# hw2 - communities C = C 1,C 2,C k Denote E X,Y the number...

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CAP5515: Computational Molecular Biology-Homework 2 Due at the beginning of the lecture on 04-07-2009 . No late assignment will be accepted. Consider the modularity function Q which is deﬁned as follows: First deﬁnition : The undirected network G = ( V,E ), with n nodes and m edges, is given by the symmetric matrix A = ( a uv ): a uv = 1 if ( u,v ) E and a uv = 0 otherwise. Let deg ( u ) denote the degree of the node u . Assume that the network is partitioned into disjoint communities. Deﬁne δ ( u,v ) = 1if u,v in a same community. Otherwise, δ ( u,v ) = 0. The modularity Q is deﬁned as Q 1 = 1 2 m ± a uv deg ( u ) deg ( v ) 2 m ² δ ( u,v ) (1) Second deﬁnition : Assume that the network G = ( V,E ) is partitioned into
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Unformatted text preview: communities C = { C 1 ,C 2 ,...,C k } . Denote E ( X,Y ) the number of edges that have one end node in X and other end node in Y with the notice that if u,v ∈ X ∩ Y the edge ( u,v ) will be counted twice. The modularity is deﬁned as Q 2 = ³ C i ∈C ´ E ( C i ,C i ) 2 m − ± E ( C i ,V ) 2 m ² 2 µ (2) Do the following questions: 1. (10 pts) Is Q 1 equivalent to Q 2 , i.e. Q 1 ( C ) = Q 2 ( C ). Justify your answer. 2. (10 pts) We have learnt that Q 2 has a resolution limit. How about Q 1 ? Justify your answer. 1...
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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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