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Complex Networks Approach
Potsdam Institute for Climate Impact Research &
Institut of Physics, Humboldt-Universität zu Berlin &
King‘s College, University of Aberdeen
Jürgen Kurths
http://www.pik-potsdam.de/members/kurths/
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Networks with Complex Topology
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1707 (Basel) – 1783 (St. Petersburg) 1741-66 Berlin
Königsberger Brückenproblem (L. Euler, 1736)
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Erdös-Renyi Random Networks (1959)
Much mathematical theory about
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Social Networks
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Biological Networks
Neural Networks
Genetic Networks
Protein interaction Ecological Webs
Metabolic Networks
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Transportation Networks
Airport Networks
Road Maps
Local Transportation
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Evolving Networks
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Network of Networks
Interconnected Networks
Interdependent Networks
Interacting Networks
Multilayer Networks
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Transportation Networks
• Romans built > 850.000 km roads (Network)
• „Silk Street“ (Network)
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Technological Networks
World-Wide Web
Power Grid
Internet
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Power grid in Japan
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How to control such networks?
Pinning Control (which nodes?)
Highly Non-trivial Task
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Papenburg: Monster Black-Out 06-11-2006
• Meyer Werft in Papenburg • Newly built ship Norwegian Pearl length: 294 m, width: 33 m • Cut one line of the power grid • Black-out in Germany ( > 10 Mio people) France (5 Mio people) Austria, Belgium, Italy, Spain
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Cerebral cortex
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Useful approaches with networks
• Immunization problems (spreading of
diseases) • Functioning of biological/physiological
processes as protein networks, brain dynamics, colonies of thermites and of social networks as network of vehicle traffic in a region, air traffic, or opinion formation etc.
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Traditional networks in Physics lattices
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Rings
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Networks with complex topology
• Random graphs/networks (Erdös, Renyi, 1959)
• Small-world (SW) networks (Watts, Strogatz, 1998 F. Karinthy hungarian writer – SW hypothesis, 1929)
• Scale-free networks (SF) (Barabasi, Albert, 1999; D. de Solla Price – number of citations – heavy tail
distribution, 1965)
Networks with Complex Topology
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Types of complex networks
fraction of nodes in the network having at least k connections to other nodes have a power law scaling – existence of hubs
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Small-world Networks
Nearest neighbour and a few long-range connections (with probability p)
Nearest neighbour Connections
Regular Complex Topology
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General description of a graph/complex network
• n nodes • m edges • ai,j = 1 if node i and j are connected; 0 otherwise • A = ((ai,j)) adjacency matrix
• Network undirected ai,j = aj,i symmetric
characterized by eigenvalues and eigenvectors
• ki - degree of node i, P(k) – degree distribution • K = <ki> mean degree
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Dynamics on the nodes - synchronization
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Universality in the synchronization of
weighted random networks
Our intention: What is the influence of weighted coupling for complete synchronization
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Weighted Network of N Identical Oscillators
F – dynamics of each oscillator
H – output function
G – coupling matrix combining adjacency A and weight W
- intensity of node i (includes topology and weights)
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General Condition for Synchronizability
Stability of synchronized state
N eigenmodes of
ith eigenvalue of G
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Main results
Synchronizability universally determined by:
- mean degree K and
- heterogeneity of the intensities
- minimum/ maximum intensities
or
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Synchronizability Ratio
Stability Interval
Synchronizability condition
Synchronizability – Master Stability Formalism (Pecora&Carrol (1998)
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Puzzle: best synchronizable = most random
However: Most real-world networks far from being random How to understand? So far: local stability studied
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Basin Stability
basin volume of a state (regime)
measures likelihood of arrival at this state (regime)
NATURE PHYSICS (in press)
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Synchronizability and basin stability inWatts-Strogatz (WS) networks of chaotic oscillators. a: Expected synchronizability R versus the WS model's parameter p. The scale of the y-axis was reversed to indicate improvement upon increase in p. b: Expected basin stability S versus p. The grey shade indicates one standard deviation. The dashed line shows an exponential fitted to the ensemble results for p > 0.15. Solid lines are guides to the eye. The plots shown were obtained for N = 100 oscillators of Roessler type, each having on average k = 8 neighbours. Choices of larger N and different k produce results that are qualitatively the same.
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Topological comparison of ensemble results with real-world networks. - Circle represents the results for Watts-Strogatz networks with N = 100, k = 10 and rewiring probability p (increasing from left to right 0.05…1.0). - Circle's area proportional to expected basin stability S. - Circle's colour indicates ex- pected synchronizability R. - Squares represent real-world networks reported to display a small-world topology.
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Social Networks: Human Communication
• Short Message correspondence (SMS) • Data record: 1.5 Mio SMS (times sent to
whom)
• User A (blue) and B (red ) – sending-response pattern
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Statistics • 50 % of users sent > 90 % of messages to
ONE partner concentrate on pairs of users • Distribution of the inter-event time P(t) bimodal:
power law on small scale (2 – 20 min) exponential tail (20 min – 6 hours)
• Not a finite-size effects (no power-law heavy tails)
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Distribution of inter-event time (one person sent)
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Phenomenon and modeling
• Typical sequence: Burst initiated by one user (exponential) Followed by a frequent mutual communication
(power law) • Model: Interacting priority queues of two persons • Parameters: processing time (adding/removal
of tasks), random initiation of tasks, interaction probability, waiting time
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empirical data (blue) vs. fitted model (red)
PNAS 107, 18803 (2010)
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System Earth
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System Earth - Horrendogram
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System Earth
Subsystem: Climate
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Network Reconstruction from a continuous dynamic system (structure vs. functionality)
New (inverse) problems arise!
Is there a backbone underlying the climate system?
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1st Step 2D:
Analysis of Earth´s Surface - Temperature Data
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Climate Networks Observation sites Earth system
Time series
Climate network
Network analysis
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Betweenness Betweenness Centrality B
Number of shortest paths that connect nodes j and k
Number of shortest paths that connect nodes j and k AND path through node i
Local betweenness of node i
(local and global aspects included!)
Betweenness Centrality B = < >
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Infer long-range connections –
Teleconnections
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Complex network approach to climate system
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Earth is 3D 2nd Step:
Include Different Layers of
the Atmosphere
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Analysing the vertical dynamical structure of the Earth´s atmosphere
• Data: pressure data at 17 different geopotential heights
• Challenge: how to describe such interacting networks? –
Network of Networks (originally used for brain modeling, PRL 2006) – but here different approach
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Network of networks
-Links inside a subnetwork
- Links between different subnetworks
-New measures: cross-degree, cross-clustering, cross-pathways…
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Modified Measures • Cross-average path length Lij – average
length of shortest paths between two subnetworks Gi and Gj
• Cross-degree centrality kv
ij – number of neighbours a node v, which is in Gi, has in Gj
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Cross-clustering coefficient Cvij –
frequency that two randomly drawn neighbours in Gj of node v, which is in Gi, are also neighbours
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Network analysis to identify large scale atmospheric motions
and as a visualisation tool
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Asian Summer Monsoon
Influence
Indian Summer Monsoon (ISM) on
Eastasian Summer Monsoon (EASM)
Clim. Dynamics (2012)
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Little Ice Age
- ISM weaker - Weaker influence
on EASM - Many short
connections (inside China)
Data White – Stalagmites Orange – Tree rings Pink – Ice core
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Recent Warm Period
- Few short connections
- Strong influence
of ISM on EASM
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Recent Monsoonal Rainfall over South Asia
Strongest daily local events
Method: Event Synchronization
and Complex Networks
Clim. Dynamics (2011), Europhys. Lett. (2012)
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Data
• Asian Rainfall highly resolved observational data integration towards the evaluation of water resources (APHRODITE)
• 1951-2007 • Resolution: 55 km • Daily data • Summer period (JJAS – June - September)
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Daily rainfall amount for
threshold 90% (1951-2007)
- strong events
orographic barriers
(Himalaya)
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Annual variability
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Network construction
• Event synchronization – m-th strong event occurs at grid positions i and j at time t(m,i) and t(m,j)
• number of times an event occurs at i after it appears in j
• strength of event synchronization
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Network construction
• Adjacency matrix
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Distribution of degree centrality
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Spatial distribution of the different „components“ of P(k)
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Local clustering coefficient Implies Predictability
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Conclusions • Complex network approach gives (new)
insights into the spatio- (temporal) organization of a complex system as climate
• Helpful vizualization tool but also enables us to identify critical (vulnerable) regions, anomalous behaviour (Monsoon years), to uncover major transport (of moisture), to evaluate predictability of strong events
• Useful for evaluation of forecast techniques? • Many open problems from methodological and
applied viewpoints
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Our papers on climate networks •Europ. Phys. J. Special Topics,174, 157-179 (2009) •Europhys. Lett. 87, 48007 (2009) •Phys. Lett. A 373, 4246 (2009) •Phys. Rev. E 81, 015101R (2010) •New J. Phys. 12, 033025 (2010) •Phys. Rev. Lett. 104, 038701 (2010) •Geoph. Res. Lett. 38, L00F04 (2011) •PNAS 108, 202422 (2011) •Europ. Phys. J. B 10797-8 (2011) •Climate Dynamics 39, 971 (2012) •Europhys. Lett. 97, 40009 (2012) •Phys. Rev. Lett. 108, 258701 (2012) •Climate Dynamics DOI 10.1007/s00382-012 (2012) •Climate Past (in press, 2012)
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Conclusions • Complex network approach gives (new)
insights into the spatio- (temporal) organization of complex systems as climate, brain, power grid,communication…
• Enables us to identify critical (vulnerable) regions, anomalous behaviour (Monsoon years), to evaluate predictability of strong events
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Outlook • Network of networks approach promising for
describing/modelling important aspects of Earth System´s Dynamics
• However, this is in the beginning and there are basic open questions, e.g.:
- How to connect different subsystems (low change in climate – fast changes in society)? - Role of critical points (tipping)? - Many more: to discuss in working group!
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Tipping points (Lenton et al., 2008)
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