Controllability of Multiplex, Multi-time-scale Networks

M. Posfai, J. Gao, S.P. Cornelius, A.-L. Barabasi, R. D’Souza.
Physical Review E 94: 3, 032316 (2016).
September 26, 2016

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 (Ni) 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: Ni decreases as the time-scale  difference increases up to a critical time-scale difference, above which Ni  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 Ni, again  up to a critical value, above which Ni 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.

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