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
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
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. -4. - Van Mieghem, P.,, "The -Intertwined SIS epidemic network model", Computing (Springer), Vol. 93, Issue, p. 47-69. 6 3
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. 484. 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 3 6 5 B L = 6 L = 9 = 4 T A = = A T Q = BB = Δ A Δ = diag ( d d d 8 ) 4
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. 643-65. 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. 563. 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. 48-49. 5
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. 643-65. 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
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 ) 6 3 + others ( ) 3 P. Van Mieghem, Graph Spectra for Complex etworks, Cambridge University Press, Comparison! 5 5 exact bound bound bound 3 3 -.4 -....4 " D Barabasi-Albert power law graph with = 5 and L=9! " 3 $ & % / 3 4 7
USA air transportation network 5 Degree-preserving rewiring USA air transport network: adjacency eig. 5 Adjacency eigenvalues!d Percentage of rewired links 3..5..5. 4 5-5 6 Percentage of rewired links 3 4 8
Degree-preserving rewiring USA air transport network: Laplacian eig..7. Laplacian eigenvalues.6.5.4.3. Pr[D = k].. Pr[D = k] ~ k -. 4 6 4 6 4 degree k.. 4 6 8 7 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
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.4...8.6.4 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.. 5 5 5 3 35 4 45 5 n Li, C., H. Wang and P. Van Mieghem,, "Degree and Principal Eigenvectors in Complex etworks", IFIP etworking, May -5, Prague, Czech Republic.
Power law networks.3 The component of the largest eigenvector.8.6.4...8.6.4...8.6.4. 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. 5 5 5 3 35 4 45 5 n Power law networks (log-scale) - The component of the largest eigenvector - -3-4 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 5 5 5 3 35 4 45 5 n
E[D] = 8 3 Outline Dynamic Processes on etworks Modifying the spectral radius Principal eigenvector ew (?) graph theoretic results Summary
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) () 35 353. 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. 39-33. 5 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
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 4 6 5 Most network problems: - shortest path - graph metrics - network algorithms X 3-ary tree : green = 8 4
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
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: http://www.nas.ewi.tudelft.nl 3 6
Thank You Piet Van Mieghem AS, TUDelft P.F.A.VanMieghem@tudelft.nl 33 7