## Small vulnerable sets determine large network cascades

The understanding of cascading failures in complex systems has been hindered by the lack of realistic large-scale modeling and analysis that can account for variable system conditions. Using the North American power grid, we identified, quantified, and analyzed the set of network components that are vulnerable to cascading failures under any out of multiple conditions. We show that the vulnerable set consists of a small but topologically central portion of the network and that large cascades are disproportionately more likely to be triggered by initial failures close to this set. These results elucidate aspects of the origins and causes of cascading failures relevant for grid design and operation and demonstrate vulnerability analysis methods that are applicable to a wider class of cascade-prone networks. For more information, please visit the original paper published in Science. A PDF version that can be downloaded without subscription can be found here.

Watch the animated summary of the paper here.

## The Network Analog of the Butterfly Effect

The relation between network structure and dynamics is determinant for the behavior of complex systems in numerous domains. An important long-standing problem concerns the properties of the networks that optimize the dynamics with respect to a given performance measure. In our recent PRX paper, we show that such optimization can lead to sensitive dependence of the dynamics on the structure of the network. Specifically, using diffusively coupled systems as examples, we demonstrate that the stability of a dynamical state can exhibit sensitivity to unweighted structural perturbations (i.e., link removals and node additions) for undirected optimal networks and to weighted perturbations (i.e., small changes in link weights) for directed optimal networks. We illustrate these findings using Turing instability in activator-inhibitor systems, synchronization in power-grid networks, network diffusion, and several other network processes. Sensitive dependence on network structure has a range of potential implications for complex systems whose functions has been or are being optimized.

Watch the video on the paper here.

## Videos

Watch members of the Motter group explain complex systems research to general audiences.

## Recent Publications

A.E. Motter and M. Timme,

Antagonistic phenomena in network dynamics,

Annu. Rev. Condens. Matter Phys. 9, 463 (2018).

doi:10.1146/annurev-conmatphys-033117-054054

Y. Zhang and A.E. Motter,

Identical synchronization of nonidentical oscillators: When only birds of different feathers flock together,

Nonlinearity **31**, R1 (2018).

arXiv:1712.03245 - doi:10.1088/1361-6544/aa8fe7

T.P. Wytock, A. Fiebig, J.W. Willett, J. Herrou, A. Fergin, A.E. Motter, and S. Crosson,

Experimental evolution of diverse Escherichia coli metabolic mutants identifies genetic loci for convergent adaptation of growth rate,

PLoS Genetics **14**(3), e1007284 (2018).

doi:10.1371/journal.pgen.1007284

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

Chimera states in continuous media: Existence and distinctness,

Phys. Rev. Lett. **119**, 244101 (2017).

arXiv:1712.00458 - doi:10.1103/PhysRevLett.119.244101

Y. Yang and A.E. Motter,

Cascading failures as continuous phase-space transitions,

Phys. Rev. Lett. **119**, 248302 (2017).

arXiv:1712.04053 - doi:10.1103/PhysRevLett.119.248302

T. Nishikawa, J. Sun, and A.E. Motter,

Sensitive dependence of optimal network dynamics on network structure,

Phys. Rev. X **7**, 041044 (2017).

arXiv:1611.01164 - doi:10.1103/PhysRevX.7.041044

Y. Yang, T. Nishikawa, and A.E. Motter,

Small vulnerable sets determine large network cascades in power grids,

Science 358 (6365), eaan3184 (2017).

doi:10.1126/science.aan3184 - PDF

Y.S. Cho, T. Nishikawa, and A.E. Motter,

Stable chimeras and independently synchronizable clusters,

Phys. Rev. Lett. **119**, 084101 (2017).

arXiv:1707.06657 - doi:10.1103/PhysRevLett.119.084101

A. Haber, F. Molnar, and A.E. Motter,

State observation and sensor selection for nonlinear networks,

IEEE Trans. Control Netw. Syst. (to appear).

arXiv:1706.05462

Y. Zhang, T. Nishikawa, and A.E. Motter,

Asymmetry-induced synchronization in oscillator networks,

Phys. Rev. E **95**, 062215 (2017).

arXiv:1705.07907 - doi:10.1103/PhysRevE.95.062215

X. Chen, T. Nishikawa, and A.E. Motter,

Slim fractals: The geometry of doubly transient chaos,

Phys. Rev. X **7**, 021040 (2017).

arXiv:1705.02349 - doi:10.1103/PhysRevX.7.021040

L Zhang, A.E. Motter, and T. Nishikawa,

Incoherence-mediated remote synchronization,

Phys. Rev. Lett. **118**, 174102 (2017).

arXiv:1703.10621 - doi:10.1103/PhysRevLett.118.174102

J.-R. Angilella, D.J. Case, and A.E. Motter,

Levitation of heavy particles against gravity in asymptotically downward flows,

Chaos **27**, 031103 (2017).

arXiv:1703.03296 - doi:10.1063/1.4978386

Y. Yang, T. Nishikawa, and A.E. Motter,

Vulnerability and cosusceptibility determine the size of network cascades,

Phys. Rev. Lett. **118**, 048301 (2017).

arXiv:1701.08790 - doi:10.1103/PhysRevLett.118.048301 - Physics Story

A.E. Motter and Y. Yang,

The unfolding and control of network cascades,

Physics Today **70**(1), 32 (2017).

arXiv:1701.00578 - doi:10.1063/PT.3.3426

T. Nishikawa and A.E. Motter,

Symmetric states requiring system asymmetry,

Phys. Rev. Lett.** 117**, 114101 (2016).

arXiv:1608.05419 - doi:10.1103/PhysRevLett.117.114101 - Synopsis

T. Nishikawa and A.E. Motter,

Network-complement transitions, symmetries, and cluster synchronization,

Chaos **26**, 094818 (2016).

arXiv:1712.06613 - doi:10.1063/1.4960617

## Adilson E. Motter

Photo by Eileen Molony

Professor Motter's research is focused on the dynamical behavior and control of complex systems and networks. More...

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## Special Issue of IEEE TCNS

Special Issue: Approaches to Control Biological and Biologically Inspired Networks in IEEE Transactions on Control of Network Systems (to appear in June 2018).

## Network Control

See the talk Advances on the Control of Nonlinear Network Dynamics by Adilson E. Motter at the 2015 SIAM Conference on Applications of Dynamical Systems and check out our featured control projects page to see a summary of our recent work in this area.

## Group News

December 2017: Group organizes the 4th edition of the Network Frontier Workshop.

May 2017: Vicky Yang receives Red Sock Award (SIAG/Dynamical Systems) for joint work at DS17.

August 2016: Adilson E. Motter receives Scialog Award jointly with Kimberly Reynolds.

April 2016: Daniel Case is awarded a Presidential Fellowship, which will support his graduate research on microfluidic networks.

March 2016: Yang (Angela) Yang receives the Topical Group on Statistical and Nonlinear Physics (GSNP) Student Speaker Award at the APS March Meeting 2016.

March 2016: Our group leads a Northwestern team receiving a $3.2 million ARPA-E Grant.

January 2016: Adilson E. Motter is selected Outstanding Referee of the APS.

## Selected Press

Sensitive Dependence on Network Structure: Analog of Chaos and Opportunity for Control

SIAM News (April 2, 2018)

http://...

Mapping the Vulnerability and Strength of the Power Grid

ISEN (November, 2017)

http://...

Scientists pinpoint 'weak spots' that cause mass blackouts

E&E News (November 17, 2017)

http://...

Power grid simulation highlights weak points in North American electrical system

Tech Xplore (November 17, 2017)

http://...

Weak links in US power grid vulnerable in event of catastrophe

New Scientist (November 16, 2017)

http://...

What Causes Cascading Power Grid Failures

IEEE Spectrum (November 16, 2017)

http://...

Looking for a Grid's Weak Spots? Scientists May Have the Answer

Bloomberg Technology (November 16, 2017)

http://...

To Converge You Must Diverge

Helix Magazine (October 7, 2017)

http://...

Power Puzzles: solutions for grid resilience

Empower Magazine (May 2017)

http://...

Visualizing the Grid: new way to imagine grid stability

Empower Magazine (May 2017)

http://...

The Behavior of Blackouts (Podcast in Portuguese)

Revista Pesquisa FAPESP (April 19, 2017)

http://...

Grid Outages from Failures of Power Line Clusters

Physics (Jan 27, 2017)

http://...

Advantage of Diversity: Consensus Because of (not Despite) Differences

SIAM News (January 17, 2017)

http://...