FINDING TIGHT-KNIT CIRCLES (I.E., TRIANGLE COUNTING)

Irene Finocchi

Network properties

¨  Locality of relationships ¤  Relationships tend to cluster: high clustering coefficient ¤  If A is related to both B and C, than B and C are

related with probability higher than average ¨  Giant connected component ¨  Sparse: m = o(n2) ¨  Small-world

¤  Small average distance between nodes ¨  Scale-free

¤  Power-law degree distribution

Counting triangles

¨  How many triangles in a graph?

¨  Triangle = smallest clique = smallest tight-knit community

¨  A lot of interest for social network analysis (not only!)

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Naïf approach

¨  List all node triples, and for each triple check if it forms a triangle

¨  triples

¨  Θ(n3) time independently of graph structure (and number of edges)

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n3

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Three case studies

¨  Clique on n nodes

¨  Star with n-1 leaves

¨  Binary tree on n nodes

Naïf approach: Θ(n3)

Algorithm NodeIterator

Γ(x) = neighbors of node x

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Ο(n3) iterations

1; 6;

NodeIterator on the three case studies

¨  Clique on n nodes: Θ(n3) ¤ Cannot improve on this bound: after all, this is the

number of triangles

¨  Star with n-1 leaves: Θ(n2)

¤  Better than naïf, but still 0 triangles and sparse (constant degree on average)

¨  Binary tree on n nodes: Θ(n)

¤ Cannot do better than linear time

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1 1 1

1 1 1

1 1 1

Reduce 1 v= 1

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1 1 1 1

1 1 1 1 $

Reduce 2

MR-NodeIterator: analysis

¨  Rounds: 2

¨  Global space: O(n3)

¨  Local reducer space: O(n)

Better algorithms exist

MR-NodeIterator performance

Algorithm NodeIterator++

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Each triangle counted only once degree d(w)>d(u) or (d(w)=d(u) and w>u)

NodeIterator++: analysis

h = # nodes of G with degree > √m h × √m ≤ 2m h ≤ 2√m

d+(v) = # neighbors of v with degree > d(v) 1)  d(v) ≤ √m: d+(v) ≤ d(v) ≤ √m 2)  d(v) > √m:

nodes counted in d+(v) have degree > d(v) > √m hence d+(v) ≤ h ≤ 2√m

Algorithm NodeIterator++

degree d(w)>d(u) or (d(w)=d(u) and w>u)

O(m3/2)

# iterations

NodeIterator++ on the three case studies

¨  Clique on n nodes: Θ(n3) ¤ Cannot improve on this bound: after all, this is the

number of triangles

¨  Star with n-1 leaves: Θ(n1.5)

¤ √n faster than before (m=Θ(n))

¨  Binary tree on n nodes: Θ(n)

¤ Cannot do better than linear time

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MR-NodeIterator++: analysis

¨  Rounds: 2

¨  Global space: O(m3/2)

¨  Local reducer space: O(n)

Local space can be further reduced

MR-NodeIterator++ performance

The curse of the last reducer

•  Very skewed degree distributions •  NodeIterator++ deals with skewness much better

References

Siddharth Suri & Sergei Vassilvitskii, Counting triangles and the curse of the last reducer, Proc. 20th International Conference on World Wide Web, 2011