Control of Multilayer Networks

The controllability of a network is a theoretical problem of relevance in a variety of contexts ranging from financial markets to the brain. Until now, network controllability has been characterized only on isolated networks, while the vast majority of complex systems are formed by multilayer networks. Here we build a theoretical framework for the linear controllability of multilayer networks by mapping the problem into a combinatorial matching problem. We found that correlating the external signals in the different layers can significantly reduce the multiplex network robustness to node removal, as it can be seen in conjunction with a hybrid phase transition occurring in interacting Poisson networks. Moreover we observe that multilayer networks can stabilize the fully controllable multiplex network configuration that can be stable also when the full controllability of the single network is not stable.

The controllability of a network is a theoretical problem of relevance in a variety of contexts ranging from financial markets to the brain.Until now, network controllability has been characterized only on isolated networks, while the vast majority of complex systems are formed by multilayer networks.Here we build a theoretical framework for the linear controllability of multilayer networks by mapping the problem into a combinatorial matching problem.We found that correlating the external signals in the different layers can significantly reduce the multiplex network robustness to node removal, as it can be seen in conjunction with a hybrid phase transition occurring in interacting Poisson networks.Moreover we observe that multilayer networks can stabilize the fully controllable multiplex network configuration that can be stable also when 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].For example, banks are linked to each other by different types of contracts and relationships, gene regulation in the cell is mediated by the different types of interactions between different kinds of molecules, brain data are described by multilayer brain networks.Studying the controllability properties of these networks is important for assessing the risk of a financial crash [3], for drug discovery [9] and for characterizing brain dynamics [4][5][6][7][8].Therefore the controllability of multilayer networks is a problem of fundamental importance for a large variety of applications.
It has been recently found that the multiplexity of networks can have profound effects on the dynamical processes taking place on them [30][31][32][33][34][35].For example, percolation processes that usually present continuous phase transitions on single networks can become discontinuous on such structures [30][31][32][33][34] and are characterized by large avalanches of disruption events.
In this Letter, we consider the elegant framework of structural controllability [11] and investigate how the multilayer structure of networks can affect their controllability.We focus on multiplex networks, which are multilayer networks in which the same set of nodes are connected by different types of interactions.Multiplex network controllability is studied under the assumption that input nodes are the same in all network layers, thus mimicking the situation in which input nodes can send different signals in the different layers of the multiplex but the position of the external signals in the layers is correlated.
We show that controlling the dynamics of multiplex networks is more costly than controlling single layers taken in isolation.Moreover, the controllability of multiplex networks displays unexpected new phenomena.In fact these networks can become extremely sensible to damage in conjunction with a discontinuous phase transition characterized by a jump in the number of input points (driver nodes).A careful investigation of this phase transition reveals that this is a hybrid phase transition with a square root singularity, therefore in the same universality class of the emergence of the mutually connected component in multiplex networks [1,30,32].The number of driver nodes in the multiplex network is in general higher than the number of driver nodes in the single layers taken in isolation.Nevertheless the degree correlations between low-degree nodes in the different layers can affect the controllability of the multiplex network and modulate the number of its driver nodes.Moreover, multilayer network can change the stability of the fully controllable configuration that can be stable in a multilayer network also if it is not stable in the isolated networks that form the multilayer structure.
Structural controllability of multiplex networks -We consider a multiplex networks [1] in which every node i = 1, 2, . . ., N has a replica node (i, α) in each layer α and every layer is formed by a directed network between the corresponding replica nodes.We assume that each replica node (i, α) is characterized by a different dynamical variable x α i ∈ R and that each layer is characterized by a possibly different dynamical process.We consider for simplicity a duplex, i.e a multiplex formed by two layers {A, B} where each layer α ∈ {A, B} is a directed network.The state of the network at time t is governed by a linear dynamical system in which the 2N -dimensional vector X(t) describes the dynamical state of each replica node, i.e.X i = x A i and X N +i = x B i for i = 1, 2, . . ., N .The matrix G is a 2N × 2N (asymmetric) matrix and K is a 2N × M matrix.They have the following block structure in which g α are the N × N matrices describing the directed weighted interactions within the layers and K α are the N ×M α matrices describing the coupling between the nodes of each layer α and M α ≤ N external signals.The latter are represented by a vector u(t) of elements u γ and γ = 1, 2 . . .M = M A + M B .Here we consider the concept of structural controllability [10,11] that guarantees the controllability of a networks for any choice of the non-zeros entries of G and K, except for a variety of zero Lebesgue measure in the parameter space.Therefore each layer of the duplex networks can be structurally controlled by identifying a minimum number of driver nodes, that are controlled nodes which do not share input vertices.If different replicas of the same node can be independently controlled, then the controllability properties of the multiplex network factorize and each layer can be studied as if was taken in isolation [10,11,15,16].Liu et al. [11] showed that in a single layer the minimum set of driver nodes can be found by mapping the problem into a matching problem.In fact the Minimum Input Theorem states that the minimum set of driver nodes that guarantees the full structural controllability of a single network is the set of unmatched nodes in a maximum matching of the same directed network.In real multiplex networks however nodes are usually univocally defined and share common properties across different layers, therefore we make the assumption that each node of the duplex network is either a driver node in each layer or it is not a driver node in any layer.The problem of finding the driver nodes of the duplex network can be thus mapped into a maximum matching problem in which every node has at most one matched incoming link and at most one matched outgoing link, with the constraint that two replica nodes either have no matched incoming links on each layer or have one matched incoming link in each layer (see Figure 1).This problem can be studied, both on ensemble of random graphs and on single network realizations which are locally tree-like, using statistical mechanics techniques, such as the cavity method and the Belief Propagation (BP) algorithm, that provide an estimate of the minimal number of driver nodes.Following [11,15], we consider the variables s α ij = 1, 0 indicating respectively if the directed link from node (i, α) to node (j, α) in layer α = A, B is matched or not.In order to have a matching in each layer of the duplex the following constraints have always to be satisfied where ∂ α + i is the set of replica nodes (j, α) in layer α that are reached by directed links from (i, α).In addition, we impose that the driver nodes in the two layers (the unmatched nodes) are replica nodes, i.e.
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 The BP equations -The BP equations of this problem are derived using the cavity method [24,27,29] in the zero-temperature limit, as described for the case of a single network problem in [11,15,25,26,28].The same approximation methods can be applied to study the maximum matching problem on directed duplex networks, as long as the structure of the interconnected layers is locally tree-like both within the layers and across them.Under the decorrelation (replica-symmetric) assumption, the energy of a maximum matching can be written in terms of cavity fields (or messages) {h α i→j } and { ĥα i→j }, defined on the directed links between neighboring nodes (i, α) and (j, α) in the same layer α = A, B and satisfying the zero-temperature limit of the BP equations, also known as Max-Sum equations, in which the fields are defined to take values in the discrete set {−1, 0, 1} and here and in the following we use the convention that the maximum over a null set is equal to −1 (see Supplemental Material [36] for details).In terms of these fields, the energy E in Eq.( 5) becomes Phase transition in Poisson duplex networks-We consider duplex networks in which the two layers are realizations of uncorrelated directed random graphs characterized by Poisson distributions for in-degree and outdegree with same average value c, i.e.
In the infinite size limit (N → ∞), the average quantities computed solving Eqs.(6) on single instances correspond to the average quantities that can be estimated at the ensemble level using the density evolution technique and discussed in the SM [36].In Figure S-3A we report the average rescaled number of driver nodes n D as function of the average degree c computed from the solutions of Eqs.(6) on single instances (blue points) and from the graphs ensemble analysis (red solid line).A comparison with two independent layers with the same topological properties shows that the controllability of a duplex network is in general more demanding in terms of number of driver nodes than the controllability of independent single layers, in particular for low average degrees.In addition, a discontinuity in the number of driver nodes at c = c = 3.2223 . . .marks a change in the controllability properties of duplex networks that is 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 (presented in the SM [36]) reveals that this is a hybrid phase transition with ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲▲ ▲▲▲ ▲▲▲ ▲▲▲▲▲ ▲▲ • BP ld p=0.5 ■ BP ld p=1 ▲ BP td p=1 a square root singularity, therefore in the same universality class of the emergence of the mutually connected component in multiplex networks [1,30,32].In correspondence to this phase transition the network responds non trivially to perturbations.This is observed by performing a numerical calculation of the robustness of the networks.Following [11] we classify the nodes into three 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 nodes decreases or is unchanged, the removed nodes are classified as redundant and ordinary respectively.Figure S-3B shows that the fraction of critical nodes reaches a maximum at the transition, revealing an increased fragility of the duplex network to random damage with respect to single layers.While an abrupt change in the number of driver nodes can result from a small change in the network topology, it is important to stress that the non-monotonic behavior of these quantities 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.k α in = k α out = c α ) a similar discontinuous phase transition is observed (see Figure S-2).Nevertheless we checked that this discontinuous phase transition is not occurring for every multiplex network structure.In particular we have checked that there is no discontinuous phase transition for multiplex networks formed by scale-free networks with the same degree distribution in both layers (see SM [36]).
Effect of degree correlations on the controllability of duplex networks -We consider a model of duplex network in which the replica nodes of the directed random graphs 3, the network have minimum in-degree equal to 2 and minimum out-degree equal to 3 and P A,in (2 in the two layers have correlated degrees.In particular, we consider a case in which only the low in-degree nodes (nodes with in-degree equal to 0, 1, 2) are correlated (replica nodes in different layers have same degree with probability p) and a case in which the in-degrees of the replica nodes are equal with probability p independently of their value (see SM [36] for details).The controllability of the network is affected by these correlations as shown in Figure 4.In fact, the number of driver nodes n D decreases as the level of correlation increases.In duplex networks with Poisson degree distribution, lowdegree correlations modify both the position of the hybrid transition and the size of the discontinuity.Once the replica nodes with low in-degree are correlated, a further correlation of the remaining replica nodes does not substantially change the number of driver nodes.This result confirms that structural controllability is essentially determined by the control of low degree nodes [15].
Stability of the fully controllable solution -A fully controllable solution, in which a single driver node is necessary to control the whole duplex network, exists if the minimum in-degree and the minimum out-degree are both greater than 1 in both layers.In random duplex networks with the same degree distribution in the two layers, the fully controllable solution is stable if and only if for α = A, B. On single networks it was instead recently found [15] that the fully controllable configuration is only stable for This implies that for multiplex networks with asymmetric in-degree and out-degree distributions it might occur that the 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 a multiplex structure can help to stabilize the fully controllable solution.
Conclusions -Within the framework of structural controllability, we have considered the controllability properties of multiplex networks in which the nodes are either driver nodes in all the layers or they are not driver nodes in any layer.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, it is possible to observe a hybrid phase transition with a discontinuity in the number of driver nodes as a function of the average degree, that is phenomenologically similar to the emergence of mutually connected components.Close to this phase transition the duplex network exhibits an increased fragility to random damage.The existence of correlations between the degrees of replica nodes in different layers, in particular between low-degree nodes, has the effect of reducing the number of driver nodes necessary to control duplex networks.Finally, multiplex structure of networks can stabilize the fully controllable solution also if this solution is not stable in the single layers that form the multiplex network.

Appendix 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 duplex network.Moreover we define the driver nodes, as the set of nodes that, if stimulated by an external signal, can drive the 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 limit known as Max-Sum equations.In Sec IV we consider the controllability of uncorrelated duplex networks, characterizing the BP equations valid for this problem, the stability conditions for the solutions of the BP equations, and the entropy of the solutions.Moreover, we consider duplex networks formed by two Poisson layers and we characterize their hybrid phase transition.Finally we consider 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 we derive the BP equations assuming either that only the low in-degrees of replica nodes are correlated or that all the in-degrees of replica nodes are correlated.

Appendix B: The structural controllability of a multiplex network
We consider a multiplex network in which every node i = 1, 2, . . ., N has a replica node in each layer and every layer is formed by a directed networks between the corresponding replica nodes [1].We assume that each replica node can have a different dynamical state and can send different signals in the different networks (each layers is characterized by a different dynamical process).In this case the controllability of the multiplex network can be treated by control theory methods used for the single layers taken in isolation [10,11,15,16].Nevertheless here we will consider an additional constraint on the number of driver nodes.In fact we impose that corresponding replica nodes are either driver 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 directed network.We call the two layers layer α = A, B. We consider a linear dynamical system determining the network dynamics in which the vector X(t) describes the dynamical state of each replica node in the duplex, and has 2N elements.The first set of N elements represents the dynamical state x A i of node i in layer A (i.e.X i = x A i for i = 1, 2 . . ., 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 B i for i = 1, 2, . . ., N .The matrix G is a 2N × 2N (asymmetric) matrix and the matrix K is a 2N × M matrix.The matrices G and K have the following block structure where g α with α = A, B are the N × N matrices describing the directed weighted interactions within each of the networks in the two layers and K α are the N × M α matrices describing the interaction between the nodes of the network α and the M α ≤ N external signals for layer α.The external signals are indicated by the vector u(t) of elements u γ and γ = 1, 2 . . .
Given block structure of both matrix G and K described in Eq. (S − 2), the problem of duplex network controllability defined by Eq. (S − 1), can be exactly recast into the problem of controllability of the single layers that form the duplex network.
Here we adopt the framework of structural controllability [10] aimed at characterizing if a given duplex network is controllable when the non-zero matrix elements of G and K given by Eq. (S − 2) are free parameters.A duplex networks in which the linear dynamics described by the Eqs.(S − 1) and (S − 2) 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 of zero Lebesgue measure in the parameter space, the Kalman's condition is fulfilled [10].Since structural controllability only distinguishes between zero and non-zero entries of the matrices g α and K α , a given directed network in layer α is structurally controllable if it is possible to determine the input nodes (i.e. the position of the non-zero entries of the matrix K α ) in a way to control the dynamics described by any realization of the matrix g α with the same non-zero elements, except for atypical realizations of zero measure.In practice, a single network can be structurally controlled by identifying a minimum number of driver nodes, that are controlled nodes which do not share input vertices in both layers.In their seminal paper [11], Liu and coworkers showed that on single networks this control theoretic problem can be reduced to a well-known optimization problem: their Minimum Input Theorem states that the minimum set of driver nodes that guarantees the full structural controllability of a network is the set of unmatched nodes in a maximum matching of the same directed network.Their result for a single network remains valid for the duplex network described by Eqs.(S − 1) − (S − 2).Therefore the structural controllability of duplex networks, in the absence of further constraints can be mapped to a Maximum Matching problem defined on the single layers of the duplex networks.Here nevertheless, we consider a further constraint to be imposed on the driver nodes, which enforce a new 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 two replica nodes are either both linked to independent and external signals or none of them is connected to external signals.In order to build an algorithm able to find the driver nodes of a duplex network we consider the variables s α ij = 1, 0 indicating respectively if the directed link from node (i, α) to node (j, α) in layer α = A, B is matched or not.In the two layers of the duplex network we want to have a matching, i.e. the following constraints must always be satisfied for α = A, B, 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 are replica nodes, i.e. in the matching problem either two replica nodes are both matched or both unmatched.Therefore the variable s α ij satisfy the following additional constraints 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 with The energy E is given by the number N D of driver replica nodes in the duplex network by 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 the variables {s α ij } associated to every directed edge i → j in layer α, such that the energy E given by the number of driver replica nodes in the duplex network is minimized provided that the conditions given by Eqs.(S − 1), (S − 2) are satisfied.Introducing as an auxiliary variable the "inverse temperature" β we cast this problem into a statistical mechanics problem where our first aim is finding the distribution P ({s ij }), parametrized by the inverse temperature β, and given by where θ(x) = 1 for x ≥ 0 and θ(x) = 0 for x < 0 , δ(x) is the Kronecker delta, and where Z is the normalization constant, that corresponds to the partition function of the statistical mechanics problem.Subsequently, we plan to perform the limit β → ∞ in order to characterize the optimal (i.e. the maximum-sized) matching in the network satisfying Eqs.(S − 1), (S − 2).The free-energy of the problem F (β) is defined as and the energy E is therefore given by (S-8) The distribution P ({s ij }) on a locally tree-like network can be (approximately) estimated by the cavity method in the replica symmetric assumption (i.e. by deriving Belief Propagation equations) [11,15,[24][25][26][27][28][29].In this respect, in each layer α of the duplex network, we define two probability marginals on each directed link, one going in the same direction of the link P α i→j (s ij ) and one in the opposite direction P α i→j (s ji ).The BP equations for these quantities are where D α i→j and Dα i→j are normalization constants.The probability marginals {P α i→j (s α ij ), P α i→j (s α ji )} can be parametrized by the cavity fields h α i→j and ĥα i→j defined by . (S-10) In terms of the cavity fields (or messages), Eqs.(S-9) reduce to the following set of finite temperature BP equations, The free energy F and the energy E = ∂βF ∂β of the model are given respectively by and by 1 + e β(h α i→j + ĥα j→i ) . (S-13)

BP Equations for β → ∞
The BP equations in the limit β → ∞ are derived from the Eqs.(S − 11).In the limit β → ∞ the solution is expressed in terms of the fields h α i→j or ĥα i→j sent from a node (i, α) to the linked node (j, α) in layer α = A, B. The cavity fields have a simple interpretation as messages between neighboring replica nodes [25]: h α i→j = ĥα i→j = 1 means "match me", h α i→j = ĥα i→j = −1 means "do not match me", and h α i→j = ĥα i→j = 0 means "do what you want".The zero-temperature BP (or Max-Sum) equations determining the values of these fields in the limit β → ∞ are are given by in which the fields are defined to take values in the discrete set {1, 0, −1} and we defined the maximum over a null set equal to −1.It follows that for k B,in i = 0 we have ĥA i→j = −1 and for k A,in i = 0 we have ĥB i→j = −1.The energy E can also be expressed in terms of these fields and is given by where < i, j > indicates pair of nodes that are nearest neighbors in the network and where we take the maximum over a null set equal to -1.
Appendix D: Controllability of uncorrelated multiplex networks with given in-degree and out-degree distribution

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 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 statistical properties, that we describe through distributions P α (h α ) and Pα ( ĥα ) of cavity fields that are defined on the support of Eqs.S-14, i.e.
where α = A, B and where the probabilities w α 1 , w α 2 , w α 3 are normalized w α 1 + w α 2 + w α 3 = 1 as well as the probabilities ŵα 1 , ŵα 2 , ŵα 3 that satisfy the equation ŵα The cavity method at the network ensemble level is also known as density evolution method [24].
It is useful to introduce the generating functions G α,in/out 0 (z), and G α,in/out 1 (z) of the multiplex network as with α = A, B. In this way, we can derive recursive equations for the probabilities {w α i } i=1,2,3 and { ŵα i } i=1,2,3 , that are the analogous of Eqs.S-14 for an ensemble of uncorrelated duplex networks The energy E of the matching problem can be also expressed in terms of the {w α i } i=1,2,3 and { ŵα i } i=1,2,3 giving (S-4)

Stability condition
The Eqs.S-3 might have multiple solutions.In order to evaluate the stability of these solutions, using a method already used in the context of single networks [15,25] here we compute the Jacobian of the system of Eqs.(S-3) and impose that all its eigenvalues have modulus less than one.We avoid to consider w α 3 and ŵα 3 because they influence only the number of null eigenvalues (4 eigenvalues upon 12).The 12 × 12 Jacobian matrix becomes 8 × 8 and it can be decomposed in four 4 × 4 blocks with and Here the generating functions G Of particular interest is the characterization of the stability of the solution w α 1 = ŵα 1 = w α 2 = ŵα 2 = 0 and w α 3 = ŵα 3 = 1, corresponding to the full controllability of the network, a configuration with E = N D = 0.This solution emerges for P in α (1) = P out α (1) = 0 for α = A, B. Therefore if the minimum in-degree and the minimum out-degree are both greater than one, the analysis at the ensemble level is consistent with the full controllability of the network.Nevertheless this solution might be not stable.By analyzing the Jacobian J for w α 1 = ŵα 1 = w α 2 = ŵα 2 = 0 and w α 3 = ŵα 3 = 1, we can determine under which condition the full controllability solution is stable.The Jacobian matrix, in this case simplify significantly and is given by Four eigenvalues of J are zero, the other four have degenerate modulus, therefore the stability conditions are we have just one stability criterion solution and it reads We observe here that on the single layers α = A, B the full controllability solution, is instead only stable [15] for 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 on the duplex network.Nevertheless, for duplex networks formed by layers in which the in-degree distribution and the out-degree distribution are not the same there can be cases in which for the duplex network the fully controllable solution is stable while for the single layers it is not stable (See main text for discussion of this phenomenon and simulation results).

Entropy
In order to evaluate the number of maximum matchings, here we evaluate the entropy of the ground state solutions in the case of uncorrelated layers.The entropy density is given by s 0 = S 0 /N and can be computed by expanding the free energy at low temperatures f (β → ∞) = e 0 − s 0 /β + O(1/β 2 ).This involves the study of the evanescent parts of the cavity field.Therefore we assume that the field can be written as From the BP equations, and the equations for P (h α ) and P ( ĥα ) we can obtain the relation between the probability distributions A α 1 (ν α ), A α 2 (µ α ), A α 3 (γ α ), and the distributions Âα The free energy density f (β) = F (β)/N = e 0 − s0 β + O(1/β 2 ) with s 0 = s 0,a,A + s 0,a,B + s 0,b + s 0,c,A + s 0,c,B (S-20) where s 0, are given by Here we consider the case of two Poisson networks with the same in/out average degree.In other words, we consider the situation in which k A,in = k A,out = k B,in = k B,out = c.The fraction n D of nodes that are driver nodes of this duplex, is always larger than the double of the fraction of driver nodes in each of the layers taken in isolation (see Fig. S-1).Moreover we observe that there is a phase transition in the controllability of these level (S − 3) read, (S-24) The energy E is given in this case by We notice that the equations for ŵ1 and ŵ2 can be rewritten to form a closed subsystem of equations,  with J indicating the Jacobian of the system of equations ŵ1 = h 1 ( ŵ1 , ŵ2 ) and ŵ2 = h 2 ( ŵ1 , ŵ2 ) given by .
For c < c we observe that w 3 = ŵ3 = 0.At c we observe a discontinuity in both w 3 and ŵ3 , but for c > c the functions h 1 ( ŵ1 , ŵ2 ) and h 2 ( ŵ1 , ŵ2 ) are analytic, and analyzing Eqs.(S − 26a) − (S − 26b) we obtain the behavior of the order parameters w 3 and ŵ3 for c > c showing that the transition is hybrid.
We further characterize this phase transition evaluating the number of maximum matchings, i.e. the entropy value of the ground state solutions in the case of two poisson uncorrelated layers.We found that the entropy has a discontinuity at c * = 3.22233.. 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 the probability of nodes of low degree (degree 0, 1, 2) that have been shown to be essential to determine the controllability of single layers [15].Therefore we consider a duplex networks with degree distributions The fraction of driver replica nodes of the duplex network is always larger that the double of the fraction of driver nodes in any single layer taken in isolation.Moreover the region in which the fully controllable solution is stable is the same for the duplex network, and for the single networks in the layers of the duplex network taken in isolation.This result is consistent with the theoretical expectations obtained in Sec.D 2. In fact the in-and out-degree distributions of the two layers are the same.

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 poisson case, the fraction n D of driver nodes of this duplex, is always larger than the double of the fraction of driver nodes in each of the layers taken in isolation (see Fig. S-5).
Moreover, low-degree nodes significantly affect the controllability of duplex networks formed by scale-free networks.We consider a duplex network with P in A (k) = P out A (k) = P in B (k) = P out B (k) = P (k) and P (k) given by S-5: Density of driver nodes ε = nD 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 red line).The minimum in/out degree 1 and the maximum in/out degree is given by the structural cutoff with N = 10 4 .In the dashed blue line we display the double of the number of driver nodes ε = 2nD for a single power-law network with the same in/out degree distributions P (k), indicating the fraction of driver nodes necessary to control separately the two layers.
In the first case we consider a joint in-degree distribution P in (k A , k B ) given by Once the low in-degree nodes are correlated, correlating also the other in-degrees of the network does not change substantially the number of driver nodes n D as discussed in the main body of the paper.

2 FIG. 1 :
FIG.1: Control of a duplex network.We assume that the dynamics in the two layer is independent but we impose that replica nodes are either both driver nodes or none of them is a driver node.The controllability of this duplex network can be mapped to a Maximum Matching Problem in which the unmatched nodes (indicated with a white circle) are the driver nodes of the duplex network.Here we have indicated with red thick links the matched links and by black thin links the unmatched links.

FIG. 2 :
FIG. 2: In panel A the fraction nD of driver nodes in a Poisson duplex network with k A,in = k A,out = k B,in = k B,out = c, plotted as a function of the average degree c.The points indicate the average BP results obtained over 5 single realizations of the Poisson duplex networks with average degree c and N = 10 4 , 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 dashed line represent twice the density of driver nodes for a single Poisson network with the same average degree.In panel B the densities nc, nr and no respectively of critical redundant and ordinary nodes are shown as functions of c for the same type of duplex networks with N = 10 3 , where each point is the average over 100 different instances.

8 FIG. 3 :
FIG. 3: Phase diagram of the Poisson multiplex network with average degrees k A,in = k A,out = cA and k B,in = k B,out = cB.The color code indicates the density of driver nodes nD = E/N in the multiplex network.

FIG. 4 :
FIG.4: 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 nD, both in the case of Poisson networks with the same in and out average degree c across the two layers (Panel A) and in the case of scale-free networks with the same in and out degree distribution across the layers, given by P (k) ∝ k −γ and minimum in/out degree 1 (Panel B).The BP data are shown for networks with N = 10 4 , and are averaged 5 times for panel A and 20 times for panel B.

FIG. 5 :
FIG.5: Histogram of the number of networks that out of 100 realizations have ND driver nodes.The results obtained for the control of a duplex networks and its two layers are compared.The duplex networks are formed by two scale-free networks with N = 10 4 and P A,in (k) = P Bin ∝ k −γ for k > 2 and P A,out (k) = P B,out (k) ∝ k −γ for k > 2, with γ = 2.3, the network have minimum in-degree equal to 2 and minimum out-degree equal to 3 and P A,in (2) = P B,in (2) = 0.3.
Appendix C: The Maximum Matching Problem for the Controllability of Duplex Networks 1. Mapping duplex controllability into a constrained Maximum Matching Problem

1 are
given by Eqs.(S − 2) and the generating functions G α,in 2

4 .
Phase transition in the controllability of Poisson duplex networks.

FIG. S- 1 :
FIG. S-1: Density of driver nodes ε = nD for a duplex network composed by two Poisson networks with k A,in = k A,out = k B,in = k B,out = c is indicated with a solid red line and it clearly shows a phase transition for c = 3.22233 . ... In the dashed blue line we display the double of the number of driver nodes ε = 2nD for a single Poisson network with the same average degree c, indicating the fraction of driver nodes necessary to control separately the two layers.

2 c = 3 . 3 FIG. S- 2 :
FIG. S-2:Plots of the functions ŵ1 = h1( ŵ1, ŵ2) and ŵ2 = h2( ŵ1, ŵ2) given by Eqs.(S − 26a)−(S − 26b).The solution of the system of these two equations, corresponds to a crossing of the two curves.We show the emergence of two new solutions of this system of equations for c > c = 3.22233 . ... The critical point c characterize an hybrid phase transition in the controllability of the duplex network.

FIG. S- 3 :
FIG. S-3:Values of the probabilities {wi}i=1,2,3 and ŵi=1,2,3 plotted as a function of the average degree c, for a duplex network formed by two Poisson layers with k A,in = k A,out = k B,in = k B,out = c.These probabilities are calculated directly from BP results obtained over 5 single realizations these multiplex networks with average degree c and N = 10 4 .
k) = P out B (k) = P (k) and with minimum degree is 2. In particular we consider P (k) given by k for k ∈ [3, ∞] (S-29) with κ indicating a normalization constant.In Fig. S-4 (on the left) we show the phase diagram of this duplex network described by the dependence of the fraction of driver nodes n D on c and P (2).The dark grey area defines the region where the zero-energy solution is stable, hence in which to control a duplex one needs only an infinitesimal fraction of driver nodes, i.e. n D = 0.These results are compared with the situation in which the two layers are controlled separately shown in Fig. S-4 (on the right).
FIG. S-5: Density of driver nodes ε = nD 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 red line).The minimum in/out degree 1 and the maximum in/out degree is given by the structural cutoff with N = 10 4 .In the dashed blue line we display the double of the number of driver nodes ε = 2nD for a single power-law network with the same in/out degree distributions P (k), indicating the fraction of driver nodes necessary to control separately the two layers.

2 ,FIG. S- 8 :
FIG. S-8:The density of the driver nodes nD in a duplex network formed by two Poisson networks with k A,in = k A,out = k B,in = k B,out = c and with correlated low in-degrees is plotted as function of c for different values of p.The result for the two separate layers is shown in black (dashed curve) while the situation for uncorrelated layers is shown in red.The value of p increases going from the red curve (p = 0) to the grey curve (p = 1).