Clustering Social Networks

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Clustering Social Networks

• Nina Mishra et al
• Presented by Nam Nguyen

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(a,ß)-Cluster

• Definition
• Given a graph G (V,E) where every vertex has a
  self-loop, C ? V is an (a,ß)-cluster if
• 1. Internally dense ?v ? V, E(v,C) ßC
• 2. Externally sparse ?u ? V\C, E(u,C) aC

u
ßC
aC
v
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Example

• a,b,c,d and d,e,f,g are (1/4, 1)-clusters
• h and i are do not fall into any (a,ß)-cluster
  for 0 alt ½ lt ß 1
• thus, they would not be clustered.
• ? (a,ß)-cluster are able for detecting
  overlapping clusters.

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Problem definition

• Objective
• Identify clusters that are internally dense,
  i.e., each vertex in the cluster is adjacent to
  at least a ß-fraction of the cluster, and
  externally sparse, i.e., any vertex outside of
  the cluster is adjacent to at most an afraction
  of the vertices in the cluster.
• Given 0 alt ß 1, find all (a,ß)-clusters in the
  network

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Contributions of the paper

• Give a bound for the overlapping of two
  (a,ß)-clusters A and B.
• They overlap in at most Cmin1-(ß- a),
  a/(2ß-1) vertices.
• If the ratio of A and B is at most (1- a)/(1-
  ß) then one cluster can not be contained in the
  other.
• Give a loose upper bound for the number of
  (a,1)-clusters of size s O( (n/s) a1 )
• Introduction of the ?-champion of a cluster and
  if ßgt ½(1 ? a), there is a simple deterministic
  algorithm for finding all such clusters in time
• O(m0.7n1.2 n2o(1))

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Some minor remarks

• ß ? 1, the cluster C ? a clique
• a ? 0, C tends to a disconnected component
• ßlt ½ then C might contain two disconnected
  components.
• We want a lt ß and ßgt ½.
• (0, ß)-clusters ? finding connected components
  output ß-connected ones.
• (1-1/n, 1)-clusters ? finding the maximal cliques
  in a graph.
• ((1-e) ß, ß)-clusters ? finding quasi-cliques.

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Result 1

• Question
• How about the intersection of 3 (or more)
  (a,ß)-clusters of the same size? different size ?
• How about the intersection of an (a,ß)-cluster
  and an (a,ß)-cluster of the same size?
  different size ?

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Result 2 Bounding the number of (a,1)-clusters

• Proof
• Two clusters of the same size s can share at most
  as vertices.
• Every subset of size (as1) must appear in at
  most one set in C.
• There are subsets of s elements from n
  elements, each of these contains
  subsets of size (as1).
• Therefore, we can have at most
  clusters in C
• ? C

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This bound is tight

• when a 0
• No overlapping ? of clusters of size s n/s.
• when a ? 1 ( a (n-1)/n )
• Consider the complement of the following graph
• Let s n N/2, then the bound is 2n.
• In fact, we do have 2n subsets of (a, 1)-clusters
  of size n by choosing from the set
• B b1b2bn bi is either xi or yi

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An algorithm for finding clusters with champions

• Why?
• In last example, each vertex has as many
  neighbors outside as within the cluster
• There is no vertex that champions the cluster
  (having more friends inside than outside)
• Why not find one who champions and start with it?

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Algorithm (contd)

• Assumption
• A big gap between ß and a/2 ß gt ½ (a?)/2
• Why?
• Recall last example We have 2n possible clusters
  of size n ? Too many
• Any algorithm that outputs more clusters than
  nodes are undesirable.
• Thus, we need some restriction to reduce the of
  returned clusters.

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Algorithm (contd)

• How many clusters with ?-champion should we have
  ?
• A big gap between ß and a/2 ß gt ½ (a?)/2
• How to find them?

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Algorithm (contd)

• If v and c have sufficient many neighbors then v
  is a part of the cluster C that c champions.
• ? thats what line 5 for
• Running time of the algorithm

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Experimental Results

• For real networks
• Do (a,ß)-clusters with ?-champion exist? ? use
  Tsukiayama algorithm
• If they do exist, do most (a,ß)-clusters have
  ?-champion?
• Results
• Able to find 90 of the maximal cliques in
  graphs where a ½.
• No strong ?-champions in missed clusters.
• Running time Weight faster than Tsukiyamas
  algorithm
• Datasets
• High Energy Physics Theory Co-Author graph (HEP)
• Theory Co-Author graph (TA)
• A subset of Live Journal graph (LP)

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Results
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Results
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Results
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Results
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References

• 1 Clustering Social Networks, Ninna Mishra,
  Robert Schreiber, Isabelle Stanton and Robert E.
  Tarjan (2007)