Presented by Ozgur D. Sahin

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Presented by Ozgur D. Sahin
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Outline

• Introduction
• Neighborhood Functions
• ANF Algorithm
• Modifications
• Experimental Results
• Data Mining using ANF
• Conclusions

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Introduction Motivation

• Graph-based data is becoming more importatnt
• Internet modeling, academic citations, phone
  records, movie databases, CAD circuits
• Example Questions
• How robust is the Internet to failures?
• What are the most influential database papers?
• What is the best opening move in tic-tac-toe?
• Are phone call patterns in Asia similar to those
  in the U.S.?
• Goal Quickly answer questions on graph-
  represented data

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Answering Questions

• We can answer these questions if we can compute
  following three properties related to
  connectivity and neighborhood structure
• Graph Similarity Decide if two graphs have
  similar connectivity/neighborhood structure
• Subgraph Similarity Compare how two subgraphs of
  a given graph are connected
• Vertex Importance Assign an importance to each
  node based on its connectivity
• This paper provides such a tool ANF (Approximate
  Neighborhood Function)

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Challenges

• Following properties should be satisfied
• Error Guarantees Accurate estimates
• Fast Scale linearly with n ( of nodes) and m (
  of edges)
• Low Storage
• Adapts to available memory
• Parallelizable
• Sequential scan of the edge file
• Estimates per node

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Definitions - Neighborhood Functions
dist(u,v) of edges on the shortest path from u
to v Define following neighborhood functions
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Definitions - Neighborhood Functions
Generalize these two definitions to deal with
subgraphs
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Basic ANF Algorithm

• N(h) can be computed by a graph traversal
• Graph traversal accesses edges in random order
• Running time is O(nm)
• Access edges in sequential order
• M(x,h) is the set of nodes within distance h of
  node x

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Basic ANF Algorithm

• How to compute the number of distinct elements in
  the set M(x,h)
• A dictionary data structure O(n2log n)
  time/space
• Use bits to mark membership O(n2) space
• Use probabilistic counting algorithm
• Approximate set sizes using log nr bits

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Probabilistic counting algorithm

• Approximate set sizes using log nr bits
• Instead of one bit per node, give half the nodes
  bit 0, a quarter of them bit 1, and so on (A node
  is given bit i with probability 1/2i1)
• The approximation of the size of a set is
  proportional to 2b, where b is the least bit that
  has not been set in the bit representation of
  this set
• Use k parallel approximations
• M(x,h) is represented by k(log nr) bits

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Basic ANF Algorithm

• Consider a ring with 5 nodes
• Example for k3 and r0
• Bit 0 is the leftmost bit in each 3-bit mask
• M(2,1) is the union of M(2,0), M(1,0), and
  M(3,0)
• M(2,1)M(2,0) OR M(1,0) OR M(3,0)
• IN(2,1) is computed from the average of the least
  zero bit positions
• Avg(211)/34/3 ? IN(2,1) (24/3)/0.77359
  3.25

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Basic ANF Algorithm
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Modifications

• M(x,h) uses M(y,h-1) but not M(y,h-2), so just
  keep the M(y,h-1) during iteration h.
• Include a mark bit to handle generalized
  neighborhood functions
• Break bit masks into smaller pieces if they are
  larger than the available memory

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Leading Ones Compression

• As ANF runs, most bit masks will have many
  leading 1s
• Compress bit masks by including a counter of the
  leading ones
• Bit shuffling of k parallel bit masks enables
  further compression
• 11010,11100 ? 1111011000
• Provides up to 23 speed-up

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Experiments

• Data Sets 3 real (Router, Cornell, Cora) and 4
  synthetic
• Evaluation Metric

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Experiments - Accuracy
k64 - ANF achieves less than 7 error - ANFs
error is independent of the data set
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Experiments - Time
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Experiments - Scalability
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Data Mining with ANF

• ANF tool can be used to answer graph mining
  problems
• Best opening move for Tic-Tac-Toe game
• Clustering movie classes
• Measuring the robustness of the Internet
• Use summarized statistics derived from
  neighborhood function
• Many real graphs follow a power law
• N(h) µ hH, where H is defined as the hop
  exponent
• Use individual hop exponent as a measure of
  importance

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Tic-Tac-Toe

• Show The best opening move is the center square
• Each possible board configuration is a node and
  there is an edge from board x to board y if it is
  a possible move
• Compute individual neighborhood functions for
  each of the 9 possible first moves

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Clustering Movies

• Consider IMDB (Internet Movie Data Base) where
  each movie is identified as being in one or more
  classes (such as documentaries, dramas, comedies,
  etc)
• Construct a graph for each class and cluster
  similar ones

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Internet Router Data

• How robust the Internet is to router failures
• Delete some number of routers and measure
  connectivity

• Random failures do not disrupt the Internet
• Targeted failures can dramatically disrupt it

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Conclusions

• ANF uses an efficient and accurate approximation
  algorithm
• ANF tool provides several advantages including
  following
• Accurate
• Fast
• Low storage requirements
• Parallelizable
• ANF makes it possible to answer many interesting
  questions