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Identification and Monitoring of Identification and Monitoring of Complex Networks Based on Complex Networks Based on Synchronization Synchronization Wallace Tang Wallace Tang Department of Electronic Engineering Department of Electronic Engineering City University of Hong Kong City University of Hong Kong International Workshop on Complex Systems and Networks 2007 July 19-21, 2007 Guilin, China

Outline Outline � Introduction Introduction � � The Target The Target � � Get Ready: Global Adaptive Observer Design Get Ready: Global Adaptive Observer Design � � Move Forward: A Proposal of Design Move Forward: A Proposal of Design � � Another Direction: Evolutionary Algorithm Another Direction: Evolutionary Algorithm � � Conclusions Conclusions �

Introduction Introduction � Complex networks: biological neural network, power system, Complex networks: biological neural network, power system, � network of chaotic oscillators, … network of chaotic oscillators, … � Statistical models: Small Statistical models: Small- -world networks, scale world networks, scale- -free networks, free networks, � random networks random networks � However, uncovering the exact topology and model of a However, uncovering the exact topology and model of a � targeted network is difficult. targeted network is difficult. � Useful for understanding the behaviour over the network, Useful for understanding the behaviour over the network, � reporting any connection failure, … reporting any connection failure, …

Example: Biological Neural Network Example: Biological Neural Network � A neuron is a nerve cell. A neuron is a nerve cell. � � Neurons have specialized projections Neurons have specialized projections � called dendrites and axons. Dendrites called dendrites and axons. Dendrites bring information to the cell body and bring information to the cell body and axons take information away from the axons take information away from the cell body. cell body. � Neurons communicate with each other Neurons communicate with each other � through through � synaptic coupling � synaptic coupling (an electrochemical (an electrochemical process) process) � electrical coupling � presynaptic electrical coupling dendrite of terminal axon receiving neuron

HR Neuron Model HR Neuron Model � H Hindmarsh indmarsh- -Rose (HR) neuron model Rose (HR) neuron model � rd order dynamical e 3 rd � 3 order dynamical equations quations � Tonic bursting = = − − − & 2 3 x ( t ) f ( x , y , z ) ax x y z x = = + α − & 2 y ( t ) f ( x , y , z ) ( a ) x y y = = µ + − & z ( t ) f ( x , y , z ) ( bx c z ) z where x(t) is the membrane potential y(t) and z(t) are the recovery variables w.r.t. fast and slow current, respectively E.M. Izhikevich, “Which model to use for cortical spiking neurons,” a=2.8, b=9, c=5, α =1.6, µ =0.001 IEEE Trans. Neural Network, 15(5), pp. 1063-1070, Sept 2004.

Model of BNN Model of BNN � Biological Neural Network: A set of connected neurons Biological Neural Network: A set of connected neurons � � E Evolution of its nodes volution of its nodes with coupling with coupling: : � N ∑ = − − − − γ 2 3 & x ( t ) a x x y z g i i i i i i ij ij = j 1 = + α − 2 & y ( t ) ( a ) x y i i i i i = µ + − & z ( t ) ( b x c z ) i i i i i i − x V γ = x − γ = i s x where for electrical coupling, or − − θ ij + ij j i v ( x ) 1 e j s > g 0 for synaptic coupling. If neurons i and j are ij = g 0 connected to each other, and otherwise. ij g = g Note: ij ji

Example: Network of Chaotic Oscillators Example: Network of Chaotic Oscillators � Collection of chaotic oscillators Collection of chaotic oscillators � 2 1 3

The Target The Target Estimate the states and parameters of Estimate the states and parameters of a complex network a complex network Imprecise models Imprecise models (Unknown parameters, Unknown topology) (Unknown parameters, Unknown topology) Limited observable states Limited observable states

Problem Formulation Problem Formulation Ω A complex network, Ω � A complex network, � observable = ⎧ x F(x, p) & output Ω ⎨ : = Cx ⎩ y x ∈ R n the state vector model F nonlinear smooth function y the observable output C a constant matrix Φ s.t. Φ and Ω are synchronized. Design another system Φ s.t. Φ and Ω � Design another system are synchronized. � & ⎧ = x F( x , p ) ⎪ ˆ ˆ ˆ , y Φ ⎨ : & → x x ˆ ⎪ = → ∞ p R( x , p ) ⎩ ˆ ˆ ˆ , y when t → p p ˆ

Possible Ways to Go Possible Ways to Go � Known p Known p � Synchronization problem Synchronization problem � Unknown p Unknown p � � adaptive observer adaptive observer design design � � Error minimization problem Error minimization problem � � Gradient descent methods (Quasi Gradient descent methods (Quasi- -Newton, Powell minimization Newton, Powell minimization � ⇒ Local optimal algorithm) ⇒ Local optimal algorithm) � Optimization: Optimization: Evolutionary Algorithms Evolutionary Algorithms, Particle Swarm Optimization , Particle Swarm Optimization �

Get Ready … Get Ready … Adaptive Observer Adaptive Observer for joint state- -parameter estimation of a targeted system parameter estimation of a targeted system for joint state

A Global Adaptive Observer Design A Global Adaptive Observer Design � Consider the following system Consider the following system � m ( ) ( ) ( ) ∑ i = + ≡ x x F x, p & x c p f i i ij ij i = 1 j F where i=1,2,…,n i=1,2,…,n and are smooth functions and are smooth functions where i � Assumptions Assumptions � m ( ( ) ) ∑ i = = ⇒ = = = x L L L r f t 0 , i 1 , n r 0 , i 1 , n , j 1 , m � � ij ij ij i = j 1 ( ) = < L is the output of the system em , , , , is the output of the syst y x x x k n � � 1 2 k = = L L are unknown parameters rs are unknown paramete p , i 1 , 2 , , k ; j 1 , 2 , , m � � ij i ( ) 1 ( ) ( ) ∃ & = + = T e e e x F x , p u x , u s.t. for the system , we have , s.t. for the system , we have , ˆ ˆ ˆ V y � � 1 2 = ˆ − e x x evaluated along the solution of error dynamics, and and evaluated along the solution of error dynamics, ( ) & e is negative definite. V is negative definite. 1

A Global Adaptive Observer Design A Global Adaptive Observer Design � Adaptive Observer: Adaptive Observer: � m ( ) ( ) ( ) ∑ i & = + + x x x ˆ ˆ ˆ ˆ ˆ x c p f u , y i i ij ij i = 1 j ( ) ( ) & = − δ − = x L ˆ ˆ ˆ p x x f i 1 , 2 , , k ij ij i i ij m n k 1 1 1 ∑ ∑∑ i = − = − = + � Let where Let where 2 2 ˆ ˆ e x x , r p p � V e r i i i ij ij ij δ i ij 2 2 = = = i 1 i 1 j 1 ij & = & By numerical cancellation, it can be proved that By numerical cancellation, it can be proved that V V 1 m n k ∑ ∑ ∑ i & & = − − x & ˆ ˆ V e ( x x ) e r f ( ) i i i i ij ij = = = i 1 i 1 j 1 m m n k k [ ] ∑ ∑ ∑ ∑ ∑ i i & = + − + − = F x , p F x, p x x ˆ ˆ ˆ e ( ) u ( ) e r f ( ) e r f ( ) V i i i i i ij ij i ij ij 1 = = = = = i 1 i 1 j 1 i 1 j 1 → � Based on the equation for the error dynamics, and Based on the equation for the error dynamics, and e 0 � i m ( ( ) ) ∑ i p = ⇒ it follows that ⇒ = x ˆ p it follows that r f t 0 ij ij ij ij = j 1

Identify and monitor and monitor Identify the topology of a BNN the topology of a BNN with limited observable states with limited observable states