Jennifer Tour Chayes

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
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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
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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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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

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Epidemic Threshold(s)

• 
  ?1 ?2
• Infinite graph extinction weak
  strong
• 
  survival survival
• Note ?1 ?2 on Zd
• ?1 lt ?2 on a tree
• Finite subset logarthmic polynomial
  exponential
• of Zd 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 ? gt 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 lt C1 lt C2 lt 1 are
independent of ? and n.
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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 ? gt 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 lt C3, C4 lt 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)

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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 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.

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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,?) gt 0.
• To prove this, we introduced a Polya Urn
  Representation of the preferential attachment
  graph.

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Polya Urn Representation of Graph

• Polyas 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.
• Polyas 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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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)!? gt ?-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
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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 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 nr required to suppress
  epidemic is bn3/2 o(1) , i.e. superlinear in n
• For power-law graphs
• bc ! 0
• , amount of antidote R required to suppress
  epidemic is superlinear in n

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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 gt 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 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 this still
  gives bc ! 0, or alternatively, it takes R n4/3
  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 rx dx. If b lt 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.

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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 aV, the number of edges joining W
  to its complement V\W is at least hW.)
• 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.

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Expanders, continued

• Theorem 3 Let e gt 0, and let Gn be a sequence
  of (a,h)-expanders on n nodes. Let rx (Xt,t)
  obey Condition. If b (1e)davg/(ah), then t
  exp(Q(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 bc 0 on a star.
• On general graphs, curing proportional to degree
  does control the epidemic in the sense that it
  gives bc gt 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 lt 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 Q(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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