Graph Spectra & the N-intertwined Mean-field SIS approximation on Networks Piet Van Mieghem work in collaboration with Eric Cator, Huijuan Wang, Rob Kooij 1 Spectral Properties of Complex Networks, ECT Workshop, Trento 23-27 July, 2012
Outline Introduction & Definitions Exact SIS model The N-intertwined MF approximation Extensions Summary
Effect of Network on Function Brain Biology Social Economy Network Science networking Man-made Infrastructures Internet, power-grid Brain/Bio-inspired Networking to design superior man-made robust networks/systems
Motivation for virus spread in networks • Digital world: • Information spread in on-line social networks • security threat to Internet ( Code Red worm: several billion $ $ in damage) • Real world: Biological epidemics (e.g. Mexican flue) • Why do we care ? • Understanding the spread of a virus is the first step in prevention • How fast do we need to disinfect nodes so that the virus dies quickly? Which nodes? 4
Algebraic graph theory Any graph G can be represented by an adjacency matrix A and an incidence matrix B, and a Laplacian Q 0 1 1 0 0 1 N = 6 L = 9 1 3 1 0 1 0 1 1 4 1 1 0 1 0 0 2 T A A = = 0 0 1 0 1 0 N N × 6 5 0 1 0 1 0 1 1 1 1 … 0 1 1 0 0 1 0 − 1 0 0 0 − T Q BB A 0 1 0 0 = = Δ − − B = N L × 0 0 0 0 diag ( d d … d ) Δ = 1 2 N 0 0 0 1 − 0 0 1 1 5
Outline Introduction & Definitions Exact SIS model The N-intertwined MF approximation Extensions Summary
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 Healthy 1 δ β 0 3 Infected 2 Infected 7
Definition of the states in SIS � • Each node j can be in either of the two states: 0 1 • “ 0 ” : healthy � • “ 1 ” : infected • Markov continuous time : • infection rate β • curing rate δ • Mathematically: • X j is the state of node j � � � � � q 1 j � = � q 1 j q 1 j q 1 j ( ) = • infinitesimal generator � � � Q j t � q 2 j � � � � � � � q 2 j 8
Governing SIS equation for node j � � N dE [ X j ] � = E � � X j + (1 � X j ) � a kj X k � � dt � � k = 1 time-change of if infected : if not infected (healthy) : E[ X j ] = Pr[ X j = 1], probability of probability of probability that curing per infection per node j is infected unit time unit time N N dE [ X j ] � � � � � � [ ] = � � E X j a kj E X k a kj E X j X k � + � � � � � � dt k = 1 k = 1 9
Joint probabilities � � � � dE X i X j N N � � � � = E � 2 � X i X j + X j � (1 � X i ) a ik X k + X i � (1 � X j ) a jk X k � � dt � � k = 1 k = 1 N N N � � � � � � � ( ) E X i X j X k � � [ ] � = � 2 � E X i X j a ik E X j X k a jk E X i X k a jk + a ik � + � � + � � � � � � k = 1 k = 1 k = 1 � � Next, we need the differential equations for E[X i X j X k ]... N � � 3 � � � � N N 2 N = In total, the SIS process is defined by � � + 1 � k linear equations � � k = 1 10 E. Cator and P. Van Mieghem, 2012, "Second-order mean-field susceptible -infected-susceptible epidemic threshold", Physical Review E, vol. 85, No. 5, May, p. 056111.
Exact SIS model 0000 Absorbing state N = 4 nodes 0 0001 0010 0100 1000 1 2 4 8 1001 0011 0101 0110 1010 1100 9 3 5 6 10 12 1011 0111 1101 1110 11 7 13 14 1111 2 N states! 15
Markov theory bi-partite Markov graph 0001 0001 0000 0000 0000 0000 1 1 0 0 0 0 0010 0010 0011 0011 2 2 3 3 0001 0001 0010 0010 0100 0100 1000 1000 1 1 2 2 4 4 8 8 0100 0100 0101 0101 4 4 5 5 0111 0111 0110 0110 1001 1001 0011 0011 0101 0101 0110 0110 1010 1010 1100 1100 7 7 6 6 9 9 3 3 5 5 6 6 10 10 12 12 1000 1000 1001 1001 8 8 9 9 0111 0111 1101 1101 1110 1110 1011 1011 1011 1011 1010 1010 11 11 7 7 13 13 14 14 11 11 10 10 1101 1101 1100 1100 13 13 12 12 1111 1111 15 15 1110 1110 1111 1111 14 14 15 15 Recursive structure of infinitesimal general Q N 12 Van Mieghem, P. and E. Cator, ε -SIS epidemics and the epidemic threshold, Physical Review E, to appear 2012
Markov Theory • SIS model is exactly described as a continuous-time Markov process on 2 N states, with infinitesimal generator Q N . • Drawbacks : • no easy structure in Q N • computationally intractable for N>20 • steady-state is the absorbing state (reached after unrealistically long time) • very few exact results... • The mathematical community (e.g. Liggett, Durrett,...) uses: • duality principle & coupling & asymptotics • graphical representation of the Poisson infection and recovery events 13
Outline Introduction & Definitions Exact SIS model The N-intertwined MF approximation Extensions Summary
N-intertwined mean-field approximation N N dE [ X j ] � � � � � � [ ] = � � E X j a kj E X k a kj E X j X k � + � � � � � � dt k = 1 k = 1 and � � � � � � [ ] � � � � E X j X k � = Pr X j = 1, X k = 1 � = Pr X j = 1 X k = 1 � Pr X k = 1 Pr X j = 1 X k = 1 �� Pr X j = 1 � � � � � � [ ] � Pr X i = 1 [ ] Pr X k = 1 [ ] = E X i [ ] E X k [ ] E X i X k N N dE [ X j ] � � � � � � [ ] [ ] � � � E X j a kj E X k � � E X j a kj E X k � + � � � � dt k = 1 k = 1 N-intertwined mean-field approx. (= equality above) upper bounds the probability of infection 15 E. Cator and P. Van Mieghem, 2012, "Second-order mean-field susceptible -infected-susceptible epidemic threshold", Physical Review E, vol. 85, No. 5, May, p. 056111.
N-intertwined non-linear equations � N dv 1 � dt = (1 � v 1 ) � v k � � v 1 � a 1 k � k = 1 where the viral probability of � N dv 2 � � dt = (1 � v 2 ) � v k � � v 2 infection is a 2 k � � � � k = 1 ( ) = E [ X k ( t )] = Pr X k t ( ) = 1 v k t � � � � N dv N � � dt = (1 � v N ) � v k � � v N a Nk � � k = 1 In matrix form: ( ) dV t ( ) � A . V t ( ) ( ) � diag v i t ( ) ( ) + � u = � A . V t dt where the vector u T =[1 1 ... 1] and V T = [v 1 v 2 ... v N ] 16 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).
Lower bound for the epidemic threshold N N dv j ( t ) � � ( ) = E [ X k ( t )] [ ] v k t = � � v j + � a kj v k a kj E X i X k � � dt k = 1 k = 1 Is the point V = 0 stable? For a very few infected nodes, we can ignore the quadratic terms dV ( t ) ( ) V ( t ) = � � I + � A dt The origin V=0 is stable attractor if all eigenvalues of � A � � I are negative (v j tends exponentially fast to zero with t ). Hence, if 1 � = � �� 1 ( A ) � � < 0 � < � 1 ( A ) < � c The N-intertwined mean-field epidemic threshold is precisely 1 1 1 � (1) � (1) � 1 ( A ) < � (2) c = � 1 ( A ) < � c c = c = � 1 ( H ) < � c 17
Exact in steady-state for large τ � τ Almost all neighbors of node j are 0 1 infected: independence � � 1 � 1 � � � � � d j = 1 + 1 = 1 + s � � Pr X j = 1 � � � � �� � � � � � � + � d j � d j d j � � � � Exact steady-state fraction of infected nodes: � 1 � � N N y � ( s ) � 1 = 1 1 + s � � � � Pr X j = 1 � � � � � � N N d j � � j = 1 j = 1 N � � dy � ( s ) = 1 1 = E 1 Slope: � � � ds N d j � D � s = 0 j = 1 18 P. Van Mieghem, 2012, “The Viral Conductance in Networks”, Computer Communications, Vol 35, No. 12, pp. 1494-1509.
What is so interesting about epidemics? β : infection rate per link δ : curing rate per node τ = β / δ : effective spreading rate • Final epidemic state • Rate of propagation • Epidemic threshold 1 � c = ( ) � 1 A 19 � � [ ] 1 + Var [ D ] � � ( ) � d max max E D 2 , d max � � � 1 A � ( ) E [ D ] � �
s = 1 Transformation & principal eigenvector � N dy � ( s ) = � 1 1 � � 1 � ds N d j 2 L s = 0 j = 1 N � ( x 1 ) j dy � ( s ) = � 1 j = 1 N ds N � 3 s = � 1 ( x 1 ) convex j j = 1 20 s c = λ 1 Van Mieghem, P., 2012, "Epidemic Phase Transition of the SIS-type in Networks", Europhysics Letters (EPL), Vol. 97, Februari, p. 48004.
Simulations 500 simulations K 10,990 � = 1 s = 0.15 ( ) y � ( s ) = mn � s 2 � � 1 1 s + m + � � � � m + n s + n 21 time
Recommend
More recommend
