Sui-Yu Wang

• 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.

• 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

• Undirected Graph G = { N, E}


• The depth of nodes other than the center node is defined as the shortest path from that node to the center

A A
2 2
B
2
C
2
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

• 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

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

•
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

• 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

Relative Frequency Set 1
Graph 1
Graph 2
Graph 3
Graph 4
Graph 5
Graph 6
75.4
100
74.5
79.5
79.5
74.5
Set 2
74.2
100
73.2
71.5
71.5
73.2

• 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

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

Graph1
Graph2
Graph3
Graph4
Graph5
Graph6
Set1
74.2
100
73.2
71.5
71.5
73.2
Set2
71.1
100
69.9
71.1
71.1
69.9

• 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

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

• Suppose the model has a graph G with two children and a
G b
G f f f b
G new isomorphism with G, the chances of finding and in are / and / a b a G
G respectively