Defining and identifying communities in networks

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Defining and identifying communities in networks

• Authors F.Radicchi, C. Castellona,
• F. Cecconi, V. Loreto, D. Parisi
• Presentation by Levent Özgür

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Goal of the paper

• The investigation of community structures in
  networks as definite and as fast as possible
• Fast algorithm
• Self contained algorithm that decides the
  resulted subraphs as community or not

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Statistical properties of network systems

• 1. Small world
• 2. Power law
• 3. Network transivity
• Thisrelated papers add the forth
• 4. Community structure

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What is community?

• Subset of nodes within the graph such that
  connections between the nodes are denser than
  connections with the rest of the networks.
• Mapping the network into a tree (dendogram).
  Leaves are nodes, branches join nodes
  (hierarchical structure).

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How to find communities?

• Agglomerative
• Traditional method to find communities
• Start with all nodes and no edges -gt come up with
  the communities with necessary edges
• Have serious drawbacks
• Fails, with some freq., to find correct
  communities in networks!
• Finds, usually, only cores of communities leave
  out periphery!

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How to find communities?

• Divisive Algorithms
• Start with all nodes edges -gt come up with the
  communities with necessary edges
• New algorithms compared to agglomerative ones
• Divide network into smaller subnetworks
• Edges are removed iteratively according to
  different algorithms

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GN Algorithm

• Calculate the edge betweenness for all edges
• Remove edge with the highest betweenness
• Recalculate values
• Repeat from step 2 until no edges remain

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GN Algorithm edge betweennes

• Selection of the edges to be cut is based on this
  value
• A generalization of the centrality betweenness.
• Betweenness of an edge is the number of shortest
  paths running through that edge.
• Idea When a graph is made of that tightly bound
  clusters, loosely interconnected, all shortest
  paths between clusters have to go through the few
  intercluster connections.

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A sample dendogram and edge betweenness
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GN Algorithm Future work

• Hope to generalize for both weighted directed
  graphs.
• Time complexity
• n number of vertices -gt not practical for very
  large graphs.
• Strength of community structure (according to
  this strength, choose networks to be divided)

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Alternatives for GN Algorithm

• Random walk betweenness
• Current flow betweenness
• Information centrality

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This study ...

• Tries to find a faster algorithm than GN, which
  is successful as GN.
• Provides an alternative divisive algorithm
  considering local quantities only -- so fast --gt
  edge clustering coefficient. ( of triangles
  (or higher order) to which a given edge belongs /
  of triangles that might potentially include)

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This study...

• Tries to cover "Community Sense" (strong sense,
  weak sense) --gt Quantative definitions of
  community (How much community?)
• Self contained algorithm --gt Accept divisions if
  two or more groups fulfill the definitions of
  community!

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Quantative Definitions of Community

• The algorithm may lead to a false community?
• We need a precise definition for community
• Define community as having nodes where internal
  connections are denser than external ones.

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Two definitions for community

• Community in a Strong Sense Each node in a
  strong community has more conections within the
  community than with the rest of the graph
• Community in a Weak Sense The sum of all degrees
  within V is larger than sum of all degrees toward
  the rest of the network.
• Strong sense gt Weak sense

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Community type formulas

• Strong Sense Community
• Weak Sense Community

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Self Contained Algorithms

• Decide whether subgraphs fulfill the definition
  of community
• If the criterioan is not reached, do not split
  into those subgraphs
• This selection is performed iteratively by a
  simple algorithm

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Self Contained Algorithms

• 1. Choose a definition of community
• 2. Compute the edge betweenness for all edges and
  remove those with the highest scores
• 3. If the removal does not split the sub-graph,
  go to 2
• 4. If there is split, test if at least two of the
  resulting subgraphs fullfil the definition
• 5. Goto 2 for all the subgraph until no edges are
  left int he network

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A Fast Algorithm

• An alternating divisive method is introduced
• Requires the consideration of local quantaties
• Much faster than GN algorithm
• Edge clustering coefficient
• of triangles to which a given edge belongs,
  divided by of triangles that might potentially
  include it
• Degrees of adjacent nodes given.

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Formula

• Edge clustering coefficient for triangle
• A more general formula (s of possible
  cyclic structures by the given degrees of nodes)

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Edge betweenness vs edge clustering coefficient
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Edge betweenness vs edge clustering coefficient

• Anticorrelation exists between two quantities,
  but not perfect
• Low values of one quantity tend to have high
  values of the ither quantity
• Expect two algorithms to yield similar community
  structures, although not perfectly coinciding.

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Time complexities
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Similar researches

• Relevant for social tasks, biological injuries or
  technological problems.
• Collaboration network of early jazz musicians
• Email meesage network in a university -- found
  formal informal levels of organizations
• Email network in a large company -- organizations
  become communities!

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Conclusion

• Two main improvements
• Self-contained algorithm that decides the
  existence of community of the splitted subgraphs
• Less time consuming algorithm based on local
  quantities that is as successful as GN algorithm

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The End...

• Thank you for your attention...
• Any Questions ?