title: Controllability of multiplex, multi-time-scale networks.
abstract: The paradigm of layered networks is used to describe many real-world systems, from biological networks to social organizations and transportation systems. While recently there has been much progress in understanding the general properties of multilayer networks, our understanding of how to control such systems remains limited. One fundamental aspect that makes this endeavor challenging is that each layer can operate at a different time scale; thus, we cannot directly apply standard ideas from structural control theory of individual networks. Here we address the problem of controlling multilayer and multi-time-scale networks focusing on two-layer multiplex networks with one-to-one interlayer coupling. We investigate the practically relevant case when the control signal is applied to the nodes of one layer. We develop a theory based on disjoint path covers to determine the minimum number of inputs (N_{i}) necessary for full control. We show that if both layers operate on the same time scale, then the network structure of both layers equally affect controllability. In the presence of time-scale separation, controllability is enhanced if the controller interacts with the faster layer: N_{i} decreases as the time-scale difference increases up to a critical time-scale difference, above which N_{i} remains constant and is completely determined by the faster layer. We show that the critical time-scale difference is large if layer I is easy and layer II is hard to control in isolation. In contrast, control becomes increasingly difficult if the controller interacts with the layer operating on the slower time scale and increasing time-scale separation leads to increased N_{i}, again up to a critical value, above which N_{i} still depends on the structure of both layers. This critical value is largely determined by the longest path in the faster layer that does not involve cycles. By identifying the underlying mechanisms that connect time-scale difference and controllability for a simplified model, we provide crucial insight into disentangling how our ability to control real interacting complex systems is affected by a variety of sources of complexity.
authors.affiliation: Complexity Science Center, University of California, Davis, California 95616, USA; Department of Computer Science, University of California, Davis, California 95616, USA; and Department of Physics of Complex Systems, Eötvös University, Budapest H-1117, Hungary.
authors.forename: Márton
authors.initials: M
authors.lastname: Pósfai
authors.affiliation: Center for Complex Network Research, Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA.
authors.forename: Jianxi
authors.initials: J
authors.lastname: Gao
authors.affiliation: Center for Complex Network Research, Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA.
authors.forename: Sean P
authors.initials: SP
authors.lastname: Cornelius
authors.affiliation: Center for Complex Network Research, Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA; Center for Cancer Systems Biology, Dana-Farber Cancer Institute, Harvard University, Boston, Massachusetts 02215, USA; Department of Medicine, Brigham and Women's Hospital, Harvard Medical School, Boston, Massachusetts 02115, USA; and Center for Network Science, Central European University, Budapest H-1051, Hungary.
authors.forename: Albert-László
authors.initials: AL
authors.lastname: Barabási
authors.affiliation: Complexity Science Center, University of California, Davis, California 95616, USA; Department of Computer Science, University of California, Davis, California 95616, USA; Department of Mechanical and Aerospace Engineering, University of California, Davis, California 95616, USA; and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA.
authors.forename: Raissa M
authors.initials: RM
authors.lastname: D'Souza