1 Epidemic Spreading in Real Networks: an Eigenvalue Viewpoint Yang Wang Deepayan Chakrabarti Chenxi Wang Christos Faloutsos

2 Introduction Computer viruses are a prevalent threat Existing defense mechanisms (eg., scanning) focus on local behaviors only  like “curing” a contagious disease in one patient Global defense strategies require the understanding of global propagation behaviors  like “prevention” of spread of a contagious disease in a population  Epidemiological models can help us do exactly that

3 Introduction Why do we care?  Understanding the spread of a virus is the first step in preventing it  How fast do we need to disinfect nodes so that the virus attack dies off?  How long will the virus take to die out?

4 Problem definition Question: How does a virus spread across an arbitrary network? Specifically, we want  a general analytic model for viral propagation  that applies to any network topology  and offers an easy-to-compute “threshold condition”

5 Framework The network of computers consists of nodes (computers) and edges (links between nodes) Each node is in one of two states  Susceptible (in other words, healthy)  Infected Susceptible-Infected-Susceptible (SIS) model  Cured nodes immediately become susceptible SusceptibleInfected Infected by neighbor Cured internally

6 Framework (Continued) Homogeneous birth rate β on all edges between infected and susceptible nodes Homogeneous death rate δ for infected nodes Infected Healthy XN1 N3 N2 Prob. β Prob. δ

7 Outline Introduction Classical models and their limitations Modeling viral propagation in arbitrary network topologies Epidemic threshold and eigenvalues Experiments Conclusions

8 Basic Homogeneous Model [Kephart-White ’91, ’93]  Homogeneous connectivity  Every node has equal probability of connecting to every other node  Many real networks deviate from this!

9 Power-law Networks Many real world networks exhibit power-law characteristics  Probability that a node has k links: P(k) = ck –γ  γ = power law exponent  The Internet: 2  γ  3…and still evolving  [Faloutsos+ ’99, Ripeanu+ ’02]

10 Power-law Networks Model for Barabási-Albert networks (PL-3)  [Pastor-Satorras & Vespignani, ’01, ’02]  Prediction limited to BA type networks  which only allow power-laws of exponent γ = 3

11 Power-law Networks Model for correlated (Markovian) networks  [Boguñá-Satorras ’02]  Additional distribution for neighbor degree correlation: P(k|k’)  Difficult to find/produce P(k|k’) in arbitrary networks  Such correlations have yet to be confirmed in real world networks

12 Outline Introduction Classical models and their limitations Modeling viral propagation in arbitrary network topologies Epidemic threshold and eigenvalues Experiments Conclusions

13 Topology-independent epidemic model Takes topological characteristics into account without being limited by them Discrete time A node is healthy at time t if it  Was healthy before t and not infected at t Infected Healthy XN1 N3 N2

14 Topology-independent epidemic model Takes topological characteristics into account without being limited by them Discrete time A node is healthy at time t if it  Was healthy before t and not infected at t OR  Was infected before t, cured and not re-infected at t Infected Healthy XN1 N3 N2

15 Topology-independent epidemic model Takes topological characteristics into account without being limited by them Discrete time A node is healthy at time t if it  Was healthy before t and not infected at t OR  Was infected before t, cured and not re-infected at t OR  Was infected before t, therefore ignored re-infection attempts and was subsequently cured at t Infected Healthy XN1 N3 N2

16 Topology-independent epidemic model Deterministic time evolution of infection  1 - p i,t : probability node i is healthy at time t  ζ k,t : probability a k-linked node will not receive infections from its neighbors at time t  Assume probability of curing before infection attempts 50%  Solve numerically Equation 1

17 Simulation evaluation of model (1/2) 1000-node homogeneous network KW model Our model Simulation

18 Simulation evaluation of model (2/2) Our model’s predictions consistently equal or outperform predictions made by models designed for specific topologies PL-3 model Our model Simulation Real-world node Oregon network

19 Outline Introduction Classical models and their limitations Modeling viral propagation in arbitrary network topologies Epidemic threshold and eigenvalues Experiments Conclusions

20 Epidemic threshold The epidemic threshold τ is the value such that  β/δ < τ  there is no epidemic  where β = birth rate, and δ = death rate What is this threshold for an arbitrary graph? [Theorem 1] τ = 1/ λ 1,A where λ 1,A is the largest eigenvalue of the adjacency matrix A of the topology λ 1,A alone captures the property of the graph!

21 Epidemic threshold for various networks Our epidemic threshold condition is accurate and general Homogeneous networks  λ 1,A = ; τ = 1/  where = average degree  This is the same result as of Kephart & White ! Star networks  λ 1,A = √d; τ = 1/ √d  where d = the degree of the central node Infinite power-law networks  λ 1,A = ∞; τ = 0 ; this concurs with previous results Finite power-law networks  τ = 1/ λ 1,A

22 Epidemic threshold [Theorem 1] The epidemic threshold is given by »τ = 1/ λ 1,A How fast will an infection die out? [Theorem 2] Below the epidemic threshold, the epidemic dies out exponentially  If β/δ < τ (β = birth rate, δ = death rate) then  any local breakout of infection dies out exponentially fast

23 Outline Introduction Classical models and their limitations Modeling viral propagation in arbitrary network topologies Epidemic threshold and eigenvalues Experiments Conclusions

24 Epidemic threshold experiments (Star) β/δ = τ (close to the threshold) β/δ < τ (below threshold) β/δ > τ (above threshold)

25 Epidemic threshold experiments (Oregon) β/δ > τ (above threshold) β/δ = τ (at the threshold) β/δ < τ (below threshold)

26 Our prediction vs. previous prediction When we do not subsume previous predictions, our predictions are much more accurate OregonStar PL-3 Our Number of infected nodes β/δβ/δβ/δβ/δ

27 Outline Introduction Classical models and their limitations Modeling viral propagation in arbitrary network topologies Epidemic threshold and eigenvalues Experiments Conclusions

28 Contributions We match our goals √ A general analytic model for viral propagation (Equation 1) √ that applies to any network topology √ and offers an easy-to-compute “threshold condition” (Theorem 1)

29 Contributions We created new topology-independent epidemic model  More accurate than previous models  More general than previous models We derived new epidemic threshold condition  Only requires one parameter (λ 1,A ) that can be calculated with existing tools  Subsumes previous theories for epidemic threshold condition  When does not subsume, our theory is more accurate

30 Halting viruses Immunization strategies must concentrate on nodes that are statistically significant Statistically significant nodes are not necessarily limited to ones that are highly connected  We are building mathematical models to identify the most significant nodes in power-law models Other system parameters may also matter

31 Summary and future work Our models will provide a theoretical basis for global defense strategies for  intelligent immunization  mechanisms to guard against distributed denial-of-service (DDOS) attacks  those that propagate via virus code – Phase transition phenomena at epidemic threshold Additional environmental factors that affect epidemic behavior

32 Basic homogeneous model - KW Homogeneous connectivity Homogeneous birth rate β on all edges between infected and susceptible nodes Homogeneous death rate δ for infected nodes Susceptible-Infected-Susceptible (SIS) model  Cured individuals immediately become susceptible Susceptible-Infected-Removed (SIR) model  Cured individuals are removed from the population

33 Homogeneous model equations Deterministic time evolution of infected population η t  Change = birth term - death term Equilibrium point of infection η, ρ’= δ/(β ) For homogeneous or Erdös-Rényi (random) networks

34 Homogeneous model η = 1-  ’= 1 -  /(  ) = = 0.9

35 Power-law networks Discrepancy between simulation results and homogeneous model predictions

36 Power-law networks There exist statistically significant nodes  Node 928 was infected 9473 times  Run #3 hits 928 around time 20  Both runs #1 and #2 hit 928 early in its run

37 Models for power-law and correlated networks Many real world networks exhibit power-law characteristics  P(k) = k -γ --- probability that a node has k links  The Internet: 2  γ  3…and still evolving Model for Barabási-Albert networks (SV)  γ = 3  Steady state: η = 2e -δ/mβ, m = minimum connectivity  Prediction limited to BA type networks Model for correlated (Markovian) networks  Additional distribution for neighbor degree correlation: P(k|k’)  Difficult to find/produce P(k|k’) in arbitrary networks  Such correlations have yet to be confirmed in real world networks

38 Epidemic threshold The epidemic threshold τ = β/δ (the ratio of birth rate to death rate) below which there is no epidemic Epidemic threshold for existing models:  Threshold of the basic homogeneous model: 1/  Threshold of SV power-law model: / We derive an epidemic threshold condition from our model  τ = 1/ λ 1,A  λ 1,A : largest eigenvalue of the adjacency matrix A of the topology Power-law networks have extremely low threshold since connectivity variance is usually high, resulting in large λ 1,A