s /β + O (1 /β ). This involves the study of the evanescent partsof the cavity field. Therefore we assume that the field can be written as h α = 1 + ln ν α β for the peak around h = 1 h α = − ln µ α β for the peak around h = − h α = ln γ α β for the peak around h = 0ˆ h α = 1 + ln ˆ ν α β for the peak around h = 1ˆ h α = − ln ˆ µ α β for the peak around h = − h α = ln ˆ γ α β for the peak around h = 0From the BP equations, and the equations for P ( h α ) and P (ˆ h α ) we can obtain the relation between the probabilitydistributions A α ( ν α ) , A α ( µ α ) , A α ( γ α ), and the distributions ˆ A α (ˆ ν α ) , ˆ A α ( µ α ) , ˆ A α (ˆ γ α ), given by6 A α ( ν ) = ∞ (cid:88) k =0 ( ˆ w α ) k w α ( k + 1) (cid:104) k outα (cid:105) P αout ( k + 1) (cid:90) (cid:34) k (cid:89) i =1 d ˆ µ αi ˆ A α (ˆ µ αi ) (cid:35) δ (cid:32) ν −
11 + (cid:80) ki =1 ˆ µ αi (cid:33) (45) A α ( µ ) = ∞ (cid:88) k =1 w α ∞ (cid:88) m = k ( m + 1) (cid:104) k outα (cid:105) P αout ( m + 1) (cid:18) mk (cid:19) ( ˆ w α ) k (1 − ˆ w α ) m − k × (cid:90) (cid:34) k (cid:89) i =1 d ˆ ν αi ˆ A α (ˆ ν αi ) (cid:35) δ (cid:32) µ − (cid:80) ki =1 ˆ ν αi (cid:33) (46) A α ( γ ) = ∞ (cid:88) k =1 ∞ (cid:88) m = k w α ( m + 1) (cid:104) k outα (cid:105) P αout ( m + 1) (cid:18) mk (cid:19) ( ˆ w α ) k ( ˆ w α ) m − k × (cid:90) (cid:34) k (cid:89) i =1 d ˆ γ αi ˆ A α (ˆ γ αi ) (cid:35) δ (cid:32) γ − (cid:80) ki =1 ˆ γ αi (cid:33) (47)ˆ A A (ˆ ν ) = ∞ (cid:88) k A =0 ( k A + 1) (cid:104) k inA (cid:105) P Ain ( k A + 1)( w A ) k A ∞ (cid:88) k B =1 ∞ (cid:88) m B = k B P Bin ( m B ) (cid:18) m B k B (cid:19) ( w B ) k B (1 − w B ) m B − k B (cid:90) k A (cid:89) i =1 dµ Ai A A ( µ Ai ) k B (cid:89) i =1 dν Bi A B ( ν Bi ) × δ ˆ ν A − (cid:80) kBi =1 ν Bi + (cid:80) k A i =1 µ Ai (48)ˆ A A (ˆ µ ) = 1ˆ w A G A,in (1 − w A ) (cid:88) k B P Bin ( k B )( w B ) k B (cid:90) k B (cid:89) i =1 dµ Bi A B ( µ Bi ) δ ˆ µ A − k B (cid:88) i =1 µ Bi + 1ˆ w A (1 − G B,in ( w B )) ∞ (cid:88) k A =1 ∞ (cid:88) m A = k A ( m + 1) (cid:104) k inA (cid:105) P Ain ( m A + 1) (cid:18) m A k A (cid:19) ( w A ) k (1 − w A ) m A − k A × (cid:90) k A (cid:89) i =1 dν Ai A ( ν Ai ) δ (cid:32) ˆ µ A − (cid:80) k A i =1 ν Ai (cid:33) + 1ˆ w A ∞ (cid:88) k A =1 ∞ (cid:88) m A = k A ( m + 1) (cid:104) k inA (cid:105) P Ain ( m A + 1) (cid:18) m A k A (cid:19) ( w A ) k (1 − w A ) m A − k A (cid:88) k B P Bin ( k B )( w B ) k B × (cid:90) k A (cid:89) i =1 dν Ai A A ( ν Ai ) k B (cid:89) i =1 dµ Bi A B ( µ Bi ) δ ˆ µ A − (cid:80) kBi =1 µ Bi + (cid:80) k A i =1 ν Ai (49)7ˆ A A (ˆ γ ) = 1ˆ w A G A,in ( w A ) ∞ (cid:88) k B =1 ∞ (cid:88) m B = k B P Bin ( k B ) (cid:18) m B k B (cid:19) ( w B ) k ( w B ) m B − k B × (cid:90) k B (cid:89) i =1 dγ Bi A B ( γ Bi ) δ (cid:32) γ A − k B (cid:88) i =1 γ Bi (cid:33) + 1ˆ w A ∞ (cid:88) k A =1 ∞ (cid:88) m A = k A ( m A + 1) (cid:104) k inA (cid:105) P Ain ( m A + 1) (cid:18) m A k A (cid:19) ( w A ) k ( w A ) m A − k A × ∞ (cid:88) k B =1 ∞ (cid:88) m B = k B P Bin ( k B ) (cid:18) m B k B (cid:19) ( w B ) k ( w B ) m B − k B (cid:90) k A (cid:89) i =1 dγ Ai A A (ˆ γ Ai ) k B (cid:89) i =1 dγ Bi A B ( γ Bi ) × δ γ A − (cid:80) kBi =1 γ Bi + (cid:80) k A i =1 γ Ai + 1ˆ w A (1 − G B,in (1 − w B )) ∞ (cid:88) k A =1 ∞ (cid:88) m A = k A ( m A + 1) (cid:104) k inA (cid:105) P Ain ( m A + 1) (cid:18) m A k A (cid:19) ( w A ) k ( w A ) m A − k A × (cid:90) k A (cid:89) i =1 dγ Ai A A (ˆ γ Ai ) δ (cid:32) γ A − (cid:80) k A i =1 γ Ai (cid:33) (50)The free energy density f ( β ) = F ( β ) /N = e − s β + O (1 /β ) with s = s ,a,A + s ,a,B + s ,b + s ,c,A + s ,c,B (51)where s ,(cid:96) are given by s ,a,α = ∞ (cid:88) k =1 ∞ (cid:88) m = k (cid:18) mk (cid:19) ( ˆ w α ) k (1 − ( ˆ w α )) m − k P αout ( m )ln k (cid:88) i =1 ˆ ν i + (cid:88) k P αout ( k )( ˆ w α ) k ln (cid:32) k (cid:88) i =1 ˆ µ i (cid:33) + ∞ (cid:88) k =1 ∞ (cid:88) m = k (cid:18) mk (cid:19) ( ˆ w α ) k ( ˆ w α ) m − k P αout ( m )ln (cid:32) k (cid:88) i =1 ˆ γ i (cid:33) (52)8 s ,b = (1 − G B,in ( w B )) ∞ (cid:88) k A =1 ∞ (cid:88) m A = k A (cid:18) m A k A (cid:19) ( w A ) k A (1 − ( w A )) m A − k A P Ain ( m A )ln k A (cid:88) i =1 ν Ai + (1 − G A,in ( w A )) ∞ (cid:88) k B =1 ∞ (cid:88) m B = k B (cid:18) m B k B (cid:19) ( w B ) k B (1 − ( w B )) m B − k B P Bin ( m B )ln k B (cid:88) i =1 ν Bi + (1 − G B,in (1 − w B )) ∞ (cid:88) k A =1 ∞ (cid:88) m A = k A (cid:18) m A k A (cid:19) ( w A ) k A ( w A ) m A − k A P Ain ( m A )ln k A (cid:88) i =1 γ Ai + (1 − G A,in (1 − w A )) ∞ (cid:88) k B =1 ∞ (cid:88) m B = k B (cid:18) m B k B (cid:19) ( w B ) k B ( w B ) m B − k B P Bin ( m B )ln k B (cid:88) i =1 γ Bi + (cid:34)(cid:88) k B P Bin ( k B )( w B ) k B ∞ (cid:88) k A =1 ∞ (cid:88) m A = k A (cid:18) m A k A (cid:19) ( w A ) k A (1 − ( w A )) m A − k A P Ain ( m A ) × ln (cid:32) k A (cid:88) i =1 ν Ai k B (cid:88) i =1 µ Bi (cid:33) + (cid:34)(cid:88) k A P Ain ( k A )( w A ) k A ∞ (cid:88) k B =1 ∞ (cid:88) m B = k B (cid:18) m B k B (cid:19) ( w B ) m B (1 − ( w B )) m B − k B P Bin ( m B ) × ln (cid:32) k B (cid:88) i =1 ν Bi k A (cid:88) i =1 µ Ai (cid:33) + (cid:34) ∞ (cid:88) k A =1 ∞ (cid:88) m A = k A (cid:18) m A k A (cid:19) ( w A ) k A ( w A ) m A − k A P Ain ( m A ) × ∞ (cid:88) k B =1 ∞ (cid:88) m B = k B (cid:18) m B k B (cid:19) ( w B ) k B ( w B ) m B − k B P Bin ( m B )ln (cid:32) k A (cid:88) i =1 γ Ai k B (cid:88) i =1 γ Bi (cid:33) (53) s ,c,α = −(cid:104) k α (cid:105) in (cid:8) ˆ w α ( w α + w α )ln ˆ ν α + w α ( ˆ w α + ˆ w α )ln ν α + ˆ w α w α ln(1 + ˆ νµ ) + w α ˆ w α ln(1 + ν ˆ µ )+ ˆ w α w α ln γ + w α ˆ w α ln ˆ γ + w α ˆ w α ln(1 + ˆ γγ ) (cid:111) . (54)
4. Phase transition in the controllability of Poisson duplex networks.
Here we consider the case of two Poisson networks with the same in/out average degree. In other words, we considerthe situation in which (cid:10) k A,in (cid:11) = (cid:10) k A,out (cid:11) = (cid:10) k B,in (cid:11) = (cid:10) k B,out (cid:11) = c . The fraction n D of nodes that are driver nodesof this duplex, is always larger than the double of the fraction of driver nodes in each of the layers taken in isolation(see Fig. 6). Moreover we observe that there is a phase transition in the controllability of these duplex networks,indicated by a discontinuity of n D for c = c (cid:63) = 3 . . . . (see Fig. 6). In order to derive these results, we assumed w Ai = w Bi for i = 1 , , w Ai = ˆ w Bi for i = 1 , ,
3. Therefore the zero-temperature BP equations at the ensemble9 c " D single network n D duplex FIG. 6: Density of driver nodes ε = n D for a duplex network composed by two Poisson networks with (cid:10) k A,in (cid:11) = (cid:10) k A,out (cid:11) = (cid:10) k B,in (cid:11) = (cid:10) k B,out (cid:11) = c is indicated with a solid red line and it clearly shows a phase transition for c = 3 . . . . . In thedashed blue line we display the double of the number of driver nodes ε = 2 n D for a single Poisson network with the sameaverage degree c , indicating the fraction of driver nodes necessary to control separately the two layers. level (34) read, w = e − c (1 − ˆ w ) ,w = (cid:2) − e − c ˆ w (cid:3) ,w = 1 − w − w , ˆ w = 1 − ˆ w − ˆ w , ˆ w = e − c (1 − w ) (cid:2) − e − cw (cid:3) , ˆ w = (cid:104) − e − cw + e − cw e − c (1 − w ) (cid:105) . (55)The energy E is given in this case by E = 2 (cid:104) e − c (1 − ˆ w ) − e − c ˆ w (cid:105) − − e − cw ][1 − e − c (1 − w ) ] + 2 c [ ˆ w (1 − w ) + w (1 − ˆ w )] . (56)We notice that the equations for ˆ w and ˆ w can be rewritten to form a closed subsystem of equations,ˆ w = h ( ˆ w , ˆ w ) = e − ce − c ˆ w (cid:104) − e − ce − c (1 − ˆ w (cid:105) (57a)ˆ w = h ( ˆ w , ˆ w ) = (cid:104) − e − ce − c (1 − ˆ w + e − ce − c (1 − ˆ w e − ce − c ˆ w (cid:105) (57b)0 w w c = w w c = w w c = FIG. 7: Plots of the functions ˆ w = h ( ˆ w , ˆ w ) and ˆ w = h ( ˆ w , ˆ w ) given by Eqs. (57 a ) − (57 b ). The solution of the systemof these two equations, corresponds to a crossing of the two curves. We show the emergence of two new solutions of this systemof equations for c > c (cid:63) = 3 . . . . . The critical point c (cid:63) characterize an hybrid phase transition in the controllability of theduplex network. c w A w w w c ^ w A ^ w ^ w ^ w c w B w w w c ^ w B ^ w ^ w ^ w FIG. 8: Values of the probabilities { w i } i =1 , , and ˆ w i =1 , , plotted as a function of the average degree c , for a duplex networkformed by two Poisson layers with (cid:10) k A,in (cid:11) = (cid:10) k A,out (cid:11) = (cid:10) k B,in (cid:11) = (cid:10) k B,out (cid:11) = c . These probabilities are calculated directlyfrom BP results obtained over 5 single realizations these multiplex networks with average degree c and N = 10 . c (cid:63) of the average degree c at which the discontinuity in the number of driver nodes n D observed in Fig.6occurs can be found by imposing that the two curves ˆ w = h ( ˆ w , ˆ w ) and ˆ w = h ( ˆ w , ˆ w ) of the plane w , w for c = c (cid:63) are tangent to each other at their interception. These functions are plotted in Figure 7 where it is possible toobserve that for c > c (cid:63) the curves cross in three points while for c < c (cid:63) they cross in one point, and at c = c (cid:63) theyare tangent to each other. The critical point c (cid:63) is found by imposing that the Eqs. (57) are satisfied together withthe condition | J | = 0 , (58)with J indicating the Jacobian of the system of equations ˆ w = h ( ˆ w , ˆ w ) and ˆ w = h ( ˆ w , ˆ w ) given by J = (cid:32) − ∂h ( ˆ w , ˆ w ) ∂ ˆ w − ∂h ( ˆ w , ˆ w ) ∂ ˆ w − ∂h ( ˆ w , ˆ w ) ∂ ˆ w − ∂h ( ˆ w , ˆ w ) ∂ ˆ w (cid:33) . Imposing that Eqs. (57) and condition (58) are simultaneously satisfied, the solution c (cid:63) = 3 . . . . is found.For c < c (cid:63) we observe that w = ˆ w = 0. At c (cid:63) we observe a discontinuity in both w and ˆ w , but for c > c (cid:63) thefunctions h ( ˆ w , ˆ w ) and h ( ˆ w , ˆ w ) are analytic, and analyzing Eqs. (57 a ) − (57 b ) we obtain the behavior of theorder parameters w and ˆ w for c > c (cid:63) w − w (cid:63) ∝ ( c − c (cid:63) ) / ˆ w − ˆ w (cid:63) ∝ ( c − c (cid:63) ) / , (59)showing that the transition is hybrid.We further characterize this phase transition evaluating the number of maximum matchings, i.e. the entropy valueof the ground state solutions in the case of two poisson uncorrelated layers. The entropy density s follows from Eq.(51) and it is plotted as a function of the average degree c in Fig. 9. The entropy density presents a small jump at c ∗ = 3 . .. marking a change in the properties of the solutions.Here we want to modify the degree distribution of the duplex network characterized in this section, by changing theprobability of nodes of low degree (degree 0 , ,
2) that have been shown to be essential to determine the controllabilityof single layers [16]. Therefore we consider a duplex networks with degree distributions P inA ( k ) = P outA ( k ) = P inB ( k ) = P outB ( k ) = P ( k ) and with minimum degree is 2. In particular we consider P ( k ) given by P ( k ) = k < P (2) for k = 2 κ k ! c k for k ∈ [3 , ∞ ] (60)with κ indicating a normalization constant. In Fig. 10 (on the left) we show the phase diagram of this duplex networkdescribed by the dependence of the fraction of driver nodes n D on c and P (2). The dark grey area defines the regionwhere the zero-energy solution is stable, hence in which to control a duplex one needs only an infinitesimal fractionof driver nodes, i.e. n D = 0. These results are compared with the situation in which the two layers are controlledseparately shown in Fig. 10 (on the right). The fraction of driver replica nodes of the duplex network is always largerthat the double of the fraction of driver nodes in any single layer taken in isolation. Moreover the region in which thefully controllable solution is stable is the same for the duplex network, and for the single networks in the layers of theduplex network taken in isolation. This result is consistent with the theoretical expectations obtained in Sec. D 2. Infact the in- and out-degree distributions of the two layers are the same.
5. Controllability of scale-free duplex networks
Following Sec. D 4 we consider now the case of two uncorrelated layers composed by two power-law networks with P ( k A,in ) = P ( k A,out ) = P ( k B,in ) = P ( k B,out ) = P ( k ) ∝ k − γ and minimal degree m = 1. Similarly to the poissoncase, the fraction n D of driver nodes of this duplex, is always larger than the double of the fraction of driver nodes ineach of the layers taken in isolation (see Fig. 11).Moreover, low-degree nodes significantly affect the controllability of duplex networks formed by scale-free networks.We consider a duplex network with P inA ( k ) = P outA ( k ) = P inB ( k ) = P outB ( k ) = P ( k ) and P ( k ) given by P ( k ) = k = 1 P (2) if k = 2 κk − γ if k ∈ [3 , M ] (61)2 c s c ∗ FIG. 9: Entropy density s for a duplex network composed by two Poisson networks with (cid:10) k A,in (cid:11) = (cid:10) k A,out (cid:11) = (cid:10) k B,in (cid:11) = (cid:10) k B,out (cid:11) = c . At c ∗ = 3 . .. , average degree corresponding to the hybrid phase transition, the entropy density s displays afinite jump. with κ indicating the normalization sum and γ >
2. We consider uncorrelated networks, therefore the cutoff M onthe degrees of the nodes will be given by M = min( √ N , { [1 − P (1) − P (2)] N } / ( γ − ) . (62)In other words, the cutoff M is given by the minimum between the structural cutoff of the network and the naturalcutoff of the degree distribution. In Fig. 12 (on the left) we present the phase diagram of a duplex network displayingthe fraction of driver nodes n D as a function of the parameters γ and P (2). The dark grey area is associatedwith the stable zero-energy solution while outside this region, the minimum fraction of driver nodes necessary for afull duplex control follows the colorcode. We compare these results with the situation in which the two layers arecontrolled separately (on the right). We observe that the number of driver replica nodes in the duplex is alwaysgreater than the total number of driver nodes of the single layer taken in isolation, provided that the duplex networkis not fully controllable. We note that for the degree distribution considered in this case, consistently with thetheoretical results obtained in Sec. D
2, we observe that the region for the stability of the full controllability solutionfor the duplex network is the same of the region for the stability of the full controllability solution in the singlelayers. Finally , in Fig. 13 we compare our theoretical results for the ensemble of duplex networks with degreedistributions P inA ( k ) = P outA ( k ) = P inB ( k ) = P outB ( k ) = P ( k ) and P ( k ) given by Eq. (61), with those obtained by themessage-passing (BP) algorithm, finding a good agreement (Eq. 43 returns a limit value for P (2) equal to 0 . c P ( ) n D duplex c P ( ) D single network FIG. 10: On the left: the density of driver nodes n D as a function of the parameters c and P (2) is plotted for duplex networkswith P inA ( k ) = P outA ( k ) = P inB ( k ) = P outB ( k ) = P ( k ) and P ( k ) given by Eq. (60). On the right: the double of the density ofdriver nodes n D for single layers with degree distribution P in ( k ) = P out ( k ) = P ( k ) and P ( k ) given by (60) is plotted as afunction of c and P (2). E. Effect of degree correlations on controllability of multiplex networks
In order to analyze the effect of degree correlations [1] on the controllability of multiplex networks, we correlate thedegree of the replica nodes in the two layers of a duplex network formed by layer A and layer B . In particular weconsider two cases: a duplex network in which only the low in-degree nodes (nodes of in-degree 0 , , P in ( k A , k B ) between layers and the corresponding expression ofthe zero-temperature BP equations in the correlated ensemble of networks.In the first case we consider a joint in-degree distribution P in ( k A , k B ) given by P in ( k A , k B ) = pδ k B ,k A P ( k A ) + (1 − p ) P ( k A ) P ( k B ) , for k A ≤ − p ) P ( k A ) P ( k B ) , for k A > k B ≤ p P ( k B ) C P ( k A ) + (1 − p ) P ( k A ) P ( k B ) , for k A > k B > , where C = 1 − (cid:80) k ≤ P ( k ) where P ( k ) is a given normalized degree distribution. The distributions of the fields overthe links of this ensemble of networks are given by P α ( h α ) = w α δ ( h α −
1) + w α δ ( h α + 1) + w α δ ( h α ) , ˆ P α (ˆ h α ) = ˆ w α δ (ˆ h α −
1) + ˆ w α δ (ˆ h α + 1) + ˆ w α δ (ˆ h α ) , (63)where α = A, B and where the probabilities w α , w α , w α are normalized w α + w α + w α = 1 as well as the probabilitiesˆ w α , ˆ w α , ˆ w α that satisfy the equation ˆ w α + ˆ w α + ˆ w α = 1. The zero-temperature BP (Max-Sum) equations (30) averaged4 γ ε D single networkn D duplex FIG. 11: Density of driver nodes ε = n D for a duplex network composed by two power-law networks with P ( k A,in ) = P ( k A,out ) = P ( k B,in ) = P ( k B,out ) = P ( k ) ∝ k − γ and minimal degree m = 1 as a function of γ (indicated with a solid redline). The minimum in/out degree 1 and the maximum in/out degree is given by the structural cutoff with N = 10 . In thedashed blue line we display the double of the number of driver nodes ε = 2 n D for a single power-law network with the samein/out degree distributions P ( k ), indicating the fraction of driver nodes necessary to control separately the two layers. over this ensemble of networks can be expressed in terms of the probabilities { w αi } i =1 , , and { w αi } i =1 , , asˆ w = p (cid:20) P (1) (cid:104) k (cid:105) w + 2 P (2) (cid:104) k (cid:105) w (1 − (1 − w ) ) + ( G ( w ) − P (1) (cid:104) k (cid:105) − P (2) (cid:104) k (cid:105) w )(1 − ˜ G (1 − w )) (cid:21) +(1 − p ) G ( w ) [1 − G (1 − w )] (64a)ˆ w = p (cid:20) P (1) (cid:104) k (cid:105) w + 2 P (2) (cid:104) k (cid:105) ( w + w (1 − w − P (1) (cid:104) k (cid:105) − P (2) (cid:104) k (cid:105)− ( G (1 − w ) − P (1) (cid:104) k (cid:105) − P (2) (cid:104) k (cid:105) (1 − w ))(1 − ˜ G ( w )) (cid:21) +(1 − p ) [1 − G (1 − w ) + G (1 − w ) G ( w )] (64b)5 γ P ( ) n D duplex γ P ( ) D single network FIG. 12: On the left: the density of driver nodes n D as a function of the parameters γ and P (2) for duplex networks of N = 10 nodes with degree distributions P inA ( k ) = P outA ( k ) = P inB ( k ) = P outB ( k ) = P ( k ) and P ( k ) given by Eq. (61). On the right:double of the density of driver nodes n D as a function of the parameters γ and P (2) for single networks of N = 10 nodes withdegree distributions P in ( k ) = P out ( k ) = P ( k ) and P ( k ) given by Eq. (61). where G ( z ) = (cid:88) k P ( k ) z k G ( z ) = (cid:88) k k (cid:104) k (cid:105) P ( k ) z k ˜ G ( z ) = (cid:88) k ≥ P ( k ) C z k . (65)Finally the energy E is given by E = 2 { G ( ˆ w ) − [1 − G (1 − ˆ w )] } + 2 (cid:104) k (cid:105) [ ˆ w (1 − w ) + w (1 − ˆ w )] − − p ) { [1 − G (1 − w )][1 − G ( w )] }− p (cid:110) P (1) w (1 − w ) + P (2)(1 − (1 − w ) )(1 − w ) + C (1 − ˜ G (1 − w ))(1 − ˜ G ( w )) (cid:111) (66)In the second case we consider the joint degree distribution P in ( k A , k B ) given by P in ( k A , k B ) = pδ k B ,k A P ( k A ) + (1 − p ) P ( k A ) P ( k B ) , (67)where P ( k ) is a given normalized degree distribution. The distributions of the fields over the links of this ensembleof duplex networks are given by P α ( h α ) = w α δ ( h α −
1) + w α δ ( h α + 1) + w α δ ( h α ) , ˆ P α (ˆ h α ) = ˆ w α δ (ˆ h α −
1) + ˆ w α δ (ˆ h α + 1) + ˆ w α δ (ˆ h α ) , (68)where α = A, B and where the probabilities w α , w α , w α are normalized w α + w α + w α = 1 as well as the probabilitiesˆ w α , ˆ w α , ˆ w α that satisfy the equation ˆ w α + ˆ w α + ˆ w α = 1. We get the equationsˆ w = p [ G ( w ) − (1 − w ) G ( w (1 − w ))] + (1 − p ) G ( w ) [1 − G (1 − w )] (69a)ˆ w = p [1 − G (1 − w ) + w G ( w (1 − w ))] + (1 − p ) [1 − G (1 − w ) + G (1 − w ) G ( w )] , (69b)6 n D theoryBP FIG. 13: Density of driver nodes n D as a function of P (2) for a duplex network with P inA ( k ) = P outA ( k ) = P inB ( k ) = P outB ( k ) = P ( k ), P ( k ) given by Eq. (61) and γ = 2 .
3. The fraction of driver nodes computed with the zero-temperature BP (Max-Sum)algorithm on a duplex network of N = 10 nodes (averaged over 25 network realizations) is compared with the theoreticalexpectation for the density n D in an ensemble of random duplex networks with the given degree distributions. where the generating functions G ( z ) and G ( z ) are defined as G ( z ) = (cid:88) k P ( k ) z k ,G ( z ) = (cid:88) k k (cid:104) k (cid:105) P ( k ) z k . (70)The energy E in this ensemble is given by E = 2 { G ( ˆ w ) − [1 − G (1 − ˆ w )] } + 2 (cid:104) k (cid:105) [ ˆ w (1 − w ) + w (1 − ˆ w )] − − p ) { [1 − G (1 − w )][1 − G ( w )] } − p { − G (1 − w ) − G ( w ) + G ( w (1 − w )) } (71)The degree correlation of low in-degree nodes can modify the number of driver nodes n D found in duplex networks (seeFig. 14 for the case of a duplex network formed by Poisson layers with (cid:10) k A,in (cid:11) = (cid:10) k A,out (cid:11) = (cid:10) k B,in (cid:11) = (cid:10) k B,out (cid:11) = c ).Once the low in-degree nodes are correlated, correlating also the other in-degrees of the network does not changesubstantially the number of driver nodes n D as discussed in the main body of the paper. [1] Boccaletti, S. et al. The structure and dynamics of multilayer networks.
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