## Network Control

### Realistic control of network dynamics

The control of complex networks is of paramount importance in areas as diverse as ecosystem management, emergency response and cell reprogramming. A fundamental property of networks is that perturbations to one node can affect other nodes, potentially causing the entire system to change behaviour or fail. Here we show that it is possible to exploit the same principle to control network behaviour. Our approach accounts for the nonlinear dynamics inherent to real systems, and allows bringing the system to a desired target state even when this state is not directly accessible due to constraints that limit the allowed interventions. Applications show that this framework permits reprogramming a network to a desired task, as well as rescuing networks from the brink of failure — which we illustrate through the mitigation of cascading failures in a power-grid network and the identification of potential drug targets in a signalling network of human cancer.

S.P. Cornelius, W.L. Kath, and A.E. Motter,

Realistic control of network dynamics,

Nature Communications **4**, 1942 (2013).

doi:10.1038/ncomms2939 -
PDF -
Supplementary Information -
Movie

arXiv:1307.0015v1

### Controllability transition and nonlocality in network control

A common goal in the control of a large network is to minimize the number of driver nodes or control inputs. Yet, the physical determination of control signals and the properties of the resulting control trajectories remain widely underexplored. Here we show that (i) numerical control fails in practice even for linear systems if the controllability Gramian is ill conditioned, which occurs frequently even when existing controllability criteria are satisfied unambiguously, (ii) the control trajectories are generally nonlocal in the phase space, and their lengths are strongly anti-correlated with the numerical success rate and number of control inputs, and (iii) numerical success rate increases abruptly from zero to nearly one as the number of control inputs is increased, a transformation we term numerical controllability transition. This reveals a trade-off between nonlocality of the control trajectory in the phase space and nonlocality of the control inputs in the network itself. The failure of numerical control cannot be overcome in general by merely increasing numerical precision — successful control requires instead increasing the number of control inputs beyond the numerical controllability transition.

J. Sun and A.E. Motter,

Controllability transition and nonlocality in network control,

Phys. Rev. Lett. **110**, 208701 (2013).

doi:10.1103/PhysRevLett.110.208701

arXiv:1305.5848

### Network observability transitions

In the modeling, monitoring, and control of complex networks, a fundamental problem concerns the comprehensive determination of the state of the system from limited measurements. Using power grids as example networks, we show that this problem leads to a new type of percolation transition, here termed a network observability transition, which we solve analytically for the configuration model. We also demonstrate a dual role of the network's community structure, which both facilitates optimal measurement placement and renders the networks substantially more sensitive to "observability attacks." Aside from their immediate implications for the development of smart grids, these results provide insights into decentralized biological, social, and technological networks.

Y. Yang, J. Wang, and A.E. Motter,

Network observability transitions,

Phys. Rev. Lett. **109**, 258701 (2012).

doi:10.1103/PhysRevLett.109.258701 -
Supplementary Information

arXiv:1301.5916

### Rescuing ecosystems from extinction cascades through compensatory perturbations

Food-web perturbations stemming from climate change, overexploitation, invasive species and habitat degradation often cause an initial loss of species that results in a cascade of secondary extinctions, posing considerable challenges to ecosystem conservation efforts. Here, we devise a systematic network-based approach to reduce the number of secondary extinctions using a predictive modelling framework. We show that the extinction of one species can often be compensated by the concurrent removal or population suppression of other specific species, a counterintuitive effect not previously tested in complex food webs. These compensatory perturbations frequently involve long-range interactions that are not evident from local predator-prey relationships. In numerous cases, even the early removal of a species that would eventually go extinct is found to significantly reduce the number of cascading extinctions. These compensatory perturbations only exploit resources available in the system, and illustrate the potential of human intervention combined with predictive modelling for ecosystem management.

S. Sahasrabudhe and A.E. Motter,

Rescuing ecosystems from extinction cascades through compensatory perturbations,

Nature Communications **2**, 170 doi:10.1038/ncomms1163 (2011).

doi:10.1038/ncomms1163 - PDF - Supplementary Information

arXiv:1103.1653

### NECO - A scalable algorithm for NEtwork COntrol

We present an algorithm for the control of complex networks and other nonlinear, high-dimensional dynamical systems. The computational approach is based on the recently-introduced concept of compensatory perturbations — intentional alterations to the state of a complex system that can drive it to a desired target state even when there are constraints on the perturbations that forbid reaching the target state directly. Included here is ready-to-use software that can be applied to identify eligible control interventions in a general system described by coupled ordinary differential equations, whose specific form can be specified by the user. The algorithm is highly scalable, with the computational cost scaling as the number of dynamical variables to the power 2.5.

S.P. Cornelius and A.E. Motter,

NECO - A scalable algorithm for NEtwork COntrol,

Protocol Exchange (2013), doi:10.1038/protex.2013.063.

doi:10.1038/protex.2013.063 -
Source Codes

arXiv:1307.2582

### Spontaneous synchrony in power-grid networks

An imperative condition for the functioning of a power-grid network is that its power generators remain synchronized. Disturbances can prompt desynchronization, which is a process that has been involved in large power outages. Here we derive a condition under which the desired synchronous state of a power grid is stable, and use this condition to identify tunable parameters of the generators that are determinants of spontaneous synchronization. Our analysis gives rise to an approach to specify parameter assignments that can enhance synchronization of any given network, which we demonstrate for a selection of both test systems and real power grids. These findings may be used to optimize stability and help devise new control schemes, thus offering an additional layer of protection and contributing to the development of smart grids that can recover from failures in real time.

A.E. Motter, S.A. Myers, M. Anghel, and T. Nishikawa,

Spontaneous synchrony in power-grid networks,

Nature Physics **9**, 191 (2013).

doi:10.1038/nphys2535 -
Supplementary Information

arXiv:1302.1914

### Mechanical metamaterials with negative compressibility transitions

When tensioned, ordinary materials expand along the direction of the applied force. Here, we explore network concepts to design metamaterials exhibiting negative compressibility transitions, during which a material undergoes contraction when tensioned (or expansion when pressured). Continuous contraction of a material in the same direction of an applied tension, and in response to this tension, is inherently unstable. The conceptually similar effect we demonstrate can be achieved, however, through destabilizations of (meta)stable equilibria of the constituents. These destabilizations give rise to a stress-induced solid-solid phase transition associated with a twisted hysteresis curve for the stress-strain relationship. The strain-driven counterpart of negative compressibility transitions is a force amplification phenomenon, where an increase in deformation induces a discontinuous increase in response force. We suggest that the proposed materials could be useful for the design of actuators, force amplifiers, micromechanical controls, and protective devices.

Z.G. Nicolaou and A.E. Motter,

Mechanical metamaterials with negative compressibility transitions,

Nature Materials **11**, 608 (2012).

doi:10.1038/nmat3331 -
Supplementary Information -
Movie

arXiv:1207.2185

### Earlier publications on network rescue and cascade control

A.E. Motter,

Cascade control and defense in complex networks,

Phys. Rev. Lett. **93**, 098701 (2004).

doi:10.1103/PhysRevLett.93.098701

arXiv:cond-mat/0401074

A.E. Motter, N. Gulbahce, E. Almaas, and A.-L. Barabási,

Predicting synthetic rescues in metabolic networks,

Molecular Systems Biology **4**, 168 (2008).

doi:10.1038/msb.2008.1 -
Supplementary Information - EMBO and Nature Publishing Group

arXiv:0803.0962

D.-H. Kim and A.E. Motter,

Slave nodes and the controllability of metabolic networks,

New J. Phys. **11**, 113047 (2009).

doi:10.1088/1367-2630/11/11/113047 -
Supplementary Information

arXiv:0911.5518

A.E. Motter,

Improved network performance via antagonism: From synthetic rescues to multi-drug combinations,

BioEssays **32**, 236 (2010) - Problems and Paradigms.

doi:10.1002/bies.200900128 -
Online Open

arXiv:1003.3391