Challenge the future

DelftUniversity ofTechnology

Role and Control of Spectral Metrics

Huijuan WangDelft University of Technology

Boston University

ECT Trento Workshop

2

Network Science“Multi-weighted Monetary Transaction Network”,Advances in Complex Systems, 2011.

“The observable Part of a Network”, IEEE/ACM Transaction on Networking, 2009.“Betweenness Centrality in Weighted Networks”, Physical Review E, 2008.

“Effect of tumor resection on the characteristics of functional brain networks”, Physical Review E. 2010.

BrainBiology

Economy

Man-made Infrastructures/SystemsInternet, power-grid

“Metabolic network destruction: relating topology to robustness”,Nano Communication Networks, 2011.

3

Network Science

BrainBiology

Economy

Man-made Infrastructures/SystemsInternet, power-grid

Effect of Network on Function

4

5

Brain Networks vs. Alzheimer’s Diseases

De Haan et al. BMC Neuroscience 2009, 10: 101.

ADControls

L (AD) < L (Controls)C (AD) < C (Controls)

6

van den Heuvel et al. J Neurosci. 2009, 29(23):7619-24.

L L

Brain Networks vs. IQ

7

Metric Correlations in Complex Networks

Cong Li, Huijuan Wang and Piet Van MieghemJournal of Statistical Mechanics: Theory and Experiment, P11018, 2011.

8

Network metrics

1

1N

1N

1

[ ]E H

GR

GC

[1/ ]E H

3 2/N N

D

9

Metric Correlation is topology dependent

1

ER random graph: [ ] is independent of

D-lattice: [ ]3

D

E H N

DE H N

Consider: Erdös-Rényi random graphsBárabasi-Albert graphs

Application: Functional brain networks (MEG, health controls)

10

Erdös-Rényi random graph

Metric correlation pattern

11

Erdös-Rényi random graph Bárabasi-Albert graph

Metric correlation pattern

12

Analytic relations between network metrics

21

2 23 3 2 2

11 1 1

1

( 1)

1

G

N

ii

R

N

N Nd

N N N

Van Mieghem, P., H. Wang, X. Ge, S. Tang and F. A. Kuipers, 2010, "Influence of Assortativity and Degree-preserving Rewiring on the Spectra of Networks", The European Physical Journal B, vol. 76, No. 4, pp. 643-652.

13

Verification in Brain Networks

T=0.019

14

Verification in Brain Networks

Erdös-Rényi random graph Functional Brain Networks

15

The Role of Spectral radius in Epidemics and Coupled Osillators

16

16

SIS model

• Homogeneous birth (infection) rate on all edges between infected and susceptible nodes

• Homogeneous death (curing) rate for infected nodes

Healthy

= / : effective spreading rate

Infected

03

2

1

Infected

17

17

• Epidemic threshold

β : infection rate per link

δ : curing rate per node

τ= β/ δ : effective spreading rate

c 1

1 A

SIS model: epidemic threshold

P. Van Mieghem, J. Omic, R. E. Kooij, “Virus Spread in Networks”,IEEE/ACM Transaction on Networking, Vol. 17, No. 1, pp. 1-14, (2009).

18

g

% c

oupl

ed

100

0

gc 2

f (0)1(A)

Coupled oscillators (Kuramoto model)

18

k˙ k k g akj sin j k

j1

N

natural frequencycoupling strength

Interaction equals sums of sinus of phase difference of each neighbor:

J. G. Restrepo, E. Ott, and B. R. Hunt. Onset of synchronization in large networks of coupled oscillators, Phys. Rev. E, vol. 71, 036151, 2005

19

Modify the spectral radius

• Degree-preserving rewiring

"Influence of Assortativity and Degree-preserving Rewiring on the Spectra of Networks", The European Physical Journal B, vol. 76, No. 4, pp. 643-652, 2010.

• Removing links/nodes (optimal way is NP-complete)

"Decreasing the spectral radius of a graph by link removals", Physical Review E, Vol. 84, 016101, 2011.

• Adding interacting links between two networks

20

Degree-preserving rewiring USA air transport network

20

150

100

50

0

-50

Adj

acen

cy e

igen

valu

es

4003002001000

Percentage of rewired links

0.20

0.15

0.10

0.05

0.00

D

4003002001000

Percentage of rewired links

2 2

3 33 2 21

1 1 1

1 N

D jj

N N NA d

N N N N

21

Link/node removal

21

Removing m links/nodes to maximally decrease 1 is NP-hard.

1 1 1 1 1

1 1 1 1 1\

1 1 1 1 1 1

( ) 2

( ) 2 ,where joins and

2 ( ) ( ) 2

m

m m

T

l ll G

Tm m l l

l G M

ml l l ll M l M

A x Ax x x

A w A w w w l l l

Lemma

w w A A x x

22

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1 1 1 1 1 1

* 1 1

[ ( ) ( )] [ ( ) ( )]

Strategy : remove a link that maximizes

i j

optimal Strategy S

x x l l

A A A A

x x

One link removal m link “greedy” removal

Link/node removal

23

Interacting link addition

S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley and S. Havlin, Nature 464, 1025 (2010 )

Two types of links: Connectivity & Dependency

1 12

2 12

1

0 0,

0 0

( )

T

A BA B

A B

A B

β : infection rate at connectivity link

δ : curing rate per node

αβ: infection rate at dependency linkNetwork A

Network B1 12

2 12

1

0 0,

0 0

( )

T

A BA B

A B

A B

β : infection rate at connectivity link

δ : curing rate per node

αβ: infection rate at dependency link

1 12

2 12

1

0 0,

0 0

( )

T

A BA B

A B

A B

β : infection rate at connectivity link

δ : curing rate per node

αβ: infection rate at dependency link

24

Conclusion

• Metric correlations show that spectral metrics seem to be essential in network characterizations to discover the association between network and function.

• Epidemics and coupled oscillators: phase transition at tc= 1/1.

• Epidemic threshold engineering:

•Degree-preserving assortative (disassortative) rewiring increases (decreases) 1.

•Removing links/nodes to maximally decrease 1 is NP-hard, which motivates heuristic strategies.

•How does adding interacting links increase 1?