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

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

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

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)

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

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

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

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

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?

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)

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

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

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.

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.

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/ )  ( )

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”).

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.

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.

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.

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

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

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

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

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?

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.

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.

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.

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.

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.

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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