Algorithmic Game Theory and Internet Computing

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Algorithmic Game Theoryand Internet Computing
On the Spread of Viruses on Networks 2nd lecture
Containment of Epidemics

• Information Networks
• MSE 337

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Previous Lecture

• Short introduction on computer viruses and worms
• Models for epidemics
• SIR (susceptible-infected-removed)
• SIS (susceptible-infected-susceptible)
• SIS model on Scale-free networks

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Modeling Spread of Viruses

• SIR model Susceptible-Infected-Removed
• susceptible infected
  at rate ( )
• infected removed
  at rate 1

infectedneighbors
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Modeling Spread of Viruses

• SIR model Susceptible-Infected-Removed
• susceptible infected
  at rate ( )
• infected removed
  at rate 1
• removed can be interpreted as either dead or
  immune

infectedneighbors
healthy
infected removed
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Modeling Mutating Worms/Viruses

• SIS model Susceptible-Infected-Susceptible
• infected healthy
  at rate 1
• healthy infected
  at rate ( )
• Known also as Contact Process
• Studied in probability theory, physics,
  epidemiology
• Kephart and White 93 modeling the spread of
  viruses in a computer network

infectedneighbors
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Epidemic Threshold

• Infinite graphs
• extinction weak survival
  strong survival

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Epidemic Threshold

• Infinite graphs
• extinction weak survival
  strong survival
• Finite graphs
• logarithmic survival time

exponential (super poly)survival time
polynomial survival time
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Epidemic Threshold

• Infinite graphs
• extinction weak survival
  strong survival
• Finite graphs
• logarithmic survival time

exponential (super poly)survival time
polynomial survival time
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Epidemic Threshold in Scale-Free Network

• In Power-law networks both thresholds are zero
  almost surely! (Pastor-Satorras,Vespignani 01)
• Rigorous proof Berger, Borgs, Chayes, S. 04
• Idea existence of high degree nodes
  (threshold is positive for bounded-degree graphs)
• high connectivity (high degree
  nodes are usually very close)

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Typical vs. Average Behavior

• The survival probability for an infection
  starting from a typical (i.e., 1 O(l2)) vertex
  is of order
• whereas, the average survival probability is of
  order

?C2
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Key Elements of the Proof

• For the contact process
• 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)
• For the preferential attachment
• Almost surely, there is a node of degree
  with distance k from a randomly chosen node v
• There is a sequence of nodes with increasing
  degrees from v to the core of the graph

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Proof of the Theorem
v

• Let k log (1/l)/loglog(1/l)
  
• the ball of radius C1k around vertex v contains a
  vertex w of degree larger than
• (C1k)!? gt l-5
  for a suitably large C1
• The infection must travel at most C1k to reach w,
  which happens with probability at least
• lC1k
• Star lemma the survival time is more than exp(C
  l-3 ).
• Iterate until we reach a vertex z of
  sufficiently high degree for exp(n1/10) survival.

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Todays lecture

• Another look at the definition of standard SIS
  model
• Find detailed behavior of in lc terms of b and
  constant recovery rate r
• Now let r vary from one vertex to the next, but
  assume total amount of antidote A rn is fixed
• Q How to distribute A most effectively?

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Containment of Epidemics

• Question
• What is the best way to distribute a fixed
  amount of antidote to contain the epidemic, i.e.
  to raise the epidemic threshold of the SIS
  process on preferential attachment (and more
  general) graphs?
• 2005 Borgs, Chayes, Ganesh, Saberi, Wilson

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Standard SIS Model

• Definition of standard SIS model
• infected to healthy at rate r
• healthy to infected at rate b infected
  nhbrs
• relevant parameter l b/r

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Re-statement of last lectures results ( with lc
bc /r and r const)

• For stars
• bc rn-1/2 o(1)
• I.e. amount of antidote A rn required to
  suppress epidemic is bn3/2 o(1) , i.e.
  superlinear in n
• For preferential attachment graphs
• bc 0
• I.e. amount of antidote A 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 A
  Sxrx to be distributed non-uniformly, dynamically
• Questions
• What is the best policy for distributing A?
• Is there a way to control the infection (i.e., to
  get lc gt 0) on a star or preferential attachment
  graph with A scaling linearly in n?

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Method I Contact Tracing

• Contact tracing is a method in epidemiology to
  diagnose and treat the contacts of infected
  individuals , cure / infected degree
• rx r r ix where ix is the number of
  infected neighbors of x.

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Method I Contact Tracing

• Contact tracing is a method in epidemiology to
  diagnose and treat the contacts of infected
  individuals , cure / infected degree
• Theorem 3 (BCGSW) Let rx r rix 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) --gt 0
• Note This is an improvement from the case r
  const
• Translation It takes A n4/3 o(1) , i.e. a
  superlinear amount, of antidote to control the
  virus via contact tracing

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Method II Cure / Degree(vs. contact tracing
with cure / infected degree)

• Theorem 4 (BCGSW) Let rx dx, where dx is the
  degree of x. If b lt 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 bdavgn, i.e. linear in
  n
• Translation Curing proportional to degree is
  enough to control epidemics on general graphs
  with bounded average degree, including
  preferential attachment graphs

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Can we do significantly better?

• Q Can we get bc--gtinfty as n --gt infty?
• A No, for expanders Roughly speaking, an
  expander has large boundary to volume, so that
  quarantine methods dont work well

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Open questions
The best curing policy? Stochastic optimization
problemE.g. Contact tracing then site
monitoring? What is the best time for
transition? What the best policy for detecting
the infections? E.g. optimal placement of honey
farms ? Better models for propagation of viruses
?
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THE END !
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THE CLASS so far

• Power-law networks as models of the Internet,
  WWW, etc... (the good, the bad, and the ugly)
• Structural properties (degree sequence,
  connectivity)
• Connectivity raison d'etre of all these
  networks
• Giant component connected
• Expansion -- Internet graph as a good expander
• Effects of structure on performance
• random walks/routing/reliability
• propagation of viruses

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The Remaining lectures

• Two more lectures on power-law networks
  (both by guest lecturers)
• limitations of the model extensions
• Spectral analysis of real networks
• The main focus will be on SOCIAL NETWORKS
• we will look at algorithmic questions
• Two important issues
• Cascading Behavior (2 lectures)
• Search and small-world phenomenon (3 lectures)

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Networks as Phenomena

• Emergence of several giants
• what are the recurring patterns? Why are they
  there? What do they mean? What are their
  consequences?

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Emerging Intersection of Social and Technological
Networks

• Social and technological networks are
  interwined (web, blogs, e-mail, IM,
  facebook)
• New technologies change our pattern of social
  interaction
• Collecting social data at unprecedented rate

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Revolution in Social Sciences

• Data can be large-scale, realistic, and
  completely mapped
• Remember Zacharys karate club

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This is a huge opportunity

• Data can be large-scale, realistic, and
  completely mapped
• Remember Zacharys karate club

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THE END !