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Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside Francesco Bullo - University of California at Santa Barbara Controllability metrics, limitations and algorithms for complex networks 1

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Page 1: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

Sandro ZampieriUniversita’ di Padova

In collaboration with Fabio Pasqualetti - University of California at RiversideFrancesco Bullo - University of California at Santa Barbara

Controllability metrics, limitations and algorithms for complex networks

1

Page 2: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

Growing Number of Applications

2

Power network Social networkBrain network

✓ Y. Y. Liu, J. J. Slotine, and A. L. Barabasi, “Controllability of complex networks,” Nature 2011. ✓ N. J. Cowan, E. J. Chastain, D. A. Vilhena, J. S. Freudenberg, and C. T. Bergstrom, “Nodal dynamics, not degree distributions, determine the structural controllability of complex networks,” PLOS ONE, 2012. ✓ G. Yan, J. Ren, Y.-C. Lai, C.-H. Lai, and B. Li, “Controlling complex networks: How much energy is needed?” Physical Review Letters, 2012. ✓ J. Sun and A. E. Motter, “Controllability transition and nonlocality in network control,” Physical Review Letters, 2013.✓ F. L. Cortesi, T. H. Summers, and J. Lygeros, “Submodularity of Energy Related Controllability Metrics,” Arxiv preprint, 2014.✓ A. Olshevsky, “Minimal Controllability Problems,” Arxiv preprint, 2014.

Page 3: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

Problem formulation

3

���� ����� ���������

(�+ 1) = �(�) + ��(�)

��� � ������� ������ ������ ������ ������������������ ����� ���������������

� = [��1 · · · ��� ]

���

�� =

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0���010���0

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Page 4: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

Problem formulation

4

�(�)

�(�) = �̄

\(X+ 1) = %\(X) + &Y(X)

'3286300%&-08= MW� XLI�TSWWMFMPMX]�SJ� WXVIIVMRK� XLI� WXEXI� JVSQ� XLI� MRMXMEPWXEXI \(0) = 0 XS�ER�EVFMXVEV]�½REP�WXEXI \(8) = \̄ F]�ETTP]MRK�E�WYMXEFPI�MRTYXWIUYIRGI Y(0), Y(1), . . . , Y(8− 1)�

Page 5: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

Need of a controllability metric

5

Controlled node

��������������������������������������������������������������������� ������������������������������������������������������������������������������������������������� �������������������������������������������������

Page 6: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

6

������� ������������ ����������������������������

Need of a controllability metric

Page 7: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

7

A controllability metric

�(�)

�(�)

3RI�QIXVMG�JSV�HIWGVMFMRK�XLI�GSRXVSPPEFMPMX]�HIKVII�MW�KMZIR�F]�XLI�GSRXVSPPE�FMPMX]�+VEQMER

;8 :=8−�!

X=�

%X&&8(%8)X

-RXIVTVIXEXMSR� *SV�HVMZMRK

\(�) = � −→ \(8) = \̄

[LIVI ||\̄||� = �� [I�RIIH�ER�MRTYX Y(�), Y(�), . . . , Y(8 − �) [MXL 0� IRIVK]\̄8;−�

8 \̄�8LI�IRIVK]�XS�HVMZI�XLI�WXEXI�JVSQ�^IVS�XS�E�RSVQ�SRI�WXEXI��MR�XLI�[SVWXGEWI �MW�KMZIR�F]

)RIVK] =�

λQMR(;8)

Page 8: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

8

A controllability metric

����� ��(��)

����� ��(��)

����� ��� �����������

�� ������� ��������� ��

�� :=�−�!

�=�

�����(��)�

���� =�

λ�(��)

����������� �������

Page 9: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

9

A controllability metricAlternative controllability metrics

;8 :=8−�!

X=�

%X&&8(%8)X

λQMR(;8)

�Rtr(;−�

8 )

�Rln det(;8)

�Rtr(;8)

Page 10: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

10

Few Nodes Cannot Control Symmetric Complex Networks

�������� ������� � ��������� ������������� � < µ < � ������

�(µ) := |{λ ∈ λ(�) : |λ|� ≤ µ}|

� ��

λ���(��) ≤�

µ(�− µ)µ�(µ)/�

�������

Page 11: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

11

Few Nodes Cannot Control Symmetric Complex Networks

µ < �

�� ��������������������

λ√µ

��(λ)

1-1õ-

A

0IX

*%(λ̄) :=|{λ ∈ λ(%) : λ ≤ λ̄}|

RERH

J%(λ̄) :=H*(λ̄)Hλ̄

8LIR

R(µ) := |{λ ∈ λ(%) : |λ|� ≤ µ}| = R! √

µ

−√µJ%(λ)Hλ = A R

,IRGI� XLI� RYQFIV R(µ) X]TMGEPP]� KVS[W� PMRIEVP]� MR� XLIRIX[SVO�GEVHMREPMX] R�

λQMR(;8) ≤�

µ(�− µ)µA R

Q

Page 12: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

12

Few Nodes Cannot Control Symmetric Complex Networks

�� ��������������������

λ√µ

��(λ)

1-1õ-

µ < �

AλQMR(;8) ≤

�µ(�− µ)

µA RQ

Page 13: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

13

✓ For fixed number m of control nodes, the controllability degree decreases exponentially in the network cardinality n.

✓ To have a fixed controllability degree, number m of control nodes must grow linearly in the network cardinality n.

Few Nodes Cannot Control Symmetric Complex Networks

�� ��������������������

λ√µ

��(λ)

1-1õ-

µ < �

λQMR(;8) ≤�

µ(�− µ)µA R

Q

A

Page 14: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

14

� =

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

� � � . . . . . . � �� � � · · · · · · � �� � � · · · · · · � ����������� � �

� � �������

���������� � �

� � �������

� � � · · · · · · � �� � � · · · · · · � �

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

λ(�) =!�+ �� cos

�+ ��#

: � = �, �, . . . , �$

Example: symmetric line graph

Page 15: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

15

Example: symmetric line graph

λ̄λ̄

��(λ̄) ��(λ̄)

��(λ̄) = �−�πarcos

!λ̄− ���

"��(λ̄) =

�/π!�−

"λ̄−���

#�

�− �� �+ �� �− �� �+ ��

Page 16: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

16

λ̄

��(λ̄)

√µ−√

µ

Example: symmetric line graph

A

Page 17: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

17

Example: Symmetric random matrix

� =

⎢⎢⎢⎢⎢⎢⎣

��� ��� . . . . . . ������ ��� · · · · · · ������

���� � �

� � ����

������� � �

� � ����

��� ��� · · · · · · ���

⎥⎥⎥⎥⎥⎥⎦

������ � ������������� ������������ ��� ��������������������� E[���] = � ��� E[���� ] = σ�/

√��

������������������ �� �������

��(λ̄) −→ �(λ̄)

λ̄

��(λ̄)

−σ√� σ

√�

√µ−√

µ

A

Page 18: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

18

Asymmetric Complex Networks

For asymmetric networks the situation is more complex

- For “isotropic” networks (networks with no preferential directions) it seems that the situation is the same as for symmetric networks, namely they are difficult to control.

- For “anisotropic” networks (networks with a preferential direction) it seems that few nodes can indeed control large scale networks.

Page 19: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

19

Asymmetric Complex Networks

Consequence: If the condition number stays bounded in the network dimension, than the network remains difficult to control.

�������%WWYQI�XLI�QEXVM\ % HMEKSREPM^EFPI�ERH�PIX : ER�IMKIRZIGXSV�QEXVM\� *M\�ER]GSRWXERX � < µ < � ERH�PIX

R(µ) := |{λ ∈ λ(%) : |λ|� ≤ µ}|

8LIR

λQMR(;8) ≤ cond(:)��

µ(�− µ)µR(µ)/Q

[LIVI cond(:) = σmax(:)/σmin(:) MW�XLI�GSRHMXMSR�RYQFIV�SJ :�

Page 20: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

20

Example: Asymmetric isotropic line graph

� :=��

⎢⎢⎢⎢⎢⎣

� �/� � � · · ·� � � � · · ·� �/� � �/� · · ·� � � � · · ·���

������

���� � �

⎥⎥⎥⎥⎥⎦.

�������� ������������������������� � �������� ���(�) = ��

λ(�) =!��+��cos

�π�+ �

, � = �, . . . , �"

������������������ �������������������������������������

λ̄

��(λ̄)

√µ−√

µ

A

Page 21: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

21

Example: Asymmetric anisotropic line graph

� =

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

� � � . . . . . . � �� � � · · · · · · � �� � � · · · · · · � ����������� � �

� � �������

���������� � �

� � �������

� � � · · · · · · � �� � � · · · · · · � �

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

λ(�) =!�+ �

√�� cos

�+ ��#

: � = �, �, . . . , �$

Given a, if c is sufficiently larger than b, than this network can be controlled with finite energy by the node on the extreme regardless the network dimension.

Exploiting spatial instability

control node

Page 22: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

22

Extension to more general graphs: controllable graphs

� =

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�� �� � . . . . . . � ��� �� �� · · · · · · � �� �� �� · · · · · · � ����������� � �

� � ����

������������� � �

� � ����

���� � � · · · · · · �ℓ−� �ℓ−�� � � · · · · · · �ℓ−� �ℓ

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

control nodes

Page 23: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

23

Extension to more general graphs

� =

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

�� �� � . . . . . . � ��� �� �� · · · · · · � �� �� �� · · · · · · � ����������� � �

� � ����

������������� � �

� � ����

���� � � · · · · · · �ℓ−� �ℓ−�� � � · · · · · · �ℓ−� �ℓ

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

� ���������� �� ��� �������� �� �� ������� ����������������������� ����� ������� �������������������������� ����� ��������������������������������������������������������� ��� ���

Page 24: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

24

"�� � � � �� !���� !�����!��%� �����" �����&����� � ��� �� " � ��� �� ��� ���� �!��!

�� = �

$���� � � �!���#��!���$�!����!��� ���"���!������ ��! � �����#��!��� "���!��!

��� = �� ��� = �

$��������� �!��! � � �!�����#�����!���� "������!��������#�������� ����!��$�!� �� !�� ����$��!��!�!�����!��� ��� � ����� ��!�!������� �����!����!&����!�����!$�����!����������!�������������!��!��

� "�!� ��!�����!��� ��� � ������� ≤ ����� ��������� ���#�� ����������!����!&��!���

!���� ����!�����!$����� ����'�"�!�!�����!����

��!��� &���!������ ��!�����!��� ��� � ��� �/��

Extension to more general graphs: uncontrollable graphs

Page 25: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

25

Controllers positioning����������������������� ��������������

����� ������������� �������� V = {�, . . . , �} ����� ���������� �� V�, . . . ,V����� ��� � ����������� ������������ �� ������ ��� ���

� =

⎢⎣�� · · · ������

������

��� · · · ��

⎥⎦ , � =

⎢⎣�� · · · ����� � �

���� · · · ��

⎥⎦ ,

�� �� ������������������ ������ ������ ���� ����� �������� � ������ ������� �����

�(�+ �) = ���(�) +∑

�∈N�

����(�) + ����(�),

�� � � ∈ {�, . . . ,�} �� N� := {� : ��� ̸= �}�

������������ �����������������������������⎧⎪⎪⎨⎪⎪⎩ !!

Page 26: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

26

Controllers positioning�� ������ ��� ��������

�� ����������������� �����������������������������

�� �����������������������������������

Page 27: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

27

Controllers positioning

�� ������ ��� ��������

�� ������� ��������

��(�) := �(�)−!

�∈N�

��� ����(�)

� ������������������� � �����������������

�(�+ �) = ���(�) + ���(�)

�� � ���� � � � ������������ ������������������� ���������������� ����������������

Page 28: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

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Controllers positioning������������������

Λ :=

⎢⎢⎢⎣

λ−����(��,�) �� · · · �� λ−�

���(��,�) · · · ����

���� � �

���� � · · · λ−�

���(��,�)

⎥⎥⎥⎦.

����������������

∆ :=

⎢⎢⎢⎣

� ∥���∥� · · · ∥���∥�∥���∥� � · · · ∥���∥����

���� � �

���∥���∥� ∥���∥� · · · �

⎥⎥⎥⎦.

Page 29: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

29

Controllers positioning������� ����������������� �������� � ��������������

λ���(��) ≥(�− λ̄���)�

∥Λ∥∞∥∆∥�∞

���

λ̄��� = max{λ���(��) : � ∈ {�, . . . ,�}} < �

Page 30: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

30

Controllers positioning

=�������� ���� ��������

��������� �������������������� ����� ��

������� ����������������� �������� � ��������������

λ���(��) ≥(�− λ̄���)�

∥Λ∥∞∥∆∥�∞

���

λ̄��� = max{λ���(��) : � ∈ {�, . . . ,�}} < �

Page 31: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

31

Controllers positioning������� ����������������� �������� � ��������������

λ���(��) ≥(�− λ̄���)�

∥Λ∥∞∥∆∥�∞

���

λ̄��� = max{λ���(��) : � ∈ {�, . . . ,�}} < �

���� �� ����������������������

�� ����������������� ∥∆∥∞ ������������������������������������� ������������������������������� ∥Λ∥∞ ������������� ���������������������

��������������������� ����������������������������������������

=�������� ���� ��������

��������� �������������������� ����� ��

Page 32: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

32

Examples���� ��������

2 4 6 8 10 12 14 16 18 20−40

−30

−20

−10

0

N 2 4 6 8 10 12 14 16 18 20

−40

−30

−20

−10

0

nb

number of subsystems size of subsystems

� �������������� ������������������

� ���� ��������� �������� �� ���������� �����

Page 33: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

33

Network partition and selection of the control nodes

�������� �������� ����� ����

������������������������� �������������������������������

�� ��������������������������������������������������������������� ���������������������������������������

Page 34: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

34

Examples

�(��) �������������� ������������������

��(��) �� ���������� �����

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−60

−50

−40

−30

−20

−10

0

10

n

���������������������

m

Page 35: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

35

Examples

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−60

−50

−40

−30

−20

−10

0

10

n

������������ ���������� ���

�(��) �������������� ������������������

��(��) �� ���������� �����

m

Page 36: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

36

Examples

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−60

−50

−40

−30

−20

−10

0

10

n

����� ��������������������

�(��) �������������� ������������������

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Page 37: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

ConclusionsSimilar results for observability

For symmetric (isotropic) networks we need to control a fixed fraction of nodes

For anisotropic networks it is enough to control a fixed number of nodes

Random positioning works pretty well

Phase transition can be noticed (critical fraction of controlled nodes)

There are a lot of open problems:

Understanding isotropic and anisotropic networks

Controllability of random and of structured graphs

Performance of random positioning

Phase transition

Different metrics

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Page 38: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

Conclusions

Controllability

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Controllability degree

Graph theory Spectral graph theory

Page 39: Controllability metrics, limitations and algorithms for ... · Sandro Zampieri Universita’ di Padova In collaboration with Fabio Pasqualetti - University of California at Riverside

Thank you

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