Finding dense components in weighted graphs

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Finding dense components in weighted graphs

• Paul Horn
• 12-2-02

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Overview

• Addressing the problem
• What is the problem
• How it differs from other already solved problems
• Building a solution
• Already existing research
• Preliminary work
• Final solution

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Overview The Sequel

• Analysis
• Testing
• Effectiveness
• Time Complexity
• Future Work
• Trimming the data set more
• Linking it with real data

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The problem

• To find dense subgraphs of a graph.
• Not just the densest
• Not necessarily all, but as many as possible of
  graphs that are dense enough
• The idea is to identify communities based on a
  communications network
• The more dense the communication is in within a
  subgraph, the more likely it is a community

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Why is it hard

• The fastest flow based methods for finding the
  single densest are cubic or worse.
• We want more than one dense subgraph
• The greedy approximation algorithm is destructive
  and thus returns only one graph
• The problem becomes harder when we allow
  subgraphs to overlap

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Weighty Ideas

• Input graphs to the algorithm are weighted
• Weights of a graph represent the intensity of a
  communication
• Intensity represents the duration and frequency
  of a communication
• Requires a new definition of density

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How dense can it get?

• Recall our old definition of
• density
• We modify it to give a
• notion of density of a
• weighted graph
• Note that if the weight of all edges is one the
  two definitions

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Done before?

• Discussed in Charikar paper presentation
• Goldberg, A.V., Finding a Maximum Density
  Subgraph. A flow based maximum density subgraph
  algorithm
• Charikar, Greedy Approximation Algorithms for
  finding Dense Components in a Graph presented a
  linear approximation algorithm

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Preliminary Work

• An implementation of Goldberg and Charikars
  algorithm
• In test data (generated in a dual-probability
  Erdos-Reyne model) Charikars algorithm
  identified close to the actual density graph
• These graphs, however were unweighted and thus
  ignored the weighted requirement, and it only had
  one dense subgraph.

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A First Attempt

• A modification of Charikars algorithm for
  weighted graphs
• At each step remove a random edge of lowest
  weight.
• Then find all connected components
• Recurse down on each component, and return the
  maximal density subgraph.
• By repeated executions of the algorithm the hope
  is that different dense components will be
  revealed, that can overlap.

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Seems Promising, but

• In test cases generated similarly to that used in
  testing Charikar and Goldbergs algorithm,
  successfully identified close to, if not the
  entire, dense portions.
• In simulated communication network data, the
  graph was dense enough that large areas of the
  graph were denser than the smaller portions, and
  they were not found.

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Partitioning?

• By partitioning optimally, by finding a cut of
  minimum size we can increase the density of the
  graph (to some extent)
• Since we cut edges of low weight, the edges of
  high weight remain on each of the partitions.
• (Obviously) doesnt work forever
• However knowing approximately what size we want
  we can find ideal candidates

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Rethinking our algorithm

• Partitioning based algorithm idea
• Uses Kernighan-Lee to find close to optimal
  partitions.
• Recurses down on the partitions until the are of
  the desired size.
• The densest of the partitions left are our output.

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Finalizing our thought

• Run the algorithm on more than one partition.
• Random partitions are likely to be close to
  orthogonal.
• Generate k partitions, and take best l partitions
  (after KL is applied) at the top level
• On each other level, generate k partitions, and
  take the top one.

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Analyzing the Situation

• The 2-approximation bound that we had for KL-is
  no longer necessarily valid.
• The algorithm has met with some success in
  identify clusters in simulated data, but needs
  more tuning with respect to size, and the
  trimming of the data set.
• By trimming out small partitions that are found
  that are similar, we reduce overlap
• Now may find too many graphs, or incorrect graphs
  but this problem can be relieved by taking only
  the small portions of a certain density (say,
  some percentage of the final)

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Time it.

• Original modification to Charikar runs in
  approximately O(VE) time
• New algorithim runs in approximately
  O(klV2logV) time.
• k, l due to generated the k partitions each time,
  and picking the top l at each step.
• V2 is a result of Kernighan-Lee
• logV is the result of continuing to partition
• In practice runs very fast. Partitioning graphs
  of size 10000 vertices is possible in a
  reasonable amount of time.

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In the future

• The algorithm still needs to better trim the
  partitions it finds, and specifically needs to
  find partitions of more variable size
• Could perhaps trim based on the density of the
  entire graph, or perhaps based on a maximum
  density subgraph (as found by the modified
  Charikar)
• Already finds graphs of many sizes, but only
  considers the smallest at the end, so could be
  modified to include more of the larger partitions

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In the Future II

• Future data will not be simulated, but instead
  come from online sources
• Running on a newsgroup induced graph, for
  instance, can hopefully help identify groups
  interested in particular topics.
• Finding graphs based on email or portions of the
  web graph, could help identify groups of friends
  or topic-related sites as well, and thus help
  predict communities

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So What?

• By looking at not just a graph, but a series of
  time based graph we can identify communities and
  how they change over time.
• Using this method we can hope to identify rules
  which govern the changes of these communities and
  make predictions on their future actions
• Simulated data used was designed with this end in
  mind.

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Summing Up

• Finding multiple dense subgraphs of a graph is a
  relatively unexplored topic, especially finding
  dense subgraphs of large graphs (so that exact
  algorithms are unreasonable)
• Prior work (such as Goldberg and Charikar)
  centered on finding a single densest subgraph

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Summing down

• First algorithm a modification of Charikar
  centered around removing edges and finding
  connected components
• Second algorithm based on Kernighan-Lee algorithm
  for finding optimal partitions, and recursing
  down to find small subgraphs that are generated
  by cutting a small number of vertices.

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The Summing

• Still work to do
• Linking it back to the real data
• Internet data from newsgroups, email, etc
• Using that to find communities over time
• Finding microlaws that govern them based on how
  the communities change over time
• Finding better ways to trim data to ensure that
  the best candidates are found