listen to the noise: bridge dynamics and topology of complex networks jie ren ( 任 捷 ) nus...
TRANSCRIPT
![Page 1: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/1.jpg)
Listen to the noise:Bridge dynamics and topology of complex networks
Jie Ren ( 任 捷 )NUS Graduate School for Integrative Sciences & Engineering
National University of Singapore
Singapore
![Page 2: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/2.jpg)
Phononics
Phonon transport (heat conduction) in nanoscale structuresHeat transport and control in quantum spin systems Functional thermal devices for energy control in low dimensional nano-systems
Complex Networks and Systems biology
Research areas:
Prof. Baowen Li:
![Page 3: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/3.jpg)
Group Photo
![Page 4: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/4.jpg)
Listen to the noise:Bridge dynamics and topology of complex networks
Jie Ren ( 任 捷 )NUS Graduate School for Integrative Sciences & Engineering
National University of Singapore
![Page 5: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/5.jpg)
Real-Life Networks
• Transportation networks: airports, highways, roads, rail, electric power…•Communications: telephone, internet, www…•Biology: protein’s residues, protein-protein, genetic, metabolic…•Social nets: friendship networks, terrorist networks, collaboration networks…
?
Dynamical patternTopological structure
It is agreed that structure plays a fundamental role in shaping the dynamics of complex systems. However, the intrinsic relationship still remains unclear.
![Page 6: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/6.jpg)
Outline Consensus Dynamics
Harmonic Oscillators
Unified View: a general approach to bridge network topology and the dynamical patterns emergent on it
Application From topology to dynamics: stability of networks
From dynamics to topology: inferring network structures
![Page 7: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/7.jpg)
Consensus Dynamics
the αth normalized eigenvector of L
Transformation to eigen-space, using
Transform back to real-space:
the corresponding eigenvalue.
P: adjacency matrixL: laplacian matrix
Examples: Internet packet traffic, information flow, opinion dynamics…
Or:
Compact form:Pseudo-inverse:
Denote
solution:
J. Ren and H. Yang, cond-mat/0703232
![Page 8: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/8.jpg)
Coupled Harmonic Oscillators
Compact form:
Examples: protein’s residues interaction, electric circuit…
Thermal noise:
second order time derivative
reduce to first order
Share the same relationship.J. Ren and B. Li, PRE 79, 051922 (2009)
Matrix form:
![Page 9: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/9.jpg)
J. Ren, W.X. Wang, B. Li, and Y.C. Lai, PRL 104, 058701 (2010)
C: dynamical correlationL: the underlying topology
Ignoring intrinsic dynamics DF=0,DH=1, symmetric coupling
A General Approach to Bridge Dynamics and Network Topology
Under noise, the dynamics of the general coupled-oscillators can be expressed as:
linearization
long time limit
compact form
covariance of noise
A general relationship:
Jacobian matrix
![Page 10: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/10.jpg)
Pseudo-inverse:Kuramoto model:
Nodes become strong correlated in groups, coherently with their topological structure.
Group structures at multi-scale are revealed clearly.
The contribution of smaller eigenvalues dominates the correlation C
smaller eigenvalues ~ smaller energy ~ large wave length ~ large length scale
Simulation TheoryExample:
~
![Page 11: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/11.jpg)
Decompose
i i j i jm1
mr
Path-integral (topology) representation of correlations (dynamics)
The correlation matrix C can thus be expressed in a series:
Path-integral representation:
Pure dynamical property. Topology associated property
PRL 104, 058701 (2010)
![Page 12: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/12.jpg)
Application
From topology to dynamics: stability of networks
From dynamics to topology: inferring network structures
J. Ren and B. Li, PRE 79, 051922 (2009)
J. Ren, W.X. Wang, B. Li, and Y.C. Lai, PRL 104, 058701 (2010)W.X. Wang, J. Ren, Y.C. Lai, and B. Li, in preparation.
![Page 13: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/13.jpg)
Small-world networks
With probability p to add random links to each node.
Define the average fluctuation as S:
Characterize the stability of networks.smaller S ~ smaller fluctuations ~ more stable
Perturbation:
Add link (i, j):
New correlation:
Adding link always decreases S
J. Ren and B. Li, PRE 79, 051922 (2009)
From topology to dynamics: stability of networks
ring chain +random links
![Page 14: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/14.jpg)
Finite Size Scaling
From topology to dynamics: stability of networks
unstable
stable
slope=-1
continuous limit
A heuristic argument for the density of state:
For small world networks (1D ring + cross-links),the ring chain is divided into quasi-linear segments.
The probability to find length l is,
Each segment l has small eigenvalue of the order of
(unstable) (stable)
A.J. Bray and G.J. Rodgers, PRB 38, 11461 (1988);R. Monasson, EPJB 12, 555 (1999).
![Page 15: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/15.jpg)
Real protein data can be download from Protein Data Bank
From topology to dynamics: stability of networks
www.pdb.org
The mean-square displacement of atomsis characterized by B factor. (B~S)
C
Each protein is a network with residue-residue interaction.
The thermodynamic stability is crucial for protein to keep its native structure for right function.
We expect that nature selection forces proteins to evolve into the stable regime:
Real protein data follow -1 scaling
J. Ren and B. Li, PRE 79, 051922 (2009)
![Page 16: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/16.jpg)
?inverse problem
![Page 17: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/17.jpg)
J. Ren, W.X. Wang, B. Li, and Y.C. Lai, PRL 104, 058701 (2010)
From dynamics to topology: inferring network structures
![Page 18: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/18.jpg)
From dynamics to topology: inferring network structures
W.X. Wang, J. Ren, Y.C. Lai, and B. Li, in preparation.
Networks with Time-delay Coupling
![Page 19: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/19.jpg)
?
?How are they canalized by Evolution?
Function
Dynamics
Structure
![Page 20: Listen to the noise: Bridge dynamics and topology of complex networks Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National](https://reader035.vdocument.in/reader035/viewer/2022062517/56649f005503460f94c15fd8/html5/thumbnails/20.jpg)
Thanks!
Collaborators:
Prof. Baowen Li (NUS)
Prof. Huijie Yang (USSC)
Dr. Wen-Xu Wang (ASU)
Prof. Ying-Cheng Lai (ASU)