Com_NP-hard - On Modularity Clustering Presented by: Group...

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On Modularity Clustering resented by: Presented by: Group III (Ying Xuan, Swati Gambhir & Ravi Tiwari)
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Modularity quality index for clustering a graph G=(VE) y A quality index for clustering a graph G=(V,E) 2 ' () (, ' ) : 2 C C C C C EC ECC q mm ⎡⎤ ⎛⎞ + =− ⎜⎟ ⎝⎠ This is equivalent to: ⎣⎦ 2 deg( ) ) C v E C : 2 C vC C C q ⎢⎥ =
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Contribution of the Paper teger Linear Program Formulation y Integer Linear Program Formulation y Fundamental Observations & Counterintuitive Behavior P l t f M i i i M d l i t P b l y NP- Completeness of Maximizing Modularity Problem y A Greedy Algorithm y Optimality Results
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Maximizing Modularity via Integer Linear programming iven a graph G=(VE) n=|V|nodes n 2 cision variables X Given a graph G=(V,E), n=|V|nodes, n decision variables X uv ={0,1} 2 ) 1 deg( ) deg( ) :() : 2 uv uv v V C uv Max q E X m ⎡⎤ =− ⎢⎥ ( , 22 : 1 uv V mm St X ⎣⎦ :1 ,: uu uv vu u uv X X = ∀= ,, : 2 1 { 0 , 1 } uv vw uw uv uvw X X X ∀+ ∀∈
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Fundamental Observations If G is an undirected and un-weighted graph and C is a clustering then: f gg p g 1 2 () 1 qc y When all the edges are inter-cluster q(C)=-1/2, eg: Bipartite graph =(X:Y,E)with cluster X and Y G (X:Y,E)with cluster X and Y y When all the clusters cliques with no inter-cluster edges q(C)=1, hen number of clusters are infinite when number of clusters are infinite
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Fundamental Observations(Contd) lustering with maximum modularity has no cluster with Clustering with maximum modularity has no cluster with single node having degree 1.
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Fundamental Observations(Contd) clustering with maximum modularity each cluster consist In clustering with maximum modularity each cluster consist of a connected sub-graph
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Counterintuitive Behavior on cality y Non-locality iti it t S t llit y Sensitivity to Satellite y Scaling Behavior
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NP-Completeness oblem 1(Modularity): y Problem 1(Modularity): Given a graph G and a number K is there a clustering C of G, for hich C)>=K which q(C)>=K b l 2 ( 3 ) y Problem 2(3-Partition): Given 3k positive integers numbers a 1 , a 2 , …, a 3k such that the sum and b/4<a i <b/2 for an integer b and for all i=1,2,…3k is there a partition of these numbers into k sets, such 3 1 k i i ak b = = that the numbers in each set sums upto b?
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NP-Completeness (contd) n instance ={ a 3 rtition can be y An instance A={ a 1 , a 2 , …, a 3k } of 3-Partition can be transformed in to an instance (G(A), K(A)) of Modularity (A) has a clustering with modularity at least (A) if and y G(A) has a clustering with modularity at least K(A) , if and only if a 1 2 3k can be partitioned into k set of sum 3 1 1 k k i i ba = =
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NP-Completeness (contd) onstruct a graph (A) ith iques H y Construct a graph G(A) with k cliques H 1 ,H 2 ,…H k of size each.
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This note was uploaded on 05/20/2011 for the course CAP 5515 taught by Professor Ungor during the Spring '08 term at University of Florida.

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Com_NP-hard - On Modularity Clustering Presented by: Group...

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