Synchronization over networks:dynamics and graph structure
Mauricio Barahona
Institute for Mathematical Sciences&
Department of Bioengineering
Mar 13, 2008
Ecosystem food webs
Protein metabolism
Genetic regulation
Internet,WWW
Networks everywhere
What do these graphs mean?
Sometimes just a ‘representation’
In other cases:• Logical relations• Observed correlations• Model-based inferred dependences• Physical connections
Recently, lots of work on characterizingstructural properties of networks
with common trends in manynatural & man-made networks
Deterministic/Constructive (Pure) Random
(Constrained) Randome.g. scale -free
Small-worlds
Exploring graph structure
Some structural graph properties
However, these data correspond to dynamical processes
Effect of structure on dynamics?
Not straightforward problem with non-intuitive results
It depends strongly on the type of dynamical processe.g. Cascades along shortest paths? Highly connected nodes
as potential lack of robustness? Not necessarily
How do properties such as, e.g.,
• Power law degree distributions• Diameter or average distance• Clustering and neighbourhoods• Number of closed paths
translate into dynamical effects?
Similarities beneath the surface?
Electric grid of Upstate New YorkPart of the mammalian cell cycle
Dynamics on networks
Interrelation between structural properties of the graphand dynamics of a process taking place on a fixed network
In particular: how does the interconnection arrangement affectthe collective dynamics of a network of vertices with internaldynamics?
Types of systems:• Systems of coupled nonlinear ODEs (oscillatory, biochemical,
ecological)• Stochastic processes on graphs• Systems of PDEs coupled through algebraic constraints
For each particular system, deduce which graph properties“matter” for stability, robustness, control
Dynamics on networks: Synchronization
Mathematical tools:
• Graph theory, especially spectral graph theory
• Dynamical systems, including Lyapunov stability theory
• Semidefinite programming and optimization
In our case, synchronization of oscillators over networks.
Specifically, described by coupled ODEs.
What properties of graphs make networks of oscillatorseasier (or more difficult) to synchronize?
Acknowledgements
Collaborators:Elias August (PhD), now at OxfordLou Pecora, NRL, WashingtonAli Jadbabaie, UPenn
Helpful comments from:Pablo Parrilo, MITSteve Strogatz, Cornell
Funding:EPSRC, UKRoyal Society
Nonlinear oscillators can entrain to external drives
Nonlinear oscillators can phase-lock and achieve completesynchronization even if they have different natural frequencies
Synchronization of coupled oscillators
Examples• Biological: yeast, algae, fireflies, crickets• Physiological: heart, brain, menstrual cycle• Biochemical: cellular clocks, genetic circuits• Engineering: electronic circuits, power networks
Huyghens (1665)
Synchronization of coupled oscillators:Kuramoto model on arbitrary, finite graphs
If connections are all-to-all (complete graph):
The mean-field approach “decouples” the problem into interaction ofindividual oscillators and the mean-field state.
When N is infinite, many results on synchronization (sharp transitionsfor onset and total synchronization).
However, few results for finite N and arbitrary connections
Synchronization of coupled oscillators:Kuramoto model on arbitrary, finite graphs
Simple case: all oscillators are identical
Theorem: When natural frequencies are the same all oscillatorswill exponentially synchronize and the rate of approach tosynchronous state is determined by λ2(L), the algebraicconnectivity of the graph
B = incidence matrix of the graph
Synchronization of coupled oscillators:Kuramoto model on arbitrary, finite graphs
• When ω=0, is an asymptotically stable fixed point.
• is a Lyapunov function,measuring velocity misalignment.
λ2(L) determines the speed of synchronization
Synchronization of coupled oscillators:Kuramoto model on arbitrary, finite graphs
• When the frequencies are non zero, no fixed point for small values ofcoupling.
i.e., N finite, small K, no partial synchronization for fixed values ofinitial frequencies.
• Theorem: For K larger than a value depending on spread of frequencies,oscillators synchronize
• For such values of K, can prove nontrivial bounds on r
Synchronization of coupled oscillators:Kuramoto model on arbitrary, finite graphs
Bound and actual onset of synchronizationN= 100, e= 2443
Coupling strength K
Ord
er P
aram
eter
, a
vera
ged
over
tim
e an
d ω
Bound on
critical coupling
Synchronization dynamics: definitions
Individual dynamics affects collective dynamics:
• Identical vs. non-identical oscillators• Complete sync vs phase locking• Periodic vs non-periodic individual systems
Network of N systems each with n-dimensional dynamics
Synchronization dynamics: definitions
Connectivity is given by:• Which nodes are connected: Laplacian matrix (NxN) = - C• What variables act as input-output: Output matrix (nxn) = D
Synchronization dynamics: definitions
If all oscillators are identical, the dynamics for the system can be written as:
and ⊗ is the Kronecker product
And the synchronized state is given by:
Necessary conditions for local stability
Idea: consider perturbations around the totally synchronized state
Construct variational equation and find the minimal perturbation that can destabilize the system
Eigendirections of the Laplacian
maximal Lyapunov exponent along attractor of uncoupled system as a function of α
Master Stability Function:
Master stability functions: typical cases
α1 α2
-0.8
-0.4
0.0
α
-1.2
λmax
151050
Criterion: The whole spectrum of the graph must fitin the negative region of the MSF
Interesting when there are two crossings!
Synchronization of oscillatory systems
• Decouple dynamics at each node from ‘graph’component (Kronecker-type structure)
• If there are two crossings, linear stability ofsynchronized state is related to an algebraiccondition on the spectrum of the Laplacian of thegraph:
• Gives bounds on the synchronizability of a network
Study: different ‘topologies’, small-world effect,scale-free synchronization
Graph structure and synchronization
Small-worlds
Small-world does notguarantee
synchronizability
Small-world does not guarantee synchronizability
Adding edges througha small-world schemeis an efficient way of
reachingsynchronizability
Small-worlds areedge-efficient
compared to othergraphs
Several other results, e.g., scale-free graphs are not easilysynchronizable
Sufficient conditions for global stability
Provide guarantees that the synchronized state will be reachedfrom any initial condition in the state space
Important for design problems and when systems are noisy
Sufficient conditions for global stability
Basic idea:
Use Lyapunov stability theory to obtain (conservative) sufficientconditions for the stability of the synchronized state based onpositivity conditions
We have set up a computational algebra methodology to searchfor Lyapunov certificates of global stability for polynomial/rationalsystems
This allows us to optimize the conditions using SDP (semidefiniteprogramming)
Sufficient conditions for global stability:Contraction
(Results from Hale, Slotine, Wu, Belykh)
Idea: Define a metric such that the flow contracts and trajectories get closer
Sufficient conditions for global stabilityContraction - computability
Maps onto a feasibility problem:
Sufficient conditions for global stabilityComputational examples
Computational bounds for Lorenz system (all-to-all)
117125529
101010
525
2
No search for
P (literature)
Sum of Squares
SOSTOOLSSDP
(YALMIP)
223251K*
22N
Sufficient conditions for global stabilityComputational examples
Computational bounds for coupled identical repressilators
Sufficient conditions for global stabilityNon-identical repressilators
Frequency synchronization
Sufficient conditions for global stability based onBendixson’s criterion for higher dimensions
Maps onto a feasibility problem:
Secondadditive
compound
Sufficient conditions for global stability based onBendixson’s criterion for higher dimensions
• Computational bounds for Lorenz system (all-to-all)
• Bounds for coupled van der Pol (all-to-all)
Summary
• Synchronization is a good example of dynamics on networks, wherenetwork properties play an important non-trivial role on the globalproperties
• The methods presented ‘factorize’ the graph and the vertex dynamicsand parameterize the effect of one on the other
• The presented conditions for synchronization (necessary local andsufficient global) have different applications depending on the systemof use
• The results involve the use of techniques from spectral graph theory,dynamical systems and matrix optimization.
• There is a very wide range of problems to pursue in this area
Thank you