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) / dj with dj = 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 attachmentWorms 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-mutatingThe 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 Zd

1 < 2 on a tree

Finite subset logarthmic polynomial exponential of Zd: 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 ’05Moreover, 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 en0.1 with probability at least

and with probability at most

where 0 < C1 < C2 < 1 are independent of and n.

log (1/) log log (1/)C1

log (1/) log log (1/)C2

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 en0.1 with probability at least

C3

and with probability at most

C4 where 0 < C3, C4 < 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 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.

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 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.

“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 graphsGanesh, Massoulie, Towsley (GMT) for “configurational” power-law graphs For stars:

c = n1/2 + o(1)

, amount of antidote R = n required to suppress epidemic is n3/2 + 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 =

xx 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 = + 0ix where ix is the number of infected neighbors of x. Then the critical infection rate on the star is

c = n1/3 + o(1) ! 0. Note: This is an improvement from the case = const,

where c = n1/2 + o(1), but this still gives c ! 0, or alternatively, it takes R = n4/3 + 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 = dx. If < 1 then the

expected survival time is = O(logn). Corollary: For graphs with a bounded average

degree davg, the total amount of antidote needed to control the epidemic is davgn, 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 Xt be the set of infected vertices at time t, and let x = x (Xt,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 Gn be a sequence of (,)-expanders on n nodes. Let x (Xt,t) obey Condition*. If ¸ (1+)davg/(), 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.

THE END

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