Abstract
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.
Highlights
The controllability of a network is a theoretical problem of relevance in a variety of contexts ranging from financial markets to the brain
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
Despite the significant interest in network controllability, all linear and non-linear approaches for the controllability of networks are still restricted to single networks while it has been recently found that the multiplexity of networks can have profound effects on the dynamical processes taking place on them[39,40,41,42,43,44]
Summary
The controllability of a network is a theoretical problem of relevance in a variety of contexts ranging from financial markets to the brain. Most of the real networks are not isolated but interact with each other forming multilayer structures[1,2]. Linear[11,12,13,14,15,16,17,18,19] and non-linear[20,21,22,23,24,25,26,27,28,29,30] approaches are providing new scenarios for the characterization of the controllability of single complex networks. We show that controlling the dynamics of multiplex networks is more costly than controlling single layers taken in isolation. The controllability of multiplex networks displays unexpected new phenomena 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 fully controllable configuration can be stable in a multilayer network even if it is not stable in the isolated networks that form the multilayer structure
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