Patterns around Gnutella Network Nodes

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Patterns around Gnutella Network Nodes

• Sui-Yu Wang

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Introduction

• Recent study shows that the distribution of
  topology in Gnutella network is not purely
  random. This might imply the possibility of the
  existence of frequent patterns around nodes in
  the network. The construction of this model not
  only help further understanding of this network
  but also possible improvement of routing
  algorithm.

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Goal

• Find out the existence of frequent patterns
• Verify the validity of the model
• Use this model to predict patterns around nodes
  that is not in the training data

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Representation of the Network (1)

• Undirected Graph G N, E
• N center, depth_1,, depth_n
• E 1, 2,, TTL
• The depth of nodes other than the center node is
  defined as the shortest path from that node to
  the center

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Representation of the Network (2)

• A
  A
• 2
  2
• 2
  2
• B C
  B
  C
• 1 1
  1 1
• D
  E D
  
• 
  
  E
• G N(A) (depth 2), N(B) (center), N(c)
  (depth 2),
• N(D) (depth 1), N(E) (depth 3),
  E(A,B) 2,
• E(B,D) 1, E(B,C) 2, E(C,E) 1
• Each G is called one transaction

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Frequent Subgraph Discovery

• Developed by Michihiro Kuramochi, George Karypis
• Able to mine patterns in a set of transaction
  give minimum frequency the patterns appear in the
  set
• Gives parent-child relation between subgraphs

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Power Law

• The frequency, , of an out degree, d, is
  proportional to the out degree to the power of
  the constant, O

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Stratified Sampling

• Principle of Stratification partitions are best
  performed by partitioning data so that samples in
  each strata are most similar to each other
• Population of nodes are partitioned into strata
• Partition by size of transaction
• Partition by the power law

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Experiment (1)

• Find out the frequent patterns in two set of data
  collected at the same time but belong to
  different connected component
• The comparison between two distributions is
  performed by comparing the relation of frequent
  subgraphs in each strata
• The maximum depth in each graph is set to be 3
• The TTL is 1
• Data is partitioned by the size of transaction

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Experiment (1)

• There are one pattern of size 3, 2 patterns of
  size 4, and 2 patterns of size 5 missing in data
  set 2
• Missing parent will cause missing child
• Grouping based on power law shows similar result
• Possible reason for difference
• Size of data
• Classification error
• Incomplete observation of the true distribution

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Experiment (2)

• Two connected component of size 591 and 524 taken
  from different time
• Data from transaction of size less than 15
• All subgraphs matches

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Set1 Set2
Graph1 74.2 71.1
Graph2 100 100
Graph3 73.2 69.9
Graph4 71.5 71.1
Graph5 71.5 71.1
Graph6 73.2 69.9
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Experiment (3)

• Grouping by size of transaction
• TTL 3
• Depth 3
• Result shown are patterns with size 6 of
  transactions of size 20 to 50
• Set 1 size 269 and set 2 size 491
• Five patterns are missing from set 1
• One patterns are missing from set 2

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Set1 Set2
Graph1 59.5 52.5
Graph2 55.4 59.0
Graph3 55.0 56.8
Graph4 52.8 55.5
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Prediction Model

• Suppose the model has a graph G with two children
  and
• The frequency of them are , and
• If a node finds a has a subgraph
  isomorphism with G, the chances of finding and
  in are / and /
  respectively