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