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

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

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

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Model for Power-Law Graphs: Preferential Attachment non-rigorous: Simon ‘55, Barabasi-Albert ‘99, measurements: Kumar et. al. ‘00, rigorous: Bollobas-Riordan ‘00, Bollobas et. al. ‘03 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) / d j with d j = degree(j)

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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

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Mutating Viruses and Worms So far, Internet viruses and worms have been non-mutating The next big threat is mutating viruses and worms, e.g. a worm equipped with 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

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

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

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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?

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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)

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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

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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 e n 0.1 with probability at least and with probability at most where 0 < C 1 < C 2 < 1 are independent of and n. log (1/ ) log log (1/ ) C 1 log (1/ ) log log (1/ ) C 2

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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.

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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 e n 0.1 with probability at least C 3 and with probability at most C 4 where 0 < C 3, C 4 < 1 are independent of and n.

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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) log (1/ ) log log (1/ ) ( )

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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”).

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

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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.

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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.

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Let By the preferential attachment lemma, the ball of radius C 1 k around vertex v contains a vertex w of degree larger than [(C 1 k)!] > -5 where the inequality follows by taking C 1 large. The infection must travel at most C 1 k to reach w, which happens with probability at least C 1 k, 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(n 1/10 ) survival. “Proof” of Main Theorem: Iterations to get to high-degree vertex log (1/ ) log log (1/ ) k = v w z

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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 a positive probability of becoming epidemic 2. calculate the survival probability for both typical and average vertices

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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 threshold of the contact process on power- law (and more general) graphs? -- work in collaboration with Borgs, Ganesh, Saberi & Wilson ‘06

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For , previous results with = const.: Our results (BBCS) for growing power-law graphs Ganesh, Massoulie, Towsley (GMT) for “configurational” power-law graphs For stars: c = n 1/2 + o(1), amount of antidote R = n required to suppress epidemic is n 3/2 + o(1), i.e. superlinear in n For power-law graphs: c ! 0, amount of antidote R required to suppress epidemic is superlinear in n

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Varying Recovery Rates = x Assume there is a fixed amount of antidote R = x x 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 c > 0) on a star or power-law graph with R scaling linearly in n?

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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 x = + 0 i x where i x is the number of infected neighbors of x. Then the critical infection rate on the star is c = n 1/3 + o(1) ! 0. Note: This is an improvement from the case = const, where c = n 1/2 + o(1), but this still gives c ! 0, or alternatively, it takes R = n 4/3 + o(1), i.e. a superlinear amount, of antidote to control the virus.

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

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

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Expanders, continued: Theorem 3: Let > 0, and let G n be a sequence of ( , )-expanders on n nodes. Let x (X t,t) obey Condition*. If ¸ (1+ )d avg /( ), then ¸ exp( (nlogn)). Corollary: For expanders, innoculating according to degree is a constant-factor competitive innoculation scheme.

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Summary of Part II: Contact tracing does not control the epidemic in the sense that it still gives c = 0 on a star. On general graphs, curing proportional to degree does control the epidemic in the sense that it gives c > 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 c < 1 as n ! 1.

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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 (1)n doses of antidote if the antidote is distributed proportionally to the degree of the nodes.

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THE END

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