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Controlling the Spread of Viruses on Power-Law Networks

This study investigates the epidemic threshold of the contact process on power-law graphs and explores methods for controlling the spread of viruses and worms in such networks.

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Controlling the Spread of Viruses on Power-Law Networks

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  1. Controlling the Spread of Viruses on Power-Law Networks Jennifer Tour Chayes Joint work with N. Berger, C. Borgs, A. Ganesh, A. Saberi, D. B. Wilson

  2. The Internet Graph appears to have a power-law degree distribution Faloutsos, Faloutsos and Faloutsos ‘99

  3. The Sex Web and Other Social Networks also appear to have a power-law degree distribution Lilijeros et. al ‘01

  4. Model for Power-Law Graphs: Preferential Attachment • Add one vertex at a time • New vertex i attaches to m¸ 1 existing vertices j chosen i.i.d. as follows: With probability , choose j uniformly and with probability 1- , choose j according to • Prob(i attaches to j) / dj • with dj = degree(j) non-rigorous: Simon ‘55, Barabasi-Albert ‘99, measurements: Kumar et. al. ‘00, rigorous: Bollobas-Riordan ‘00, Bollobas et. al. ‘03

  5. Computer Viruses and Worms • Viruses are programs that • attach themselves to a host program (executable) • cannot spread unless you run an infected application or attachment • Worms are programs that • break into your computer using some vulnerability • do not require user actions to spread

  6. Mutating Viruses and Worms • So far, Internet viruses and worms have been non-mutating • The next big threat is mutatingviruses and worms, e.g. a worm equippedwith a list of vulnerabilities that changes the vulnerability exploited as a deterministic or random function of time, or in response to a command from a central authority

  7. Model for of Mutating Viruses & Worms: Contact Process • Definition of model: infected ! healthy at rate r healthy ! infected at rate b (# infected nhbrs) relevant parameter: l = b/r • Studied in probability theory, physics, epidemiology • Kephart and White ’93: modelling the spread of viruses in a computer network

  8. Epidemic Threshold(s) 12 • Infinite graph: extinction weak strong survival survival Note: 1 = 2 on Zd 1 < 2 on a tree • Finite subset logarthmic polynomial exponential of Zd: survival survival (super-poly) time time survival time

  9. The Internet Graph What is the epidemic threshold of the Internet graph, and is there a way of increasing the threshold, i.e. controlling the spread of the epidemic?

  10. Part I:Epidemics of Mutating Viruses and Worms Question: What is the epidemic threshold of the contact process on power-law graphs? -- work in collaboration with Berger, Borgs & Saberi (SODA ’05)

  11. Epidemic Threshold in Scale-Free Network • In power-law networks both thresholds are zero asymptotically almost surely, i.e. • 1 = 2 = 0 a.a.s. • Physics argument:Pastarros, Vespignani ‘01 • Rigorous proof: Berger, Borgs, C., Saberi ’05 Moreover, we get detailed estimates (matching upper and lower bounds) on the survival probability as a function of 

  12. log (1/) log (1/) C1 C2 log log (1/) log log (1/) Theorem 1.For every  > 0, and for all n large enough, if the infection starts from a uniformly random vertex in a sample of the scale-free graph of size n, then with probability 1-O(2),v is such that the infection survives longer than en0.1 with probability at leastand with probability at mostwhere 0 < C1 < C2 < 1 are independent of  and n.

  13. Typical versus average behavior • Notice that we left out O(2 n) vertices in Theorem 1. • Q: What are the effect of these vertices on the average survival probability? • A: Dramatic.

  14. Theorem 2.For every  > 0, and for all n large enough, if the infection starts from a uniformly random vertex in a sample of the scale-free graph of size n, then the infection survives longer than en0.1 with probability at leastC3and with probability at mostC4where 0 < C3, C4 < 1 are independent of  and n.

  15. log (1/) ( )   log log (1/) Typical versus average behavior • The survival probability for an infection starting from a typical (i.e., 1 – O(2) ) vertex is • The average survival probability is (1)

  16. Key Elements of the Proof: 1. For the contact process • If the maximum degree is much less than 1/, then the infection dies out very quickly. • On a vertex of degree much more than 1/2, the infection lives for a long time in the neighborhood of the vertex (“star lemma”).

  17. Star Lemma If we start by infecting the centerof a star of degree k ,with high probability, the survival time is more than Key Idea: The center infects a constant fractionof vertices before being cured.

  18. Key Elements of the Proof: 2. For preferential attachment graphs Lemma: With high probability, the largest degree in a ball of radius k about a vertex v is at most (k!)10 and at least (k!)(m,) where (m,) > 0. To prove this, we introduced a Polya Urn Representation of the preferential attachment graph.

  19. Polya Urn Representation of Graph • Polya’s Urn: At each time step, add a ball to one of the urns with probability proportional to the number of balls already in that urn. • Polya’s Theorem: This is equivalent to choosing a number p according to the -distribution, and then sending the balls i.i.d. with probability p to the left urn and with probability 1– p to the right urn. • Use this and some work to show that the addition of a new vertex can be represented by adding a new urn to the existing sequence of urns and adding edges between the new urn and m of the old ones.

  20. log (1/) k = log log (1/) “Proof” of Main Theorem: v • Let By the preferential attachment lemma, the ball of radius C1k around vertex v contains a vertex w of degree larger than [(C1k)!] > -5 where the inequality follows by taking C1 large. The infection must travel at most C1k to reach w, which happens with probability at least C1k , at which point, by the star lemma, the survival time is more than exp(C -3 ). Iterate until we reach a vertex z of sufficiently high degree for exp(n1/10) survival. w z Iterations to get to high-degree vertex

  21. Summary of Part I: • Developed a new representation of the preferential attachment model: Polya Urn Representation. • Used the representation to: 1. prove that any virus with a positive rate of spread has apositive probability of becoming epidemic 2. calculate the survival probability for both typical and average vertices

  22. Part II:Control of Mutating Viruses and Worms: Question: What is the best way to distribute antidote to control the epidemic, i.e. to raise the thresholdof the contact process on power-law (and more general) graphs? -- work in collaboration with Borgs, Ganesh, Saberi & Wilson ‘06

  23. For l = b/r, previous results with r = const.:Our results (BBCS) for growing power-law graphsGanesh, Massoulie, Towsley (GMT) for “configurational” power-law graphs • For stars: • bc = rn-1/2 + o(1) ,amount of antidote R = nrrequired to suppress epidemic isbn3/2 + o(1) , i.e. superlinear in n • For power-law graphs: • bc! 0 ,amount of antidote R required to suppress epidemicis superlinear in n

  24. Varying Recovery Rates r = rx • Assume there is a fixed amount of antidote R = Sxrx to be distributed non-uniformly among the sites, even depending on the current state of the infection • Questions: • What is the best policy for distributing R? • Is there a way to control the infection (i.e., to get lc > 0) on a star or power-law graph with R scaling linearly in n?

  25. Method 1: Contact Tracing • Contact tracing is a method in epidemiology to diagnose and treat the contacts of infected individuals , augmenting the cure rate of neighbors of infected nodes,cure / infected degree • Theorem 1: Let rx = r + r0ix where ix is the number of infected neighbors of x. Then the critical infection rate on the star is bc = rn-1/3 + o(1) ! 0. • Note: This is an improvement from the case r = const, where bc = rn-1/2 + o(1), but thisstill gives bc! 0, or alternatively, it takes R = n4/3 + o(1), i.e. a superlinear amount, of antidote to control the virus.

  26. Method 2: Cure / Degree(vs. contact tracing with cure /infected degree) • Theorem 2: Let rx = dx. If b < 1 then the expected survival time is t = O(logn). • Corollary: For graphs with a bounded average degree davg, the total amount of antidote needed to control the epidemic is bdavgn, i.e. linear in n. • Thus, curing proportional to degree is enough to control epidemics on power-law graphs.

  27. Q:Can we do significantly better? I.e., can we get bc! 1 as n ! 1 ? • A: No, for expanders. (Recall a graph G = (V,E) is an (a,h)-expander if for each subset W of V of size at most a|V|, the number of edges joining W to its complement V\W is at least h|W|.) • Condition* (for comparison): Let Xt be the set of infected vertices at time t, and let rx = rx (Xt,t) be an arbitrary non-uniform allocation of antidote obeying the condition that the sum of rx over any subset of V is less than the sum of the degrees over that subset.

  28. Expanders, continued: • Theorem 3: Let e > 0, and let Gn be a sequence of (a,h)-expanders on n nodes. Let rx (Xt,t)obey Condition*. If b¸ (1+e)davg/(ah), then t¸ exp(Q(nlogn)). • Corollary: For expanders, innoculating according to degree is a constant-factor competitive innoculation scheme.

  29. Summary of Part II: • Contact tracing does not control the epidemic in the sense that it still gives bc = 0 on a star. • On general graphs, curing proportional to degree does control the epidemicin the sense that it gives bc > 0. • For expanders with bounded average degree, no other (inhomogenous, configuration-dependent, time-dependent) innoculation scheme works more than a constant factor better than curing proportional to degree in the sense that any such scheme gives bc < 1 as n ! 1.

  30. Overall Summary • Mutating viruses and worms with any positive rate of transmission to neighbors become epidemic with positive probability. • These epidemics can be controlled with Q(1)n doses of antidote if the antidote is distributed proportionally to the degree of the nodes.

  31. THE END

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