Transcript Hans Herrmann, Theories for Extreme Events

Theories for
Extreme Events
Hans J. Herrmann
Computational Physics, IfB, ETH
Zürich, Switzerland
New Views on Extreme Events
Workshop of the Risk Center at SwissRe
Adliswil, October 24-25, 2012
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
ETH Risk Center
LSA
Technology
(Kröger)
Systemic
Risks
(Schweitzer)
ZISC
Entrepreneurial Risks
(Sornette)
Innovation
Policy
(Gersbach)
Decision
Making
Information Security
(Basin)
(Murphy)
Sociology
Integrative
Risk Mgmt.
RiskLab
(Bommier)
Finance & Insurance
(Embrechts)
Math.
Finance
ETH
Risk Center
(Embrechts)
Conflict
Research
(Cederman)
Traffic
Systems
Comp.
Physics
(Herrmann)
(Axhausen)
Forest
Engineering
(Heinimann)
HazNETH
(Helbing)
CSS
Center for
Security Studies
(Wenger)
Natural Hazards
(Faber)
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
ETH Risk Center
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
The three types of flooding
flooding landscapes
braided rivers
breaking
dam
AnnualEvents,
CSP Workshop,
UGA,at
Athens,
GA, February
21-25,
2011 24-25, 2012
New Views on24th
Extreme
Workshop
SwissRe,
Adliswil,
October
The braided river
The river carries sediments which
deposit on the bottom of the bed until
they reach the level of the water and
create a natural dam clogging the
branch. So this branch dies and a new
branch is created somewhere else.
Basic principle is a conservation law (here the mass of
water) and the formation of local bottlenecks.
+ randomness
Other examples: traffic, fatigue, electrical networks.
AnnualEvents,
CSP Workshop,
UGA,at
Athens,
GA, February
21-25,
2011 24-25, 2012
New Views on24th
Extreme
Workshop
SwissRe,
Adliswil,
October
Traffic
density
fundamental
diagram
flux
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Classical Probability Theory
Poisson distribution
Gaussian distribution
Black-Scholes Model
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Flooding landscapes
When the water level of a lake
rises in a random landscape it
spills over into the
neighboring basin and the
sizes of these invasions follow
a power law distribution.
Basic principle is the existence of a local threshold at which
discharging occurs.
+ randomness
Other examples are earthquakes, brain activity.
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Earthquakes
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Frequency Distribution of Earthquakes
Gutenberg-Richter law
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Conclusion
Paul Pierre Levy
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Earthquake Model
Spring-Block Model
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Self-Organized Criticality (SOC)
Per Bak
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Sandpile Model
Applet
http://www.cmth.bnl.gov/~maslov/Sandpile.htm
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Size distribution of avalanches
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Avalanches on the Surface of a Sandpile
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Self-Organized Criticality (SOC)
The lazy burocrats
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
The Stockmarket
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
SOC Model for the Stockmarket
Dupoyet et al 2011
Comparison with NASDAQ
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Model for the distribution of price
fluctuations
Stauffer + Sornette, 1999
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Examples for SOC
•
•
•
•
•
•
•
•
Earthquakes
Stockmarket
Evolution
Cerebral activity
Solar flares
Floodings
Landslides
......
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Breaking a dam
Each time a dam is in danger to
break it is repaired and made
stronger. When finally the dam
does one day break all the land
is flooded at once.
Basic principle is that the catastrophe is avoided by local
repairs until it can not be withhold anymore.
+ randomness
Other examples are volcanos
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Volcano eruption
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Branch pipes
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
The Black Swan
Nassim Nicholas Taleb
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
The Black Swan
Didier Sornette
Dragon King
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Explosive Percolation
Product Rule (PR)
• Consider a fully connected graph
• Select randomly two bonds and occupy the
one which creates the smaller cluster
classical percolation
product rule
Dimitris Achlioptas
D. Achlioptas, R. M. D’Souza, and J. Spencer, Science 323, 1453 (2009)
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Largest Cluster Model
Nuno Araújo
• Select randomly a bond
• if not related with the
largest cluster occupy it
• else, occupy it with
probability
  s  s 2 
q  exp 
 
  s  
Nuno Araújo and HJH, Phys. Rev. Lett. 105, 035701 (2010)
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Largest Cluster Model
order parameter: P∞ = fraction of sites in largest cluster
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Phase transition of 1st order
Sudden jump with our previous warning
Its consequences touch the entire system.
It is the worst case scenario.
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Complex Systems
antropoz metron
Protagoraz
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Internet
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Scale-free networks
P(k )  k
g
Internet
g  2.4
actors
g  2.3
WWW:
g out  2.4
g in  2.1
Model: Barabasi-Albert g = 3
HEP
neuroscience
g  2.1
scientific collaborations
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Terrorist network
September 11
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Random Attack
MaliciousAttack
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
European Power Grid
The changes in the EU power grid (red lines are replaced by green ones) and
the fraction of nodes in the largest connected cluster s(q) after removing a
fraction of nodes q for the EU powergrid and its improved network
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Coupled Networks
Collapse of the power grid in
Italy and Switzerland, 2003
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Collapse of two coupled networks
Largest
connected
cluster
Number of
iterations
Fraction of attacked nodes
 Phase transition of 1st order
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Reducing the risk by decoupling the
networks through autonomous nodes
Largest
connected
cluster
Fraction of attacked nodes
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Proposal to improve robustness
The blackout in Italy and Switzerland, 2003
Original networks
4 autonomous nodes
39 communication servers (stars) + 310 power stations (circles)
Random failure of 14 communication servers
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012
Outlook
• There exist unmeasurable risks.
• Mending is dangerous, because the risk
becomes more brittle.
• Usually one can substantially reduce the
risk in a network through rather minor
changes.
• Autonomous nodes make coupled
networks more robust.
New Views on Extreme Events, Workshop at SwissRe, Adliswil, October 24-25, 2012