Spectral Graph Theory for. Dynamic Processes on Networks
|
|
- Alvin Oliver
- 5 years ago
- Views:
Transcription
1 Spectral Graph Theory for Dynamic Processes on etworks Piet Van Mieghem in collaboration with Huijuan Wang, Dragan Stevanovic, Fernando Kuipers, Stojan Trajanovski, Dajie Liu, Cong Li, Javier Martin-Hernandez, Ruud van de Bovenkamp and Rob Kooij Outline Dynamic Processes on etworks Modifying the spectral radius Principal eigenvector ew (?) graph theoretic results Summary
2 Effect of etwork on Function Brain Biology Economy Social etwork Science networking Man-made Infrastructures Internet, power-grid Brain/Bio-inspired etworking to design superior man-made robust networks/systems Coupled oscillators (Kuramoto model) Interaction equals sums of sinus of phase difference of each neighbor:! k = " k + g$ akj sin(! j! k ) θk j= coupling strength natural frequency % coupled gc! "f () $ (A) g 4 J. G. Restrepo, E. Ott, and B. R. Hunt. Onset of synchronization in large networks of coupled oscillators, Phys. Rev. E, vol. 7, 365, 5
3 Simple SIS model Homogeneous birth (infection) rate β on all edges between infected and susceptible nodes Homogeneous death (curing) rate δ for infected nodes τ = β /δ : effective spreading rate δ Healthy 3 β β Infected Infected 5 Epidemics in networks: modeling & immunization network protection self-replicating objects (worms) propagation errors rumors (social nets) epidemic algorithms (gossiping) cybercrime : network robustness & security! " X j j!neighbor(i)!! c =! = " " ( A) Epidemic threshold - Van Mieghem, P., J. S. Omic and R. E. Kooij, 9, "Virus Spread in etworks", IEEE/ACM Transaction on etworking, Vol. 7, o., February, pp Van Mieghem, P.,, "The -Intertwined SIS epidemic network model", Computing (Springer), Vol. 93, Issue, p
4 Transformation s =! & principal eigenvector dy! (s) ds s= = " $ j= " d j L dy! (s) ds s=! = " j= j= (x ) j (x ) j 3 7 Van Mieghem, P.,, "Epidemic Phase Transition of the SIS-type in etworks", Europhysics Letters (EPL), Vol. 97, Februari, p s c = λ Algebraic graph theory: notation Any graph G can be represented by an adjacency matrix A and an incidence matrix B, and a Laplacian Q B L = 6 L = 9 = 4 T A = = A T Q = BB = Δ A Δ = diag ( d d d 8 ) 4
5 Outline Dynamic Processes on etworks Modifying the spectral radius Principal eigenvector ew (?) graph theoretic results Summary Affecting the epidemic threshold Degree-preserving rewiring (assortativity of the graph) Van Mieghem, P., H. Wang, X. Ge, S. Tang and F. A. Kuipers,, "Influence of Assortativity and Degree-preserving Rewiring on the Spectra of etworks", The European Physical Journal B, vol. 76, o. 4, pp Van Mieghem, P., X. Ge, P. Schumm, S. Trajanovski and H. Wang,, "Spectral Graph Analysis of Modularity and Assortativity", Physical Review E, Vol. 8, ovember, p Li, C., H. Wang and P. Van Mieghem,, "Degree and Principal Eigenvectors in Complex etworks", IFIP etworking, May -5, Prague, Czech Republic. Removing links/nodes (optimal way is P-complete) Van Mieghem, P., D. Stevanovic, F. A. Kuipers, C. Li, R. van de Bovenkamp, D. Liu and H. Wang,,"Decreasing the spectral radius of a graph by link removals", Physical Review E, Vol. 84, o., July, p. 6. Quarantining and network protection Omic, J., J. Martin Hernandez and P. Van Mieghem,, "etwork protection against worms and cascading failures using modularity partitioning", nd International Teletraffic Congress (ITC ), September 7-9, Amsterdam, etherlands. Gourdin, E., J. Omic and P. Van Mieghem,, "Optimization of network protection against virus spread", 8th International Workshop on Design of Reliable Communication etworks (DRC ), October -, Krakow, Poland. E.R. van Dam, R.E. Kooij, The minimal spectral radius of graphs with a given diameter, Linear Algebra and its Applications, 43, 7, pp
6 Assortativity Reformulation of ewman s definition into algebraic graph theory =! d j =, =! d j = L, = d T d =! d j j =! D = E [ D D l + l " ] " E [ D l + ]E D l " Var[ D l + ]Var[ D l + ] j = [ ] A network is (degree) assortative if ρ D > A network is (degree) disassortative if ρ D < = 3 " 3 d j " where k = u T A k u is the total number of walks with k hops: j = j = k! " j = d j k Van Mieghem, P., H. Wang, X. Ge, S. Tang and F. A. Kuipers,,"Influence of Assortativity and Degree-preserving Rewiring on the Spectra of etworks", The European Physical Journal B, vol. 76, o. 4, pp Degree-preserving rewiring a c a c a c b d b d b d ( ) d i " d j i~ j! D =" 3 d j " $ j = L & % j = d j ' ) ( only two terms change degree-preserving rewiring algorithm 6
7 Bounds largest eigenvalue adjacency matrix Classical bounds: Walks based: E [ D] = L! " ( A)! d max d max!! (A)! d max! ( A) " /(k) & k % ( " / k & k % ( $ ' $ ' / k = :! ( A) " dt d& % ( = L $ ' + Var[ D] E D ( [ ]) k = 3 :! 3 ( A) " 3 = ) 3 D * d j + & % $ j = ( + & % $ ' ( ' Optimized:! ( A) " 3 + ( 3 ) others ( ) 3 P. Van Mieghem, Graph Spectra for Complex etworks, Cambridge University Press, Comparison! 5 5 exact bound bound bound " D Barabasi-Albert power law graph with = 5 and L=9! " 3 $ & % / 3 4 7
8 USA air transportation network 5 Degree-preserving rewiring USA air transport network: adjacency eig. 5 Adjacency eigenvalues!d Percentage of rewired links Percentage of rewired links 3 4 8
9 Degree-preserving rewiring USA air transport network: Laplacian eig..7. Laplacian eigenvalues Pr[D = k].. Pr[D = k] ~ k degree k Percentage of rewired links Decreasing the spectral radius Which set of m removed links (or nodes) decreases the spectral radius most? [P-complete problem] Scaling law:! (A)!! (A m ) = c G m" What is the best heuristic strategy? remove the link l with maximum product of the eigenvector components: x l + x l - 8 Van Mieghem, P., D. Stevanovic, F. A. Kuipers, C. Li, R. van de Bovenkamp, D. Liu and H. Wang,,"Decreasing the spectral radius of a graph by link removals", Physical Review E, Vol. 84, o., July, p. 6. 9
10 Outline Dynamic Processes on etworks Modifying the spectral radius Principal eigenvector ew (?) graph theoretic results Summary.6 Binomial networks The component of the largest eigenvector Binomial etworks =5, L=984 ρ=.8 mean=.64 ρ=.4 mean=.375 ρ= mean=.4 ρ=.4 mean=.47 ρ=.8 mean=.43 without rewiring normalized degree The mean of the components decreases with the assortativity The normalized, ranked degree is between the green and purple line n Li, C., H. Wang and P. Van Mieghem,, "Degree and Principal Eigenvectors in Complex etworks", IFIP etworking, May -5, Prague, Czech Republic.
11 Power law networks.3 The component of the largest eigenvector Power law etwork: =5, L=984 ρ=.4 mean=.85 ρ=. mean=.433 ρ= mean=.3 ρ=. mean=.3458 ρ=.4 mean=.3793 without rewiring normalized degree The mean of the components decreases with the assortativity n Power law networks (log-scale) - The component of the largest eigenvector Power law etwork: =5, L=984 ρ=.4 mean=.85 ρ=. mean=.433 ρ= mean=.3 ρ=. mean=.3458 ρ=.4 mean=.3793 without rewiring normalized degree n
12 E[D] = 8 3 Outline Dynamic Processes on etworks Modifying the spectral radius Principal eigenvector ew (?) graph theoretic results Summary
13 Upper bound principal eigenvector of A Let x denote the principal eigenvector of A belonging to the largest eigenvalue λ (A). Let m denote the set of nodes that are removed from G x n! $ & + (& % x j ) '&! (A) *& n" m If m =, equality holds a ij x i j" m i" m If m =, then x n! S.M. Cioaba, A necessary and sufficient eigenvector condition for a connected graph to be bipartite, Electron. J. Linear Algebra (ELA) () Li, C., H. Wang and P. Van Mieghem,, "Bounds for the spectral radius of a graph when nodes are removed", Linear Algebra and its Applications, vol. 437, pp Lower bound principal eigenvector of A Let $ n j=; jm Then x m x m T = n! " ( z! x j ) = b k (m)z k and all eigenvalues are different k= This follows after inversion of A k = (! m!! ) k=;k"m k! k x k x k T! $ k= b k (m)a k = ( A!! k I ) k=;k"m (! m!! ) k=;k"m k using the inversion of the Vandermonde matrix, which can be related to Lagrange s interpolation theorem! j= Corollary: for any connected graph, it holds that ( x ) j! max & ( ( ' (! "! ) k= k " % max $ j$ k=h ij b k () ( A k ) ij, " " % b k () A k e! k=h ij ( ) 6 jj ) + + > * 3
14 Spectral identity For any symmetric matrix A ( ) = det A \ i det A!!I ( { }!!I )det A \{ k}!!i ( )! ( det( A \ rowi col k!!i)) det( A \{ i,k}!!i) Is this identity new? Using this identity, we can deduce (x k ) j = det A \{ j}!! ki P. Van Mieghem, Graph Spectra for Complex etworks, Cambridge University Press, second edition in preparation. " n= ( ) ( ) det A \{ n}!! k I 7 Topology domain 3 = 6 L = 9 Spectral domain A = X!X T Most network problems: - shortest path - graph metrics - network algorithms X 3-ary tree : green = 8 4
15 Spectral invariants The number of eigenvector components with positive (negative) sign and those with zero value umerical results (Javier Martin-Hernandez): about 5 % of the eigenvector components are positive invariant? due to orthonormal matrix properties? What does it mean? 9 Outline Dynamic Processes on etworks Modifying the spectral radius Principal eigenvector ew (?) graph theoretic results Summary 5
16 Summary Real epidemics: phase transition at τc > /λ, but the Intertwined mean-field approximation is very useful in network engineering via spectral analysis Epidemic threshold engineering: Degree-preserving assortative rewiring increases λ, while degree-preserving disassortative rewiring decreases λ Removing links/nodes to maximally decrease λ is Phard What do eigenvalues and eigenvectors "physically" mean? 3 Books Articles: 3 6
17 Thank You Piet Van Mieghem AS, TUDelft 33 7
SIS epidemics on Networks
SIS epidemics on etworks Piet Van Mieghem in collaboration with Eric Cator, Ruud van de Bovenkamp, Cong Li, Stojan Trajanovski, Dongchao Guo, Annalisa Socievole and Huijuan Wang 1 EURADOM 30 June-2 July,
More informationNetwork Science & Telecommunications
Network Science & Telecommunications Piet Van Mieghem 1 ITC30 Networking Science Vision Day 5 September 2018, Vienna Outline Networks Birth of Network Science Function and graph Outlook 1 Network: service(s)
More informationOn Susceptible-Infected-Susceptible Epidemic Spreading: an Overview of Recent Study
The 1 st Net-X (2017) On Susceptible-Infected-Susceptible Epidemic Spreading: an Overview of Recent Study Cong Li Adaptive Networks and Control Lab, Fudan University Collaborators: Xiaojie Li, Jianbo Wang,
More informationDie-out Probability in SIS Epidemic Processes on Networks
Die-out Probability in SIS Epidemic Processes on etworks Qiang Liu and Piet Van Mieghem Abstract An accurate approximate formula of the die-out probability in a SIS epidemic process on a network is proposed.
More informationOptimally decreasing the spectral radius of a graph by link removals
Optimally decreasing the spectral radius of a graph by link removals Piet Van Mieghem, Dragan Stevanović y, Fernando Kuipers, Cong Li, Ruud van de Bovenkamp, Daijie Liu and Huijuan Wang Delft University
More informationSocial Influence in Online Social Networks. Epidemiological Models. Epidemic Process
Social Influence in Online Social Networks Toward Understanding Spatial Dependence on Epidemic Thresholds in Networks Dr. Zesheng Chen Viral marketing ( word-of-mouth ) Blog information cascading Rumor
More informationInfluence of assortativity and degree-preserving rewiring on the spectra of networks
Eur. Phys. J. B 76, 643 651 DOI: 1.114/epjb/e1-19-x Regular Article THE EUROPEA PHYSICAL JOURAL B Influence of assortativity and degree-preserving rewiring on the spectra of networks P. Van Mieghem 1,H.Wang
More informationITTC Science of Communication Networks The University of Kansas EECS SCN Graph Spectra
Science of Communication Networks The University of Kansas EECS SCN Graph Spectra Egemen K. Çetinkaya and James P.G. Sterbenz Department of Electrical Engineering & Computer Science Information Technology
More informationTopology-Driven Performance Analysis of Power Grids
Topology-Driven Performance Analysis of Power Grids Hale Çetinay, Yakup Koç, Fernando A. Kuipers, Piet Van Mieghem Abstract Direct connections between nodes usually result in efficient transmission in
More informationThe minimal spectral radius of graphs with a given diameter
Linear Algebra and its Applications 43 (007) 408 419 www.elsevier.com/locate/laa The minimal spectral radius of graphs with a given diameter E.R. van Dam a,, R.E. Kooij b,c a Tilburg University, Department
More informationA new centrality measure for probabilistic diffusion in network
ACSIJ Advances in Computer Science: an International Journal, Vol. 3, Issue 5, No., September 204 ISSN : 2322-557 A new centrality measure for probabilistic diffusion in network Kiyotaka Ide, Akira Namatame,
More informationLinear Algebra Methods for Data Mining
Linear Algebra Methods for Data Mining Saara Hyvönen, Saara.Hyvonen@cs.helsinki.fi Spring 2007 2. Basic Linear Algebra continued Linear Algebra Methods for Data Mining, Spring 2007, University of Helsinki
More informationNonnegative and spectral matrix theory Lecture notes
Nonnegative and spectral matrix theory Lecture notes Dario Fasino, University of Udine (Italy) Lecture notes for the first part of the course Nonnegative and spectral matrix theory with applications to
More informationA network approach for power grid robustness against cascading failures
A network approach for power grid robustness against cascading failures Xiangrong Wang, Yakup Koç, Robert E. Kooij,, Piet Van Mieghem Faculty of Electrical Engineering, Mathematics and Computer Science,
More informationDecentralized Protection Strategies against SIS Epidemics in Networks
Decentralized Protection Strategies against SIS Epidemics in etworks Stojan Trajanovski, Yezekael Hayel, Eitan Altman, Huijuan Wang, and Piet Van Mieghem Abstract Defining an optimal protection strategy
More informationCharacterization and Design of Complex Networks
Characterization and Design of Complex Networks Characterization and Design of Complex Networks Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van
More informationLink Operations for Slowing the Spread of Disease in Complex Networks. Abstract
PACS: 89.75.Hc; 88.80.Cd; 89.65.Ef Revision 1: Major Areas with Changes are Highlighted in Red Link Operations for Slowing the Spread of Disease in Complex Networks Adrian N. Bishop and Iman Shames NICTA,
More informationToward Understanding Spatial Dependence on Epidemic Thresholds in Networks
Toward Understanding Spatial Dependence on Epidemic Thresholds in Networks Zesheng Chen Department of Computer Science Indiana University - Purdue University Fort Wayne, Indiana 4685 Email: chenz@ipfw.edu
More informationThe Laplacian Spectrum of Complex Networks
1 The Laplacian Spectrum of Complex etworks A. Jamakovic and P. Van Mieghem Delft University of Technology, The etherlands {A.Jamakovic,P.VanMieghem}@ewi.tudelft.nl Abstract The set of all eigenvalues
More informationThe N-intertwined SIS epidemic network model
Computing 0 93:47 69 DOI 0.007/s00607-0-055-y The N-intertwined SIS epidemic network model Piet Van Mieghem Received: September 0 / Accepted: 7 September 0 / Published online: 3 October 0 The Authors 0.
More informationChapter 2 Topology-Driven Performance Analysis of Power Grids
Chapter 2 Topology-Driven Performance Analysis of Power Grids Hale Çetinay, Yakup Koç, Fernando A. Kuipers and Piet Van Mieghem Abstract Direct connections between nodes usually result in efficient transmission
More informationIntegral equations for the heterogeneous, Markovian SIS process with time-dependent rates in time-variant networks
Integral equations for the heterogeneous, Markovian SIS process with time-dependent rates in time-variant networks P. Van Mieghem Delft University of Technology 2 April 208 Abstract We reformulate the
More informationEpidemics in Complex Networks and Phase Transitions
Master M2 Sciences de la Matière ENS de Lyon 2015-2016 Phase Transitions and Critical Phenomena Epidemics in Complex Networks and Phase Transitions Jordan Cambe January 13, 2016 Abstract Spreading phenomena
More informationPotential Game approach to virus attacks in network with general topology
Potential Game approach to virus attacks in network with general topology François-Xavier Legenvre, Yezekael Hayel, Eitan Altman To cite this version: François-Xavier Legenvre, Yezekael Hayel, Eitan Altman.
More informationAnalytically tractable processes on networks
University of California San Diego CERTH, 25 May 2011 Outline Motivation 1 Motivation Networks Random walk and Consensus Epidemic models Spreading processes on networks 2 Networks Motivation Networks Random
More informationWE FOCUS ON A simple continuous-time model for the
IEEE/ACM TRANSACTIONS ON NETWORKING, VOL 17, NO 1, FEBRUARY 2009 1 Virus Spread in Networks Piet Van Mieghem, Member, IEEE, Jasmina Omic, and Robert Kooij Abstract The influence of the network characteristics
More informationAn Optimal Control Problem Over Infected Networks
Proceedings of the International Conference of Control, Dynamic Systems, and Robotics Ottawa, Ontario, Canada, May 15-16 214 Paper No. 125 An Optimal Control Problem Over Infected Networks Ali Khanafer,
More informationMarkov Chains, Random Walks on Graphs, and the Laplacian
Markov Chains, Random Walks on Graphs, and the Laplacian CMPSCI 791BB: Advanced ML Sridhar Mahadevan Random Walks! There is significant interest in the problem of random walks! Markov chain analysis! Computer
More informationSpectral Graph Theory Tools. Analysis of Complex Networks
Spectral Graph Theory Tools for the Department of Mathematics and Computer Science Emory University Atlanta, GA 30322, USA Acknowledgments Christine Klymko (Emory) Ernesto Estrada (Strathclyde, UK) Support:
More informationDecentralized Protection Strategies against SIS Epidemics in Networks
Decentralized Protection Strategies against SIS Epidemics in etworks Stojan Trajanovski, Student ember, IEEE, Yezekael Hayel, ember, IEEE, Eitan Altman, Fellow, IEEE, Huijuan Wang, and Piet Van ieghem,
More informationNetwork Theory with Applications to Economics and Finance
Network Theory with Applications to Economics and Finance Instructor: Michael D. König University of Zurich, Department of Economics, Schönberggasse 1, CH - 8001 Zurich, email: michael.koenig@econ.uzh.ch.
More informationKristina Lerman USC Information Sciences Institute
Rethinking Network Structure Kristina Lerman USC Information Sciences Institute Università della Svizzera Italiana, December 16, 2011 Measuring network structure Central nodes Community structure Strength
More informationThe Spreading of Epidemics in Complex Networks
The Spreading of Epidemics in Complex Networks Xiangyu Song PHY 563 Term Paper, Department of Physics, UIUC May 8, 2017 Abstract The spreading of epidemics in complex networks has been extensively studied
More informationECS 289 F / MAE 298, Lecture 15 May 20, Diffusion, Cascades and Influence
ECS 289 F / MAE 298, Lecture 15 May 20, 2014 Diffusion, Cascades and Influence Diffusion and cascades in networks (Nodes in one of two states) Viruses (human and computer) contact processes epidemic thresholds
More informationDiffusion of Innovation
Diffusion of Innovation Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Social Network Analysis
More informationTHE near-threshold behavior, i.e. the behavior around the
etwork localization is unalterable by infections in bursts Qiang Liu and Piet Van Mieghem arxiv:80.0880v [physics.soc-ph] Dec 08 Abstract To shed light on the disease localization phenomenon, we study
More informationA New Spectral Technique Using Normalized Adjacency Matrices for Graph Matching 1
CHAPTER-3 A New Spectral Technique Using Normalized Adjacency Matrices for Graph Matching Graph matching problem has found many applications in areas as diverse as chemical structure analysis, pattern
More informationShortest Paths & Link Weight Structure in Networks
Shortest Paths & Link Weight Structure in etworks Piet Van Mieghem CAIDA WIT (May 2006) P. Van Mieghem 1 Outline Introduction The Art of Modeling Conclusions P. Van Mieghem 2 Telecommunication: e2e A ETWORK
More informationFast Linear Iterations for Distributed Averaging 1
Fast Linear Iterations for Distributed Averaging 1 Lin Xiao Stephen Boyd Information Systems Laboratory, Stanford University Stanford, CA 943-91 lxiao@stanford.edu, boyd@stanford.edu Abstract We consider
More informationData Mining and Analysis: Fundamental Concepts and Algorithms
Data Mining and Analysis: Fundamental Concepts and Algorithms dataminingbook.info Mohammed J. Zaki 1 Wagner Meira Jr. 2 1 Department of Computer Science Rensselaer Polytechnic Institute, Troy, NY, USA
More informationA Study on Relationship between Modularity and Diffusion Dynamics in Networks from Spectral Analysis Perspective
A Study on Relationship between Modularity and Diffusion Dynamics in Networks from Spectral Analysis Perspective Kiyotaka Ide, Akira Namatame Department of Computer Science National Defense Academy of
More informationORIE 4741: Learning with Big Messy Data. Spectral Graph Theory
ORIE 4741: Learning with Big Messy Data Spectral Graph Theory Mika Sumida Operations Research and Information Engineering Cornell September 15, 2017 1 / 32 Outline Graph Theory Spectral Graph Theory Laplacian
More informationLecture VI Introduction to complex networks. Santo Fortunato
Lecture VI Introduction to complex networks Santo Fortunato Plan of the course I. Networks: definitions, characteristics, basic concepts in graph theory II. III. IV. Real world networks: basic properties
More informationNumerical evaluation of the upper critical dimension of percolation in scale-free networks
umerical evaluation of the upper critical dimension of percolation in scale-free networks Zhenhua Wu, 1 Cecilia Lagorio, 2 Lidia A. Braunstein, 1,2 Reuven Cohen, 3 Shlomo Havlin, 3 and H. Eugene Stanley
More informationConsensus, Flocking and Opinion Dynamics
Consensus, Flocking and Opinion Dynamics Antoine Girard Laboratoire Jean Kuntzmann, Université de Grenoble antoine.girard@imag.fr International Summer School of Automatic Control GIPSA Lab, Grenoble, France,
More informationEffect of the interconnected network structure on the epidemic threshold
PHYSICAL REVIEW E 8883 Effect of the interconnected network structure on the epidemic threshold Huijuan Wang * Qian Li Gregorio D Agostino 3 Shlomo Havlin 4 H. Eugene Stanley and Piet Van Mieghem Faculty
More informationNetwork observability and localization of the source of diffusion based on a subset of nodes
Network observability and localization of the source of diffusion based on a subset of nodes Sabina Zejnilović, João Gomes, Bruno Sinopoli Carnegie Mellon University, Department of Electrical and Computer
More informationEstimating network degree distributions from sampled networks: An inverse problem
Estimating network degree distributions from sampled networks: An inverse problem Eric D. Kolaczyk Dept of Mathematics and Statistics, Boston University kolaczyk@bu.edu Introduction: Networks and Degree
More informationDiffusion of Innovation and Influence Maximization
Diffusion of Innovation and Influence Maximization Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics
More informationThe Hierarchical Product of Graphs
The Hierarchical Product of Graphs Lali Barrière Francesc Comellas Cristina Dalfó Miquel Àngel Fiol Universitat Politècnica de Catalunya - DMA4 April 8, 2008 Outline 1 Introduction 2 Graphs and matrices
More informationComplex Networks, Course 303A, Spring, Prof. Peter Dodds
Complex Networks, Course 303A, Spring, 2009 Prof. Peter Dodds Department of Mathematics & Statistics University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.
More informationKalavakkam, Chennai, , Tamilnadu, INDIA 2,3 School of Advanced Sciences. VIT University Vellore, , Tamilnadu, INDIA
International Journal of Pure and Applied Mathematics Volume 09 No. 4 206, 799-82 ISSN: 3-8080 (printed version); ISSN: 34-3395 (on-line version) url: http://www.ijpam.eu doi: 0.2732/ijpam.v09i4.4 PAijpam.eu
More informationStructural Vulnerability Assessment of Electric Power Grids
Structural Vulnerability Assessment of Electric Power Grids Yakup Koç 1 Martijn Warnier 1 Robert E. Kooij 2,3 Frances M.T. Brazier 1 1 Faculty of Technology, Policy and Management, Delft University of
More informationMATH 56A: STOCHASTIC PROCESSES CHAPTER 0
MATH 56A: STOCHASTIC PROCESSES CHAPTER 0 0. Chapter 0 I reviewed basic properties of linear differential equations in one variable. I still need to do the theory for several variables. 0.1. linear differential
More informationSpectral Graph Theory and You: Matrix Tree Theorem and Centrality Metrics
Spectral Graph Theory and You: and Centrality Metrics Jonathan Gootenberg March 11, 2013 1 / 19 Outline of Topics 1 Motivation Basics of Spectral Graph Theory Understanding the characteristic polynomial
More informationInformation Propagation Analysis of Social Network Using the Universality of Random Matrix
Information Propagation Analysis of Social Network Using the Universality of Random Matrix Yusuke Sakumoto, Tsukasa Kameyama, Chisa Takano and Masaki Aida Tokyo Metropolitan University, 6-6 Asahigaoka,
More informationAn Introduction to Spectral Graph Theory
An Introduction to Spectral Graph Theory Mackenzie Wheeler Supervisor: Dr. Gary MacGillivray University of Victoria WheelerM@uvic.ca Outline Outline 1. How many walks are there from vertices v i to v j
More informationAlgebraic Representation of Networks
Algebraic Representation of Networks 0 1 2 1 1 0 0 1 2 0 0 1 1 1 1 1 Hiroki Sayama sayama@binghamton.edu Describing networks with matrices (1) Adjacency matrix A matrix with rows and columns labeled by
More informationLecture 1 and 2: Introduction and Graph theory basics. Spring EE 194, Networked estimation and control (Prof. Khan) January 23, 2012
Lecture 1 and 2: Introduction and Graph theory basics Spring 2012 - EE 194, Networked estimation and control (Prof. Khan) January 23, 2012 Spring 2012: EE-194-02 Networked estimation and control Schedule
More informationSparse Linear Algebra Issues Arising in the Analysis of Complex Networks
Sparse Linear Algebra Issues Arising in the Analysis of Complex Networks Department of Mathematics and Computer Science Emory University Atlanta, GA 30322, USA Acknowledgments Christine Klymko (Emory)
More information1 Complex Networks - A Brief Overview
Power-law Degree Distributions 1 Complex Networks - A Brief Overview Complex networks occur in many social, technological and scientific settings. Examples of complex networks include World Wide Web, Internet,
More informationA lower bound for the spectral radius of graphs with fixed diameter
A lower bound for the spectral radius of graphs with fixed diameter Sebastian M. Cioabă Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA e-mail: cioaba@math.udel.edu Edwin
More informationEpidemics and information spreading
Epidemics and information spreading Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Social Network
More informationCommunities, Spectral Clustering, and Random Walks
Communities, Spectral Clustering, and Random Walks David Bindel Department of Computer Science Cornell University 26 Sep 2011 20 21 19 16 22 28 17 18 29 26 27 30 23 1 25 5 8 24 2 4 14 3 9 13 15 11 10 12
More informationApplied Mathematics Letters
Applied Mathematics Letters (009) 15 130 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Spectral characterizations of sandglass graphs
More informationData Analysis and Manifold Learning Lecture 9: Diffusion on Manifolds and on Graphs
Data Analysis and Manifold Learning Lecture 9: Diffusion on Manifolds and on Graphs Radu Horaud INRIA Grenoble Rhone-Alpes, France Radu.Horaud@inrialpes.fr http://perception.inrialpes.fr/ Outline of Lecture
More informationPerformance Analysis of Complex Networks and Systems
Performance Analysis of Complex Networks and Systems PIET VAN MIEGHEM Delft University of Technology in this web service University Printing House, Cambridge CB2 8BS, United Kingdom is part of the University
More informationarxiv: v2 [physics.soc-ph] 6 Aug 2012
Localization and spreading of diseases in complex networks arxiv:122.4411v2 [physics.soc-ph] 6 Aug 212 A. V. Goltsev, 1,2 S. N. Dorogovtsev, 1,2 J. G. Oliveira, 1,3 and J. F. F. Mendes 1 1 Department of
More informationSpectral Graph Theory
Spectral Graph Theory Aaron Mishtal April 27, 2016 1 / 36 Outline Overview Linear Algebra Primer History Theory Applications Open Problems Homework Problems References 2 / 36 Outline Overview Linear Algebra
More informationON THE WIENER INDEX AND LAPLACIAN COEFFICIENTS OF GRAPHS WITH GIVEN DIAMETER OR RADIUS
MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 63 (2010) 91-100 ISSN 0340-6253 ON THE WIENER INDEX AND LAPLACIAN COEFFICIENTS OF GRAPHS WITH GIVEN DIAMETER
More informationThe Distance Spectrum
The Distance Spectrum of a Tree Russell Merris DEPARTMENT OF MATHEMATICS AND COMPUTER SClENCE CALlFORNlA STATE UNlVERSlN HAYWARD, CAL lfornla ABSTRACT Let T be a tree with line graph T*. Define K = 21
More informationDiffusion and random walks on graphs
Diffusion and random walks on graphs Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Structural
More informationDiffusion of information and social contagion
Diffusion of information and social contagion Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics
More informationKernels of Directed Graph Laplacians. J. S. Caughman and J.J.P. Veerman
Kernels of Directed Graph Laplacians J. S. Caughman and J.J.P. Veerman Department of Mathematics and Statistics Portland State University PO Box 751, Portland, OR 97207. caughman@pdx.edu, veerman@pdx.edu
More informationEmergent Phenomena on Complex Networks
Chapter 1 Emergent Phenomena on Complex Networks 1.0.1 More is different When many interacting elements give rise to a collective behavior that cannot be explained or predicted by considering them individually,
More informationA Modified Earthquake Model Based on Generalized Barabási Albert Scale-Free
Commun. Theor. Phys. (Beijing, China) 46 (2006) pp. 1011 1016 c International Academic Publishers Vol. 46, No. 6, December 15, 2006 A Modified Earthquake Model Based on Generalized Barabási Albert Scale-Free
More informationMachine Learning for Data Science (CS4786) Lecture 11
Machine Learning for Data Science (CS4786) Lecture 11 Spectral clustering Course Webpage : http://www.cs.cornell.edu/courses/cs4786/2016sp/ ANNOUNCEMENT 1 Assignment P1 the Diagnostic assignment 1 will
More informationThe Hierarchical Product of Graphs
The Hierarchical Product of Graphs Lali Barrière Francesc Comellas Cristina Dalfó Miquel Àngel Fiol Universitat Politècnica de Catalunya - DMA4 March 22, 2007 Outline 1 Introduction 2 The hierarchical
More informationGraphs with given diameter maximizing the spectral radius van Dam, Edwin
Tilburg University Graphs with given diameter maximizing the spectral radius van Dam, Edwin Published in: Linear Algebra and its Applications Publication date: 2007 Link to publication Citation for published
More informationData Mining and Analysis: Fundamental Concepts and Algorithms
: Fundamental Concepts and Algorithms dataminingbook.info Mohammed J. Zaki 1 Wagner Meira Jr. 2 1 Department of Computer Science Rensselaer Polytechnic Institute, Troy, NY, USA 2 Department of Computer
More informationLecture 10. Under Attack!
Lecture 10 Under Attack! Science of Complex Systems Tuesday Wednesday Thursday 11.15 am 12.15 pm 11.15 am 12.15 pm Feb. 26 Feb. 27 Feb. 28 Mar.4 Mar.5 Mar.6 Mar.11 Mar.12 Mar.13 Mar.18 Mar.19 Mar.20 Mar.25
More informationGéza Ódor MTA-EK-MFA Budapest 16/01/2015 Rio de Janeiro
Griffiths phases, localization and burstyness in network models Géza Ódor MTA-EK-MFA Budapest 16/01/2015 Rio de Janeiro Partners: R. Juhász M. A. Munoz C. Castellano R. Pastor-Satorras Infocommunication
More informationEvolutionary Games on Networks. Wen-Xu Wang and G Ron Chen Center for Chaos and Complex Networks
Evolutionary Games on Networks Wen-Xu Wang and G Ron Chen Center for Chaos and Complex Networks Email: wenxuw@gmail.com; wxwang@cityu.edu.hk Cooperative behavior among selfish individuals Evolutionary
More informationLECTURE NOTE #11 PROF. ALAN YUILLE
LECTURE NOTE #11 PROF. ALAN YUILLE 1. NonLinear Dimension Reduction Spectral Methods. The basic idea is to assume that the data lies on a manifold/surface in D-dimensional space, see figure (1) Perform
More informationMoment-Based Analysis of Synchronization in Small-World Networks of Oscillators
Moment-Based Analysis of Synchronization in Small-World etworks of Oscillators Victor M. Preciado and Ali Jadbabaie Abstract In this paper, we investigate synchronization in a small-world network of coupled
More informationThe non-backtracking operator
The non-backtracking operator Florent Krzakala LPS, Ecole Normale Supérieure in collaboration with Paris: L. Zdeborova, A. Saade Rome: A. Decelle Würzburg: J. Reichardt Santa Fe: C. Moore, P. Zhang Berkeley:
More informationCS168: The Modern Algorithmic Toolbox Lectures #11 and #12: Spectral Graph Theory
CS168: The Modern Algorithmic Toolbox Lectures #11 and #12: Spectral Graph Theory Tim Roughgarden & Gregory Valiant May 2, 2016 Spectral graph theory is the powerful and beautiful theory that arises from
More informationSynchronization Transitions in Complex Networks
Synchronization Transitions in Complex Networks Y. Moreno 1,2,3 1 Institute for Biocomputation and Physics of Complex Systems (BIFI) University of Zaragoza, Zaragoza 50018, Spain 2 Department of Theoretical
More informationTHE unavailability of electrical power can severely disrupt
2524 IEEE SYSTEMS JOURNAL, VOL. 12, NO. 3, SEPTEMBER 2018 A Topological Investigation of Power Flow Hale Cetinay, Fernando A. Kuipers, Senior Member, IEEE, and Piet Van Mieghem Abstract This paper combines
More informationLecture 1: Graphs, Adjacency Matrices, Graph Laplacian
Lecture 1: Graphs, Adjacency Matrices, Graph Laplacian Radu Balan January 31, 2017 G = (V, E) An undirected graph G is given by two pieces of information: a set of vertices V and a set of edges E, G =
More informationQuantitative model to measure the spread of Security attacks in Computer Networks
Quantitative model to measure the spread of Security attacks in Computer Networks SAURABH BARANWAL M.Sc.(Int.) 4 th year Mathematics and Sci. Computing Indian Institute of Technology, Kanpur email: saurabhbrn@gmail.com,
More informationShannon Entropy and Degree Correlations in Complex Networks
Shannon Entropy and Degree Correlations in Complex etworks SAMUEL JOHSO, JOAQUÍ J. TORRES, J. MARRO, and MIGUEL A. MUÑOZ Instituto Carlos I de Física Teórica y Computacional, Departamento de Electromagnetismo
More informationAttack Strategies on Complex Networks
Attack Strategies on Complex Networks Lazaros K. Gallos 1, Reuven Cohen 2, Fredrik Liljeros 3, Panos Argyrakis 1, Armin Bunde 4, and Shlomo Havlin 5 1 Department of Physics, University of Thessaloniki,
More informationStability and topology of scale-free networks under attack and defense strategies
Stability and topology of scale-free networks under attack and defense strategies Lazaros K. Gallos, Reuven Cohen 2, Panos Argyrakis, Armin Bunde 3, and Shlomo Havlin 2 Department of Physics, University
More informationCS224W: Analysis of Networks Jure Leskovec, Stanford University
Announcements: Please fill HW Survey Weekend Office Hours starting this weekend (Hangout only) Proposal: Can use 1 late period CS224W: Analysis of Networks Jure Leskovec, Stanford University http://cs224w.stanford.edu
More informationNetworks and Their Spectra
Networks and Their Spectra Victor Amelkin University of California, Santa Barbara Department of Computer Science victor@cs.ucsb.edu December 4, 2017 1 / 18 Introduction Networks (= graphs) are everywhere.
More informationv iv j E(G) x u, for each v V(G).
Volume 3, pp. 514-5, May 01 A NOTE ON THE LEAST EIGENVALUE OF A GRAPH WITH GIVEN MAXIMUM DEGREE BAO-XUAN ZHU Abstract. This note investigates the least eigenvalues of connected graphs with n vertices and
More informationSpectra of Adjacency and Laplacian Matrices
Spectra of Adjacency and Laplacian Matrices Definition: University of Alicante (Spain) Matrix Computing (subject 3168 Degree in Maths) 30 hours (theory)) + 15 hours (practical assignment) Contents 1. Spectra
More informationIntrinsic products and factorizations of matrices
Available online at www.sciencedirect.com Linear Algebra and its Applications 428 (2008) 5 3 www.elsevier.com/locate/laa Intrinsic products and factorizations of matrices Miroslav Fiedler Academy of Sciences
More information6.207/14.15: Networks Lectures 4, 5 & 6: Linear Dynamics, Markov Chains, Centralities
6.207/14.15: Networks Lectures 4, 5 & 6: Linear Dynamics, Markov Chains, Centralities 1 Outline Outline Dynamical systems. Linear and Non-linear. Convergence. Linear algebra and Lyapunov functions. Markov
More information