Protocol and Connectivity based Overlay Level Capacity Calculation of P2P Networks Kasim ztoprak and

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Protocol and Connectivity based Overlay Level
Capacity Calculation of P2P Networks Kasim
Öztoprak and Hürevren KiliçComputer Engineering
DepartmentAtilim University, Ankara,
TURKEYkasim, hurevren_at_atilim.edu.tr
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CONTENT

• Introduction
• Combinatorial Capacity Metric
• Experimental Results
• Conclusions

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INTRODUCTION

• Paradigm shift in programming From stand-alone,
  single bodies to simple, primitive, interacting
  many entities
• Designed protocols and run-time connectivity
• Run-time dynamics, efficiency
• Fault-tolerance
• Self-organization ability
• Metrics (P2P) based on run-time measurements
  only Hit ratio, hit time, traffic overhead

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INTRODUCTION

• A hybrid metric both design-time (e.g. protocol)
  and run-time (e.g. connectivity) consideration
• P2P system as a discrete noiseless channel
• - Connection topology
• - Labelled transitions
• with durations

A Shannon Language
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INTRODUCTION

• Shannons L-channel capacity calculation idea
• Combinatorial Capacity maximum amount of
  information (in bits per second) that can be
  transmitted over P2P network
• Example applications of the metric
• Pure Gnutella 0.6 (known traffic explosion
  problem)
• Timebased clustered Gnutella
• Potential correlations between the metric and the
  of query hits and query-hit response time.

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COMBINATORIAL CAPACITY METRIC

• Definitions and a theorem compiled from (Shannon,
  1948) by (Khandekar et.al., 2000)
• Definition 1 A discrete noiseless channel is
  a channel which allows the noiseless transmission
  of a sequence of symbols chosen from a finite
  alphabet A (called q-letter alphabet) each
  symbol, say , having a certain duration
  t(a) in time, possibly different for different
  symbols.

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COMBINATORIAL CAPACITY METRIC

• Analogies

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COMBINATORIAL CAPACITY METRIC

• Definition 2 A word of length k over A is a
  finite string of k letters from A. If a a1a2
  ak is such a word, its duration is defined to be
  t(a) t(a1) t(a2) t(ak)
• Definition 3 A language L over A is a
  collection of words over A. The discrete
  noiseless channel associated with L (the
  L-channel for short) is the channel which is only
  allowed to transmit sequences from L without
  error.
• Question What is the capacity of such
  L-channel to transmit information ?
• Capacity The maximum rate (in bits per second)
  that information can be transmitted over the P2P
  network.

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COMBINATORIAL CAPACITY METRIC

• Definition 4 Shannon language is defined by
  a directed graph whose edges are labeled with
  letters from the alphabet A. It is the set of
  words that result by reading off the edge labels
  on paths of the graph.
• P2P setup with its connection topology protocol
  vs. Shannon language.

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COMBINATORIAL CAPACITY METRIC
Fig. 1 An example Gnutella based P2P network
setup describing a Shannon Language.
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COMBINATORIAL CAPACITY METRIC

• Points and assumptions
• 1. t(imn) t(omn) t(umn) t(hmn) where m and
  n are any two different peers (for computational
  efficiency)
• 2. For any message type x and different peers m
  and n
• If xmn is a label then xnm is also a label
  and t(xmn) t(xnm)
• 3. No self ping, pong,query, query-hit i.e. m ?
  n.
• 4. Number of peers may change during system
  evolution.
• The graph does not describe any messaging
  scenario but considers the potential of sending a
  message from a peer to another !!!

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COMBINATORIAL CAPACITY METRIC

• Definition 5 Let L be a Shannon language,
  the combinatorial capacity of the L-channel is
  defined as
• Ccomb lim sup (1/t)log(N(t))
• t?8
• where N(t) is the total number of words in L
  of duration t.
• An easy algebraic method to calculate Ccomb
  developed by (Shannon) using the notion of
  partition function.

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COMBINATORIAL CAPACITY METRIC

• Definition 6 Let s be nonnegative real
  number and for a given pair of vertices
  describing an edge, branch duration partition
  function is defined as
• where b is a member of the set Bv,w showing
  the edges whose starting node is v and ending
  node is w.
• The functions constitute the entries of
  M by M matrix P(s) where M is the number of peers
  at the time of capacity calculation.

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COMBINATORIAL CAPACITY METRIC

• For Fig. 1, we obtain the following matrix
• The partition function for the language defined
  by given graph G (i.e. LG,?) is called spectral
  radius of the matrix P(s) and it is represented
  by ?(s).

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COMBINATORIAL CAPACITY METRIC

• Theorem The combinatorial capacity of the LG,?
  language is given by
• Ccomb ln(s0)
• where s0 is the unique solution to the equation
  ?(s)1.
• (for original proof see (Shannon, 1948)
• for simplified version see (Khandekar et.al.,
  2000))
• An alternative way of computing s0
• Find s0 as the greatest positive solution
  to the equation
• det(I-P(s))0

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EXPERIMENTAL RESULTS

• Aim of the experiments
• To figure out the effect of clustering on the P2P
  system by using the proposed capacity metric
• To observe (if exists) potential correlations
  between proposed metric results and the known
  metrics (like of query hits)
• Tools
• Matlab 7.2.0 for capacity calculation.
• GnutellaSim (overlay level)
• NS-2 (network simulator)
• GT-ITM (backbone generation) (Nguyen Zakhor,
  2002)

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EXPERIMENTAL RESULTS

• Experimental Set-up
• Gnutella Network
• Configurations (Identical initial topology setup
  for both)
• Pure-Gnutella overlay network
• Time-based clustered version
• Initially 27 peers
• Uniformly distributed random peer arrivals
  (Sripanidkulchai et. al., 2003)
• 3-node transit-stub network topology using
  GT-ITM
• 1 ultra-peer and 8 leaf peers for each stub node

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EXPERIMENTAL RESULTS

• Modeled real world activities
• Join, leave (adding or deleting a new graph node)
• Request, serve, forward or block a query (edges
  umn and hmn)
• Ping, pong (edges imn and omn)
• Delays are assumed to be equal for all message
  types
• High probability of not flushed but dependent
  queries (typical user behavior)
• Experiments
• Snapshots Starting at 100th second ending at
  1000th second, once in every 5 minute intervals.
• 3 experiments for each configuration, total
  number of capacity calculations 2x3x424
• How to pick-up and merge connectivities ?

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EXPERIMENTAL RESULTS
Table 1 Capacity calculation results for
pure-Gnutella experimentation
Table 2 Capacity calculation results for
time-based clustered Gnutella experimentation
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EXPERIMENTAL RESULTS

• Observations
• Similar capacity values for both configurations
• Capacity decrease trend in pure-Gnutella
• Time-based clustered version Initiated 150
  queries and get 140 hits. Pure-Gnutella
  Initiated 86 queries and get 76 hits. Similar hit
  ratio!
• However, because of high channel capacity,
  doing more jobs (i.e. query initiation) is
  possible.

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CONCLUSIONS

• A metric for P2P networks based on Shannons
  L-channel capacity calculation idea
• Able to observe and figure out the efficiency
  obtained through clustering
• Cannot be used directly for run-time
  self-organization of peers connectivity
  (peer-side partial observability).
• Solution Capacity calculation for partial graphs
  ? Use of the metric for efficient clustering ?
• High and explosive computational time for
  capacity calculation.
• Solution Problem specific shortcuts using graph
  properties?

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• ! Thanks !
• ? Questions ?