Controllability and Identification of Complex Networks€¦ · Controllability metrics: how much energy to steer the network, strategies G. Yan et al(PRL 12): how much energy is needed

Controllability and Identification of Complex Networks

Jorge CortesUNIVERSITY OF CALIFORNIAUNOFFICIAL SEAL

Attachment B - “Unofficial” SealFor Use on Letterhead

Mechanical and Aerospace EngineeringUniversity of California, San Diegohttp://carmenere.ucsd.edu/jorge

IMA Annual Program Year WorkshopAnalysis and Control of Network Dynamics

University of MinnesotaOctober 23, 2015

Joint work with Yingbo Zhao

http://carmenere.ucsd.edu/jorge

Complex interconnected systems

Networked systems play increasingly essential role in modern society

power networks

wireless communication networks

social networks

biological networks

transportation networks

sensor&robotic networks

US power grid Google Loon Ocean science

Seek to understand how complex networks work, how individual(s) affect networkbehavior, how robust network is against external disruption and shocks,...

J. Cortes (UC San Diego) Controllability and Identification of Complex Networks October 23, 2015 2 / 40

Network science and distributed control

Challenging goals, different flavors

Modeling behavior and understanding interactions in large-scale networks

Design provably correct distributed coordination algorithms to make networkachieve desired task

Analyze properties of complex networks, mutual effect between dynamics andtopology

Focus today on analysis of network reachability and identification

Understanding how to drive the network state efficiently by controlling only afew selected components

Identifying network structure from input/output data

Plethora of other relevant issues: understanding noise propagation, fragility,optimizing placement of limited resources for enhancing network properties


Network science and distributed control

Challenging goals, different flavors

Modeling behavior and understanding interactions in large-scale networks

Design provably correct distributed coordination algorithms to make networkachieve desired task

Analyze properties of complex networks, mutual effect between dynamics andtopology

Focus today on analysis of network reachability and identification

Understanding how to drive the network state efficiently by controlling only afew selected components

Identifying network structure from input/output data

Plethora of other relevant issues: understanding noise propagation, fragility,optimizing placement of limited resources for enhancing network properties


Complex network as linear control system

Linear control systems provide simplest (but not simple!) model

x(k + 1) = Ax(k) + Bu(k)

each component of x corresponds to the state of a node

network adjacency matrix A

each column of B only affects a specific node




x(k + 1) = Ax(k) + Bu(k)




Deciding controllability: how many nodes, how robust, influence of topology

Y. Liu, J. Slotine, and A. Barabasi (Nature 2011): structuralcontrollability and degree distribution

A. Olshevsky (TAC 14): minimal controllability problems

A. Rahmani, M. Ji, M. Mesbahi, and M. Egerstedt (SIAM JCO, 09):controllability of consensus-type linear networks

C. Aguilar and B. Gharesifard (ACC 14): controllability ofconsensus-type nonlinear networks




x(k + 1) = Ax(k) + Bu(k)




Controllability metrics: how much energy to steer the network, strategies

G. Yan et al (PRL 12): how much energy is needed to control complexnetworks

F. Pasqualetti, S. Zampieri, and F. Bullo (IEEE TCNS 14):Controllability metrics, asymptotic regimes




x(k + 1) = Ax(k) + Bu(k)




Actuator and sensor placement: node location

T. Summers and J. Lygeros (IFAC World Congress 14): optimal sensorand actuator placement

V. Tzoumas, M. A. Rahimian, G. J. Pappas, A. Jadbabaie (ACC15): minimal actuator placement with optimal control constraints

...


Effective connectivity in the brain

Brain areas with different functions represented as nodes, brain modeled asnetwork of nodes

Effective connectivity: external inputs not only have direct effect on the state ofthe brain in a particular area, but can also activate the connections amongdifferent brain areas


Complex network as bilinear control system

Bilinear control systems provide next (in the complexity scale) simplest model

x(k + 1) = Ax(k) + Bu(k) + x(k)TFu(k)

Inputs not only affect the state of individual nodes, but can also modulateagent-to-agent interconnection

1 11 1

13 2 3 11 3

( 1) ( )

( ) ( )) )( (u k f

x k a x k

a x k u k

+ = +

+ +

21a

2 22 2

21 1 2

( 1) ( )

( ) ( )

x k a x k

a x k u k

+ = +

+

3 32 2

33 3 3

( 1) ( )

( ) ( )

x k a x k

a x k u k

+ = +

+

32a

13a2( )u k

13f

Akin to “dynamic causal modeling” in neuroscience: K. Friston, L.

Harrison, W. Penny (Neuroimage, 03)


Outline

1 Reachability metrics for bilinear control systemsReachability GramianGramian-based reachability metrics and actuator selectionAdding bilinear inputs to linear networks

2 Network identification with latent nodesManifest and latent nodesAsymptotically exact identification via auto-regressive modelsLeast-squares auto-regressive estimation


Reachability metrics for bilinear control systems

How easy/difficult to control is a given bilinear control system?How much energy is needed to steer the system to desirable state?Do bilinear systems offer any advantage compared to linear ones?

6 7

67 7( ) or ( )?u k u k

1

5 3

4

21( )u k4( )u k

Given desired performance and constraints (sparsity, limited inputenergy), is it better to control a node state directly or modulate(strengthen, weaken, or create) an interconnection?


Controllability of bilinear systemsIncomplete literature review

Most available results study controllability as a binary property

W. Boothby and E. Wilson (SIAM JCO, 79): determination of transitivity

D. Koditschek and K. Narendra (IEEE TAC 85): controllability of planarcontinuous-time bilinear systems

U. Piechottka and P. Frank (Automatica 92): uncontrollability test for3-d continuous-time systems

M. Evans and D. Murthy (IEEE TAC 77), T. Goka, T. Tarn, and J.

Zaborszky (Automatica 73): controllability of discrete-timehomogeneous-in-the-state bilinear system using rank-1 controllers

L. Tie and K. Cai (IEEE TAC 10): near controllability of discrete-timesystems

Missing quantitative measures to ascertain energy amount required to steer state


Minimum-energy optimal control problem

Minimum energy required to steer state from 0n to desired xf in K time steps

min{u}K−1

∑K−1k=0 uT (k)u(k)

s.t. ∀k = 0, . . . ,K − 1,x(k + 1) = Ax(k) +

∑mi=1(Fix(k) + Bi )ui (k)

x(0) = 0n, x(K ) = xf .

Necessary optimality conditions for solution lead to nonlinear two-pointboundary-value problem without known analytical solution

For linear systems (F = 0), optimal solution and value

u∗(k) = BT

(AT

)K−k−1W−11,Kxf

K−1∑k=0

u∗(k)Tu∗(k) = xTf W−11,Kxf

(W1,K ,∑K−1

k=0 AkBBT (AT )k is K -step controllability Gramian)


Gramian-based reachability metrics for linear networks

Define controllability Gramian

W1 = limK→∞

W1,K =∞∑k=0

AkBBT (AT )k

For arbitrary time horizon K , lower bound on minimum input energy

K−1∑k=0

(u∗(k))Tu∗(k) = xTf W−11,Kxf > xTf W−1

1 xf

Gramian-based quantitative reachability metrics

λmin(W1): worst-case minimum input energytr(W1): average minimum control energy over {x ∈ Rn|‖x‖= 1}det(W1): volume of ellipsoid reachable using unit-energy control inputs


Reachability Gramian for bilinear control systems

Even though bilinear systems are not linear, analogous concept exist

For stable bilinear system (A,F ,B), define recursively

W1 =∞∑

k1=0

Ak1BBT (Ak1 )T

Wi =∞∑

ki=0

Aki( m∑j=1

FjWi−1FTj

)(Aki )T , i ≥ 2

Reachability Gramian is

W =∞∑i=1

Wi


Properties of the reachability Gramian

Reachability Gramian W is positive semi-definite

Reachability Gramian W satisfies generalized Lyapunov equation

AWAT −W +m∑j=1

FjWFTj + BBT = 0n×n

Unique positive semi-definite solution exists to Lyapunov equation iff

ρ(A⊗ A +m∑j=1

Fj ⊗ Fj) < 1

Im(W) is invariant under bilinear control dynamics (A,F ,B)—hence any state xf reachable from the origin belongs to Im(W)


Lower bound on minimum input energy∗

Gramian-based lower bound for bounded inputs

For arbitrary time horizon K , if

‖u(k)‖∞ ≤1

2

( m∑i,j=1

‖FTj ΨFi‖

)−1β, k = 0, 1, . . . ,K − 1

then input energy is lower bounded as

K−1∑k=0

uT (k)u(k) ≥ xT (K )W−1x(K )

input u

state x

Ψ ,W−1 −W−1B(BTW−1B − Im)−1BTW−1

β , −m∑j=1

‖AT ΨFj + FTj ΨA‖+(( m∑

j=1

‖AT ΨFj + FTj ΨA‖)2 − 4

m∑i,j=1

‖FTj ΨFi‖·λmax(AT ΨA −W−1))1/2


Lower bound on minimum input energy – cont’d

input u state x

Similar bound for continuous-time bilinear systems

W. Gray and J. Mesko (IFAC NCSDS 98) underintegrability condition

P. Benner and T. Damm (IJC 11): underdiagonal reachability Gramian

Bound on input energy does not hold if input norm is unconstrained

input u state x

inf

∑K−1k=0 uT (k)u(k)

‖x(K )‖2= 0


Gramian-based reachability metrics

Lower bound on minimum input energy justifies following reachability metrics

λmin(W): minimum input energy in the worst casetr(W): average minimum input energy over unit hypersphere in state spacedet(W): volume of the ellipsoid containing the reachable states using

inputs with no more than unit energy

Plenty of (open) questions

how does Gramian and metrics depend on network?

what networks are better?

how robust are these properties against malfunctions/failures?

how to select inputs to make network better?

. . .


Actuator selection

Consider M candidate actuators, labeled V = {1, . . . ,M}Actuator i associated to pair (bi ,Fi ) ∈ Rn × Rn×n

Actuator selection: select m ≤ M actuators S = {s1, . . . , sm} for

maxS⊆V

f (W(S)) (f is either tr, λmin or det)

Increasing returns property of map from S to W(S)

For any S1 ⊆ S2 ⊆ V and s ∈ V \S2,

W(S2 ∪ {s})−W(S2) ≥ W(S1 ∪ {s})−W(S1)

Maximization of supermodular functions under cardinality constraints is NP hard


Actuator selection

Consider M candidate actuators, labeled V = {1, . . . ,M}Actuator i associated to pair (bi ,Fi ) ∈ Rn × Rn×n

Actuator selection: select m ≤ M actuators S = {s1, . . . , sm} for

maxS⊆V

f (W(S)) (f is either tr, λmin or det)

Increasing returns property of map from S to W(S)

For any S1 ⊆ S2 ⊆ V and s ∈ V \S2,

W(S2 ∪ {s})−W(S2) ≥ W(S1 ∪ {s})−W(S1)

Maximization of supermodular functions under cardinality constraints is NP hard


Bounds on optimal value

Upper bound on optimal value: Lagrangian dual and continuous relaxationsolvable in polynomial time [G. Gallo and B. Simeone (Math. Prog. 89)]

Lower bound on reachability metrics

Let f be either tr, λmin or det. For any m actuators S ,

f (W(S)) ≥∑s∈S

f (W(s))

Greedy algorithm: maximize lower bound by

1 compute f (W(s)) for every s ∈ V

2 order results in decreasing order

3 select actuators sequentially starting with the one with largest value


Example: controller selection via greedy algorithm

Baseline system on R5: (A,F0, b0)

b = (0.8, 0.6, 0.4, 0.2, 0.5), F = diag(0.1, 0.2, 0.3, 0.4, 0.5)

Additional actuator candidates: {(Bi ,Fi )}3i=1, Bi = ei ,

F1 =

0 0.02 0 0 00 0 0.02 0 00 0 0 0 00 0 0 0 00 0 0 0 0

, F2 =

0 0 0 0 00 0 0 0 0.010 0 0 0 00 0.05 0 0 00 0 0 0 0

, F3 =

0.05 0 0 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0.020 0 0 0 0

S tr(W (S)) λmin(W (S)) det(W (S)) S tr(W (S)) λmin(W (S)) det(W (S))

{0} 14.42 0.027 0.242 {0, 2} 19.91 0.07 3.32

{1} 5.03 0.023 0.025 {0, 3} 18.69 0.05 1.13

{2} 4.04 3 × 10−5 9 × 10−7 {0, 1, 2} 26.50 0.137 46.15

{3} 3.03 1.6 × 10−6 4 × 10−11 {0, 1, 3} 25.28 0.125 28.68

{0, 1} 20.98 0.09 11.704 {0, 2, 3} 24.19 0.103 8.34

Lower bound is tight for tr, but not for λmin and det

Actuators with a large individual contribution are involved in combinatorialchoices with large values


Outline




Asymptotic behavior of worst-case minimum input energy

Difficult-to-control (DTC) networks

A class of networks is difficult to control if, for fixed number of inputs,

limn→∞

supxf∈Rn

inf{u}∞:u(k)∈Rm

‖{u}∞‖2

‖xf ‖2→∞

i.e., worst-case minimum input energy grows unbounded with network size

For linear networks (A(n), 0n×nm,B(n)),

supxf∈Rn:‖xf ‖2=1

inf{u}∞:u(k)∈Rm

‖{u}∞‖2 = λ−1min(W1(n))

Stable symmetric linear networks are DTC [Pasqualetti et al (TCNS 14)]

λ−1min(W1(n)) increases exponentially with rate n

m for any choice ofB(n) ∈ Rn×m whose columns are canonical vectors in Rn


Asymptotic behavior of worst-case minimum input energy

Difficult-to-control (DTC) networks

A class of networks is difficult to control if, for fixed number of inputs,

limn→∞

supxf∈Rn

inf{u}∞:u(k)∈Rm

‖{u}∞‖2

‖xf ‖2→∞

i.e., worst-case minimum input energy grows unbounded with network size

For linear networks (A(n), 0n×nm,B(n)),

supxf∈Rn:‖xf ‖2=1

inf{u}∞:u(k)∈Rm

‖{u}∞‖2 = λ−1min(W1(n))

Stable symmetric linear networks are DTC [Pasqualetti et al (TCNS 14)]

λ−1min(W1(n)) increases exponentially with rate n

m for any choice ofB(n) ∈ Rn×m whose columns are canonical vectors in Rn


Does interconnection modulation help with DTC networks?

1mu− m

u1u

2u

Symmetric linear line networks with finitely many controlled nodes are DTC

1v

pv

1u

2u

m pu−

Symmetric linear line networks with finitely many controlled nodesAND finitely many controlled interconnections


Does interconnection modulation help? Negative results

Symmetric linear networks remain difficult to control after

addition of finitely many bilinear inputs

with bounded homogeneous self-loop modulation






DTC symmetric linear networks w/ finitely many bilinear inputs

Consider a class of DTC symmetric linear networks (A(n), 0n×nm,B(n))and let the number of nonzero entries in F (n) ∈ Rn×nm and‖F (n)‖max = maxi,j |Fij(n)| be uniformly bounded (with respect to n)

Then the class of bilinear networks (A(n),F (n),B(n)) is DTC






DTC symmetric linear networks w/ self-loop modulation

Consider a class of DTC symmetric linear networks (A(n), 0n×nm,B(n))

with ρ(A(n)) <√

1− T−1m (Tm ,

⌈nm

⌉− 1)

Then the reachability Gramian of the class of bilinear networks(A(n), αIn,B(n)) with |tr(αIn)| ≤ µ satisfies, for any n > m−1µ2,

λmin(W) ≤ (1− Tmα2)−1

1− ρ2(A)− T−1m

ρ2Tm(A)

The term(1−Tmα

2)−1

1−ρ2(A)−T−1m

decreases in n and limn→∞(1−Tmα

2)−1

1−ρ2(A)−T−1m

= (1 − ρ2(A))−1


Line network with self-loop modulation

Line networks with adjacency matricesA(n) with

aij =

{0.25 if |i − j | ≤ 1

0 otherwise

m = 1 and | tr(αIn)| = µ = 0.9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15−18

−16

−14

−12

−10

−8

−6

−4

−2

0

2

Number of nodes (n)

log 1

0(λ

min(W

))

(A,0n×n, B) : log10(λmin (W1))

(A,αIn,B) : log10(λmin(W ))


Does interconnection modulation help? Positive example

Line networks (A(n),B(n)) with

aij =

{0.05 if |i − j | ≤ 1

0 otherwiseB(n) = e1

plus single bilinear input F (n) with

fij =

{1 if i = j + 1

0 otherwise1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

−40

−35

−30

−25

−20

−15

−10

−5

0

5

Number of nodes

log 1

0(λ

min(W

))

(A,0n×n, B) : log10(λmin (W1))

(A,F,B) : log10(λmin(W))

DTC linear network can be made easy to control by adding single bilinear inputaffecting infinite number of interconnections (with constant strength)


Outline




Identification of network topology from measured data

In many applications,network itself is unknown

Aim is to identify underlying dynamics or topology

1 2

1u 2u

3 4

3u 4u

1

2

3

4

( )

( )

( )

( )

u t

u t

u t

u t

1

2

3

4

( )

( )

( )

( )

x t

x t

x t

x t

Identify network using time series of stimulated node states

user may choose some of the inputs (control nodes) and measure some of thestates (observed nodes)

determine causal relationships among nodes: identify network transferfunction or adjacency matrix


Network topology identificationIncomplete literature review

If every node manipulated&measured, large number of experiments allowed

M. Timme (PRL 07): identify networks of coupled phase oscillators

M. Nabi-Abdolyousefi, M. Mesbahi (IEEE TAC 12): identifyconsensus-type networks using node knockout

S. Shahrampour, V. Preciado (IEEE TAC 15): identify directed LTInetworks using node knockout

W/o ability to manipulate every node, perform large number of experiments, it isdifficult to accurately identify network topology

J. Goncalves, S. Warnick (IEEE TAC 08): not possible to identify evenBoolean structure of a general network using its transfer function


Network topology identification-cont’dIncomplete literature review

Focus on particular network realizations (e.g., sparsest) that explain data

A. Julius, M. Zavlanos, S. Boyd, G. J. Pappas (Systems Biology09): genetic network identification using convex programming

D. Materassi, G. Innocenti, L. Giarre, M. Salapaka (SCL 13):trade-off between accuracy and sparsity

D. Materassi, G. Innocenti (IEEE TAC 10): identify tree networks

Our work inspired by wide use of auto-regressive (AR) models and time series toanalyze brain data via Granger causality

E. Bullmore, O. Sporns (Nature Reviews Neuroscience, 09): graphtheoretical analysis of structural and functional brain systems

S. L. Bressler, A. K. Seth (Neuroimage, 11): Wiener–Grangermethodology for determining causality

J. R. Iversen et al: (IEEE Engineering in Medicine and Biology, 14)Causal analysis of cortical networks involved in reaching to spatial targets


Network model: linear system

Discrete-time, linear time-invariant state-space representation

x(k + 1) = Ax(k) + Bu(k)

y(k) = Cx(k)

manifest nodes: those whose state can be observedlatent nodes: those whose state cannot be observed

Assume manifest nodes can be controlled individually and latent nodes are passive(no control input)

[xm(k + 1)xl(k + 1)

]=

[A11 A12

A21 A22

] [xm(k)xl(k)

]+

[um(k)

0

]y(k) = xm(k)


Network identification goal

Objective: reconstruct transfer function Txmum(ω) (from um to xm) formanifest subnetwork, absent knowledge of latent nodes

Characterize network behavior via auto-regressive (AR) model

Txmum(ω) : xm(k + 1) =τ−1∑i=0

Ai xm(k − i) + um(k)

How well can Txmum(ω) approximate original Txmum(ω)?


Asymptotically exact identification via AR model

AR transfer function converges to manifest transfer function

If (1− ‖A11‖)(1− ‖A22‖) > ‖A12‖ · ‖A21‖, then there exists γ ∈ R>0 such that,for any τ ∈ Z≥0, there is a matrix sequence A∗0 , · · · , A∗τ−1 ∈ Rnm×nm satisfying

‖Txmum(ω, τ)− Txmum(ω)‖∞ ≤ γ‖A22‖τ

In fact, one such sequence is given by

A∗0 = A11, A∗i = −A12Ai−122 A21, i ≥ 1







A∗0 = A11, A∗i = −A12Ai−122 A21, i ≥ 1

Stability margin vs interaction strength between manifest&latent subnetworks

as stability margin increasingly small, (1− ‖A11‖)(1− ‖A22‖) gets small

interaction strength between subnetworks (‖A12‖ · ‖A21‖) should beincreasingly small as well







A∗0 = A11, A∗i = −A12Ai−122 A21, i ≥ 1

Exact identification: for acyclic latent subnetworks, matrix sequence ensures

Txmum = Txmum


Direct versus latent interactions between manifest nodes

From linear state-space representation[xm(k + 1)xl(k + 1)

]=

[A11 A12

A21 A22

] [xm(k)xl(k)

]+

[um(k)

0

]

Given manifest nodes p and q, interaction is

direct iff A∗0(q, p) = A11(q, p) 6= 0

indirect (through latent nodes) iff A∗i (q, p) 6= 0 for some i ≥ 1

From A∗i = −A12Ai−122 A21, p first affects latent nodes (through

A21), then propagates through latent subnetwork (reflected byAi−1

22 ), and finally affects q (through A12)

If latent subnetwork is acyclic, A∗i (q, p) 6= 0 implies there are exactly i latentnodes in a path connecting p to q



From linear state-space representation

xm(k + 1) =k∑

i=0

A∗i xm(k − i) + A12Ak22xl(0) + um(k)










From linear state-space representation

xm(k + 1) =k∑

i=0

A∗i xm(k − i) + A12Ak22xl(0) + um(k)









Outline




Least-squares AR estimation

Manifest subnetwork can be reconstructed

without knowledge of passive latent nodes, but

computation of matrix sequence requires network adjacency matrix

Given input/output data

persistently exciting input: {um}N1 zero-mean stochastic process withi.i.d. random vectors

measure response data sequence {y}N1

Least-squares AR estimation: given measurement data {y = xm}N1 ⊂ Rnm ,

determine {A}τ−10 minimizing 2-norm of residual sequence {e}N−1

τ

e(k) = y(k + 1)−τ−1∑i=0

Aiy(k − i)


Least-squares AR estimation

Manifest subnetwork can be reconstructed

without knowledge of passive latent nodes, but

computation of matrix sequence requires network adjacency matrix

Given input/output data

persistently exciting input: {um}N1 zero-mean stochastic process withi.i.d. random vectors

measure response data sequence {y}N1

Least-squares AR estimation: given measurement data {y = xm}N1 ⊂ Rnm ,

determine {A}τ−10 minimizing 2-norm of residual sequence {e}N−1

τ

e(k) = y(k + 1)−τ−1∑i=0

Aiy(k − i)


Convergence of least-squares estimate

Least-squares estimate converges in probability

Least-squares estimate satisfies

‖plimN→∞ Aτ ({y}N1 )− A∗τ‖max ≤ βτ‖A22‖τ

β only depends on network adjacency matrix and input covariance σ2um

Direct/indirect interactions between manifest nodes can be ascertained from

A0 → A∗0 = A11 and Ai → A∗i = −A12Ai−122 A21, i ≥ 1

Least-squares estimate converges in mean square

If latent subnetwork is acyclic (Aτ2222 = 0nl×nl ), then for τ ≥ τ22 + 1,

least-squares estimate satisfies

limN→∞

E[(Aτ ({y}N1 )− A∗τ )T (Aτ ({y}N1 )− A∗τ )] = 0nmτ×nmτ


Convergence of least-squares estimate

Least-squares estimate converges in probability

Least-squares estimate satisfies

‖plimN→∞ Aτ ({y}N1 )− A∗τ‖max ≤ βτ‖A22‖τ

β only depends on network adjacency matrix and input covariance σ2um

Direct/indirect interactions between manifest nodes can be ascertained from

A0 → A∗0 = A11 and Ai → A∗i = −A12Ai−122 A21, i ≥ 1

Least-squares estimate converges in mean square


least-squares estimate satisfies

limN→∞

E[(Aτ ({y}N1 )− A∗τ )T (Aτ ({y}N1 )− A∗τ )] = 0nmτ×nmτ


Least-squares recovers manifest transfer function

Least-squares consistently estimates manifest transfer function

There exist positive scalars β, γ and τ0 such that, for τ ≥ τ0,

‖plimN→∞ Tye({y}N1 , τ)− Txmum‖∞ ≤ (βτ 2 + γ)‖A22‖τ

As a consequence, plimN→∞,τ→∞ Tye({y}N1 , τ) = Txmum

Critical model order τ0 increases with ‖A22‖:less stable latent subnetworks require higher order of the AR model

Exact identification for acyclic latent subnetworks


plimN→∞ Tye({y}N1 , τ) = Txmum


Least-squares recovers manifest transfer function

Least-squares consistently estimates manifest transfer function

There exist positive scalars β, γ and τ0 such that, for τ ≥ τ0,

‖plimN→∞ Tye({y}N1 , τ)− Txmum‖∞ ≤ (βτ 2 + γ)‖A22‖τ

As a consequence, plimN→∞,τ→∞ Tye({y}N1 , τ) = Txmum

Critical model order τ0 increases with ‖A22‖:less stable latent subnetworks require higher order of the AR model

Exact identification for acyclic latent subnetworks


plimN→∞ Tye({y}N1 , τ) = Txmum


Example: Erdos–Renyi random network

Group of G (10, 0.25) Erdos–Renyi random networks

3 randomly chosen manifest nodes each

edge weight with uniform distribution in (0, 0.35)

N = 105 data points

24

68

24

68

10

0.1

0.2

0.3

0.4

0.5

Model orderNetwork index

H∞ n

orm

err

or

1 2 3 4 5 6

0.05

0.1

0.15

0.2

0.25

Model candidate order

H∞ n

orm

err

or

LSAR estimationAR model in Theorem 1

‖Txmum − Txmum‖∞ Network w/ index 1 Comparison w/ opt matrix seq


Example: Erdos–Renyi random network

Group of G (10, 0.25) Erdos–Renyi random networks

3 randomly chosen manifest nodes each

edge weight with uniform distribution in (0, 0.35)

N = 105 data points

24

68

24

68

10

0.1

0.2

0.3

0.4

0.5

Model orderNetwork index

H∞ n

orm

err

or

1 2 3 4 5 6

0.05

0.1

0.15

0.2

0.25

Model candidate order

H∞ n

orm

err

or

LSAR estimationAR model in Theorem 1

‖Txmum − Txmum‖∞ Manifest network Comparison w/ opt matrix seq


Summary

Reachability for bilinear control systems

inputs might not only have effect on state of nodes but also on theinteractions among nodes

Gramian-based reachability metrics capture ease of control

actuator selection

linear networks that remain difficult to control after addition of bilinear inputs

Network identification with latent nodes

linear model with manifest and latent nodes

existence of AR models that asymptotically identify manifest transfer function

reconstruction of direct/indirect interactions among manifest nodes

least-squares auto-regressive estimate: performance in probability

Y. Zhao and J. Cortes. Gramian-based reachability metrics for bilinear networks.IEEE Transactions on Control of Network Systems, 2015.Submitted

Y. Zhao and J. Cortes. Identification of linear networks with latent nodes.In American Control Conference, Boston, MA, July 2016.Submitted


Plethora of open questions

Reachability for bilinear control systems

dependence of Gramian and metrics on network. Asymptotic results withnetwork scale

identify networks for which bilinear control does quantitatively better

robustness of reachability metrics against malfunctions/failures

optimal selection of actuators

. . .

Network identification with latent nodes

active latent nodes with unknown control inputs

linear network models of higher (unknown, time-varying) order

identification of properties that make networks easier to identify

applications to network science: brain and social networks

. . .


Documents

Controllability and Identification of Complex Networks€¦ · Controllability metrics: how much energy to steer the network, strategies G. Yan et al(PRL 12): how much energy is needed