## 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.

## Network Frontier Workshop 2017

Network Frontier Workshop — Virtual Edition. Now you can attend from the comfort of your office.

The workshop will be held on Dec 12-13 and broadcast via WebEx to registered participants.

## 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).

## Asymmetry-Induced Symmetry in Network Dynamics

It is generally assumed that individual entities are more likely to exhibit the same or similar behavior if they are equal to each other: think of lasers pulsing at the same frequency, animals using the same gait, agents reaching consensus. In a recent PRL paper we have shown that this assumption is in fact false in networks of interacting entities. Our central discovery is a network phenomenon we term “asymmetry-induced symmetry” (AIS), in which the state of the system can be symmetric only when the system itself is not symmetric. Using synchronization as a model process, we demonstrate that the state in which all nodes synchronize and exhibit identical dynamics (a state of maximum symmetry, as it remains unchanged after swapping any two nodes), such as when lasers pulse together, can only be realized when the nodes themselves are not identical. Asymmetry-induced symmetry can be seen as the converse of the well-studied phenomenon of symmetry breaking, where the state has less symmetry than the system. AIS has far-reaching implications for processes that involve converging to uniform states; in particular, it offers a mechanism for yet-to-be-explained convergent forms of pattern formation, in which an asymmetric structure develops into a symmetric one. AIS also has implications for consensus dynamics, where it gives rise to the scenario in which interacting agents only reach consensus when they are sufficiently different from each other. See also AIS Demo and AIS-Inspired Dance Outreach Video.

## Videos

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

## Mechanical Networks and Negative Compressibility Materials

When tensioned, ordinary materials expand along the direction of the applied force. In our recent Nature Materials paper, 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 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. We suggest that the proposed materials could be useful for the design of actuators, force amplifiers, micromechanical controls, and protective devices.

- Movie: Simulated response of the material to uniform, diagonal, pinched, and splayed stress profiles.

## Recent Publications

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

Small vulnerable sets determine large network cascades in power grids,

Science 358 (6365), eaan3184 (2017).

http://science.sciencemag.org/content/358/6365/eaan3184

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

Chimera states in continuous media: Existence and distinctness,

Phys. Rev. Lett. (to appear).

https://journals.aps.org/prl/

A.E. Motter and M. Timme,

Antagonistic phenomena in network dynamics,

Annu. Rev. Condens. Matter Phys. (to appear).

http://www.annualreviews.org/journal/conmatphys

Y. Zhang and A.E. Motter,

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

Nonlinearity (to appear).

http://iopscience.iop.org/journal/0951-7715

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

Sensitive dependence of optimal network dynamics on network structure,

Phys. Rev. X (to appear).

arXiv:1611.01164

Y. Yang and A.E. Motter,

Cascading failures as continuous phase-space transitions,

Phys. Rev. Lett. (to appear).

https://journals.aps.org/prl/

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).

http://aip.scitation.org/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.

## Positions

(August 2017) The group has openings for postdoctoral researchers in the area of network dynamics. One position is available immediately. Deadline for applications: September 30 and November 15, 2017.

## 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

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.

December 2015: Group organizes the 3rd edition of the Network Frontier Workshop.

November 2015: Adilson E. Motter is elected Fellow of the American Association for the Advancement of Science (AAAS).

October 2015: Takashi Nishikawa and Corey Brady present "Networks All Around!" at the Jr. Science Café series of the Museum of Science and Industry of Chicago.

June 2015: Xiaowen Chen receives the Prize for Distinguished Honors Thesis for her senior thesis entitled "Fractal Geometry of Undriven Dissipative Systems".

March 2015: Adilson E. Motter is awarded a 2015 Simons Fellowship in Theoretical Physics.

## Selected Press

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://...