Nina Mishra et al
Presented by Nam Nguyen
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Definition
◦ Given a graph G = (V,E) where every vertex has a
self-loop, C ⊂ V is an (α,β)-cluster if
 1. Internally dense: ∀v ∈ V, |E(v,C)| ≥ β|C|
 2. Externally sparse: ∀u ∈ V\C, |E(u,C)| ≤ α|C|
u
≥ β|C|
≤ α|C|
v
{a,b,c,d} and {d,e,f,g} are (1/4, 1)-clusters
 h and i are do not fall into any (α,β)-cluster
for 0≤ α< ½ < β ≤1
thus, they would not be clustered.
 (α,β)-cluster are able for detecting
overlapping clusters.
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Objective
Identify clusters that are internally dense, i.e., each
vertex in the cluster is adjacent to at least a β-fraction
of the cluster, and externally sparse, i.e., any vertex
outside of the cluster is adjacent to at most an α–
fraction of the vertices in the cluster.
Given 0≤ α< β ≤1, find all (α,β)-clusters in the network
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Give a bound for the overlapping of two (α,β)clusters A and B.
◦ They overlap in at most |C|*min{1-(β- α), α/(2β-1)}
vertices.
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If the ratio of |A| and |B| is at most (1- α)/(1- β)
then one cluster can not be contained in the
other.
Give a loose upper bound for the number of (α,1)clusters of size s: O( (n/s) α+1 )
Introduction of the ρ-champion of a cluster and if
β> ½(1+ ρ+ α), there is a simple deterministic
algorithm for finding all such clusters in time
O(m0.7n1.2 + n2+o(1))
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β  1, the cluster C  a clique
α  0, C tends to a disconnected component
β< ½ then C might contain two disconnected
components.
We want α < β and β> ½.
(0, β)-clusters  finding connected components &
output β-connected ones.
(1-1/n, 1)-clusters  finding the maximal cliques
in a graph.
((1-ε) β, β)-clusters  finding quasi-cliques.
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Question:
◦ How about the intersection of 3 (or more) (α,β)-clusters of the same size?
different size ?
◦ How about the intersection of an (α,β)-cluster and an (α’,β’)-cluster of the
same size? different size ?
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Proof
◦ Two clusters of the same size s can share at most
αs vertices.
◦ Every subset of size (αs+1) 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 (αs+1).
◦ Therefore, we can have at most
clusters in C
◦  |C| ≤
=
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when α = 0
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◦ No overlapping  # of clusters of size s = n/s.
when α  1 ( α = (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 (α, 1)-clusters of
size n by choosing from the set
B = {b1b2…bn | bi is either xi or yi}
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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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Assumption:
◦ A big gap between β and α/2: β > ½ + (α+ρ)/2
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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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How many clusters with ρ-champion should we have ?
◦ A big gap between β and α/2: β > ½ + (α+ρ)/2
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How to find them?
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If v and c have sufficient many neighbors then v is a
part of the cluster C that c champions.
 that’s what line #5 for
Running time of the algorithm
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For real networks
◦ Do (α,β)-clusters with ρ-champion exist?  use Tsukiayama algorithm
◦ If they do exist, do most (α,β)-clusters have ρ-champion?
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Results
◦ Able to find ~90% of the maximal cliques in graphs where α ≤ ½.
◦ No strong ρ-champions in missed clusters.
◦ Running time: Weight faster than Tsukiyama’s algorithm
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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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[1] Clustering Social Networks, Ninna Mishra, Robert Schreiber, Isabelle
Stanton and Robert E. Tarjan (2007)