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Listen to the noise:

Listen to the noise:. Singapore. Bridge dynamics and topology of complex networks. Jie Ren ( 任 捷 ) NUS Graduate School for Integrative Sciences & Engineering National University of Singapore. Prof. Baowen Li:. Research areas:. Group Photo. Listen to the noise:.

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Listen to the noise:

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  1. Listen to the noise: Singapore Bridge dynamics and topology of complex networks Jie Ren (任 捷) NUS Graduate School for Integrative Sciences & Engineering National University of Singapore

  2. Prof. Baowen Li: Research areas:

  3. Group Photo

  4. 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

  5. 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… It is agreed that structure plays a fundamental role in shaping the dynamics of complex systems. However, the intrinsic relationship still remains unclear. ? Dynamical pattern Topological structure

  6. 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

  7. Consensus Dynamics Examples: Internet packet traffic, information flow, opinion dynamics… P: adjacency matrix L: laplacian matrix Or: Denote the αth normalized eigenvector of L the corresponding eigenvalue. Transformation to eigen-space, using solution: Transform back to real-space: Compact form: Pseudo-inverse: J. Ren and H. Yang, cond-mat/0703232

  8. Coupled Harmonic Oscillators Examples: protein’s residues interaction, electric circuit… second order time derivative Thermal noise: Matrix form: reduce to first order Compact form: Share the same relationship. J. Ren and B. Li, PRE 79, 051922 (2009)

  9. A General Approach to Bridge Dynamics and Network Topology Under noise, the dynamics of the general coupled-oscillators can be expressed as: linearization compact form Jacobian matrix covariance of noise long time limit A general relationship: Ignoring intrinsic dynamics DF=0, DH=1, symmetric coupling C: dynamical correlation L: the underlying topology J. Ren, W.X. Wang, B. Li, and Y.C. Lai, PRL 104, 058701 (2010)

  10. Simulation Theory Example: Kuramoto model: Pseudo-inverse: Group structures at multi-scale are revealed clearly. Nodes become strong correlated in groups, coherently with their topological structure. ~ The contribution of smaller eigenvalues dominates the correlation C smaller eigenvalues ~ smaller energy ~ large wave length ~ large length scale

  11. Path-integral (topology) representation of correlations (dynamics) Decompose The correlation matrix C can thus be expressed in a series: i j m1 i j i mr Path-integral representation: PRL 104, 058701 (2010) Topology associated property Pure dynamical property.

  12. 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.

  13. From topology to dynamics: stability of networks Define the average fluctuation as S: Small-world networks ring chain + random links Characterize the stability of networks. smaller S ~ smaller fluctuations ~ more stable With probability p to add random links to each node. Perturbation: New correlation: Adding link always decreases S Add link (i, j): J. Ren and B. Li, PRE 79, 051922 (2009)

  14. From topology to dynamics: stability of networks Finite Size Scaling continuous limit A heuristic argument for the density of state: slope=-1 For small world networks (1D ring + cross-links), the ring chain is divided into quasi-linear segments. unstable The probability to find length l is, stable Each segment l has small eigenvalue of the order of A.J. Bray and G.J. Rodgers, PRB 38,11461 (1988); R. Monasson, EPJB 12, 555 (1999). (unstable) (stable)

  15. From topology to dynamics: stability of networks Each protein is a network with residue-residue interaction. Real protein data follow -1 scaling The thermodynamic stability is crucial for protein to keep its native structure for right function. The mean-square displacement of atoms is characterized by B factor. (B~S) We expect that nature selection forces proteins to evolve into the stable regime: Real protein data can be download from Protein Data Bank www.pdb.org J. Ren and B. Li, PRE 79, 051922 (2009)

  16. ? inverse problem

  17. From dynamics to topology: inferring network structures J. Ren, W.X. Wang, B. Li, and Y.C. Lai, PRL 104, 058701 (2010)

  18. From dynamics to topology: inferring network structures Networks with Time-delay Coupling W.X. Wang, J. Ren, Y.C. Lai, and B. Li, in preparation.

  19. Function ? Dynamics Structure ? How are they canalized by Evolution?

  20. Collaborators: Prof. Baowen Li (NUS) Prof. Huijie Yang (USSC) Dr. Wen-Xu Wang (ASU) Prof. Ying-Cheng Lai (ASU) Thanks!

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