On Unbiased Sampling for Unstructured PeertoPeer Networks

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On Unbiased Sampling for Unstructured
Peer-to-Peer Networks

• Daniel Stutzbach, Rea Rejaie, Nick Duffield,
  Subhabrata Sen, Walter Willinger

IMC 2006
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Outline

• Introduction
• Existing Sampling Techniques
• MRWB Sampling Technique
• Evaluation
• Discussions

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Sampling in P2P Systems

• Difficult to capture global behavior of p2p
  systems
• Large scale - Millions of peers
• Dynamics
• Sampling in p2p systems
• Explore part of system
• Select representative samples of peer properties

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Challenges in Sampling P2P Systems

• Obtain representative samples
• Unbiased sampling
• Select samples uniformly at random
• Biased sampling in p2p systems due to
• Temporal dynamics
• Topological structure

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Sampling Bias Temporal Dynamics

• General model of p2p system
• Overlay network G(V, E)
• Time t Gt(Vt, Et)
• Sampling time ?
• Sample set
• Peer sessions are highly skewed
• Bias towards short-lived peers (lifetime lt?)
• 50 long-lived peers stay all the time
• 50 short-lived peers lifetime lt?, replaced by
  short-lived peers
• Uniform sampling, times
• More than 50 short-lived peers in sample set

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Sampling Bias Temporal Dynamics

• Bias can be avoid!
• Sample peer property instead of peer
• It must be possible to sample from the same peer
  more than once at different points in time
• Vi(t1) ltgt Vi(t2)

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Sampling Bias Topological Structure

• Basic operations in p2p sampling
• Query known peers about neighbors
• Bias towards nodes with high degrees

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Existing Sampling Techniques-BFS

• Breath-first Search (BFS)
• Crawl a portion of overlay topology
• Samples are unique peers
• Problems
• Bias towards short-lived peers
• Bias towards high degree peers
• Sample peers are correlated by neighboring
  relationship

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Existing Sampling Techniques-RW

• Random Walk (RW)
• P transition matrix

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Existing Sampling Techniques-RW

• Properties
• Bias towards high degree peers
• No correlation between selected peers
• Bias due to temporal dynamics is not significant

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MRWB Sampling Technique

• A variant of random walk
• No correlation between selected peers
• Less bias due to temporal dynamics
• Decrease bias toward high degree peers by
  designing new transition matrix

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Adjusting Degree Bias

• Terminology
• P transition matrix of regular random walk
• Q new transition matrix
• Vector target stationary distribution
• Goal
• Design Q based on Metropolis-Hastings method
  (Hastings 1970)

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Metropolis-Hastings Method

• Given

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Metropolized Random Walk (MRW)

• Node x selects neighbor y based on P(x,y)
• Query y for ys degree
• Accept y as the next step with Prob.
• How to handle dynamics?
• Node y may not response
• Backtracking (MRWB)

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Metropolized Random Walk with Backtracking (RMWB)

• Maintain stack of visited peers
• Latest visited peer is on top of stack
• If no neighbor of x response
• Refresh xs neighbor list
• If neighbor list is empty, pop x out of stack
• If stack is empty, random walk fails

No response from y
Query new neighbor z
Query y

x is last step
Push y to stack
Pop y out of stack
Push z to stack
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Performance Evaluation

• Fundamental properties that interact with walk
• Degree
• Session lengths
• Query latency
• Expected property get from crawling
• Summary statistic D
• S(x) CDF of sampled property x
• E(x) CDF of expected property x
• D max(S(x)-E(x))
• D gt 0.1 ? significant bias
• D lt 0.01 ? very little bias

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Performance Under Different Session Length

• Three distributions of session lengths
• Pareto
• Exponential
• Weibull

MRWB is affected primarily by the rate of local
variation in the degree ratio relative to the
time required to query peers
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Performance Under Different Topologies

• Peer discovery mechanisms
• Random Oracle
• FIFO
• Soft State
• History
• 90 invalid info.

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Empirical Results
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Determine Walk Length
r O(log V )
Suggest r 25
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Discussions

• Low efficiency
• 4 if r 25
• Bias correlated to large scale property
• Clustering
• Can MRWB discover global properties?
• Network diameter?
• Number of peers?

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Thanks!