Role and Control of Spectral Metrics · Erdös-Rényi random graph Bárabasi-Albert graph Metric...
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Challenge the future
DelftUniversity ofTechnology
Role and Control of Spectral Metrics
Huijuan WangDelft University of Technology
Boston University
ECT Trento Workshop
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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.
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Network Science
BrainBiology
Economy
Man-made Infrastructures/SystemsInternet, power-grid
Effect of Network on Function
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Brain Networks vs. Alzheimer’s Diseases
De Haan et al. BMC Neuroscience 2009, 10: 101.
ADControls
L (AD) < L (Controls)C (AD) < C (Controls)
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van den Heuvel et al. J Neurosci. 2009, 29(23):7619-24.
L L
Brain Networks vs. IQ
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Metric Correlations in Complex Networks
Cong Li, Huijuan Wang and Piet Van MieghemJournal of Statistical Mechanics: Theory and Experiment, P11018, 2011.
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Network metrics
1
1N
1N
1
[ ]E H
GR
GC
[1/ ]E H
3 2/N N
D
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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)
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Erdös-Rényi random graph
Metric correlation pattern
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Erdös-Rényi random graph Bárabasi-Albert graph
Metric correlation pattern
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Analytic relations between network metrics
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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.
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Verification in Brain Networks
T=0.019
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Verification in Brain Networks
Erdös-Rényi random graph Functional Brain Networks
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The Role of Spectral radius in Epidemics and Coupled Osillators
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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
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• 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).
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g
% c
oupl
ed
100
0
gc 2
f (0)1(A)
Coupled oscillators (Kuramoto model)
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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
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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
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Degree-preserving rewiring USA air transport network
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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
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Link/node removal
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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
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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
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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
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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
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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?