Costeffective Outbreak Detection in Networks

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Cost-effective Outbreak Detection in Networks

• (Leskovec, Krause, Guestrin, Faloutsos,
  VanBriesen, Grance, KDD 2007)

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What this paper is about

• Task Given a network, which node should we
  monitor to optimally catch outbreak of important
  events while minmizing the cost?
• Approach Greedily, by leveraging submodularity
  property
• This is a set optimization problem. Many
  realistic problem (e.g., sampling strategy in AL)
  can be reduced to this paradigm

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• Example Blogosphere

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Key Idea in this paper

• The paper demonstrate the steps in solving the
  problem using Submodularity optimization
• Show the objectives are submodular
• Devise greedy algorithm (submodularity give a
  good lower bound)
• Submodularity also give bound to special cases
  (non-uniform cost, on-line selection)
• Submodularity allow Efficient algorithm

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Submodularity

• What is submodularity?
• A property of set function.
• Informally - If a function has the diminishing
  returns property, it is submodular
• Law of diminishing return - Adding an element to
  a smaller set returns bigger utility than adding
  it to a larger set
• Has nice properties w.r.t optimization (like
  convex function, we will see later).

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Submodularity

• Formally - A set function with the following
  property

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• Example Blogosphere

What if your objective is to read the most
effective set of blogs. Is this problem
submodular?
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Submodularity

• Why this is important in this context?
• Turned out that many realistic objectives in
  outbreak detection is submodular (I.e., exhibit a
  diminishing return property)
• Many other optimization problems are also
  submodular ( See tutorial from Select lab
  http//www.select.cs.cmu.edu/tutorials/icml08submo
  dularity.html )

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Submodularity

• Examples
• A lot of machine learning algorithms !
• Set cover
• Forward feature selection
• Mutual Information
• Factorization in structure learning
• And there is nice connection to convexity
  anyone?

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What to Optimize here?

• One can think of many things
• Fraction of events detected by the certain
  placement of sensors (Detection Likelihood)
• Time passed from outbreak till detection
  (Detection Time)
• Number of node touched by the event before the
  detection (Population Affected)
• Any of them, if expressed as Expected Penalty,
  would satisfy submodularity property.

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What is the trick?

• (how to elicit submodular function?)
• Any of them, if expressed as Expected Penalty,
  can satisfy submodularity.
• So, our next task is to express those objectives
  in such a way.

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Objectives

• Present objective as Maximize Expected Penalty
  Reduction (how much saving in penalty you can get
  by betting on set A)

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Property of the objective

• Is this Submodular?
• Do they meet the criteria?
• R(0) 0
• R is non-decreasing
• Marginal gain diminishes as the set get bigger.
• Hence, this function is Submodular

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What this means?

• In uniform cost case (all node has the same
  placement cost), a greedy algorithm is only
  constant factor away from the optimal solution.
  Nice.
• Notice that objective function have to be
  expressed in terms of expected penalty in order
  to be submodular.

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• By the way, Submodular function is in general
  NP-hard.
• But it has a nice theorem that guarantee lower
  bounding for greedy algorithm (so go greedy,)

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Greedy Algorithm

• The authors proposed the following greedy
  algorithm for uniform cost case
• Start with A , (null set)
• At step k, adds the node sk which maximizes the
  marginal gain, stop when Budge max out.

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Non-uniform case CEF

• Non-uniform cost case is also bounded

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Computing the online bound

• You can also compute the tighter upper bound for
  submodular function
• (This is useful for evaluation)

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Efficient Algorithm - CELF
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Efficient Algorithm - CELF
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Multicriterion Optimization

• If there no set A s.t. Ri(A) Ri(A) for all i
  and strictly better for at least one event, j
  (Rj(A) Rj(A) for some j) te solution is called
  pareto-optimal
• For some positive weight ? consider objective
  R(A) ? ?i Ri(A). Any solution maximizing R(A)
  is guaranteed to be Pereto-optimal
• Under submodularity assumption, this objective
  also is submodular. Nice.

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Experiments

• 45K blogs, 1M links, 17,589 information cascades
• True optimal somewhere between green and blue
• 13.8 away from the optimal (much closer than
  the guaranteed bound)
• Note this is the evaluation on the training data
  - this is the utility score ( penalty reduction)
  based on the labeled data.

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Experiments

• Cost sensitive algorithm achieve the same level
  of utility score at 1500 posts, as opposed to
  10,710 posts.
• Under non-uniform cost, summarization type
  blogs score much higher than large one.

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Experiments

• Comparizon with other heuristics. The proposed
  model does much better.
• (Is this fair comparizon?)

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Experiments

• How well does it generalize? Save 50 of data for
  test. CELF appear to overfit.
• The problem is alleviated when filter out very
  small blog sites.

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Summary of the paper