Liaoruo Wang and John E. Hopcroft Dept. of Computer Engineering & Computer Science, Cornell University In Proc. 7th Annual Conference on Theory and Applications of Models of Computation (TAMC), June 2010 Presented by Nam Nguyen

 Motivation  Introduction  Contributions of the paper  Definitions  WHISKER is NP-Complete.  Algorithms.

 C.S is a classical but still-hot topic in complex networks.  Previous studies: Communities were assumed to be densely connected inside but sparsely connected outside.  A different point of view: We should disregard “whiskers” and elaborate “cores” in the networks.

 Roughly speaking ◦ Whiskers: Subsets of vertices that are barely connected to the rest of the network. ◦ Cores: Connected subgraphs that are densely connected inside and well-connected to the rest of the network, i.e., “real communities”  Why??? ◦ For real-world societies, communities are also well connected to the rest of the network. ◦ Imagine a close-nit community, CISE Dept., with only one connection with the outer world.  Definitions come right away.

 More concrete definitions of “whiskers” and “cores” in a networks.  WHISKER is NP-Complete  Three heuristic algorithms for finding approximate cores.  Simulation results.

 Graph G = (V,E) undirected, A = (A i,j ). For S⊆V, let S C = V\S.  Conduction of S where  A suitable cut

 A k-whisker  A maximal k-whisker

 A whisker  A maximal whisker

 A core

Proof The only suitable cut of size = 26 |S ⋃ T| = 25 >

Proof (1a) e xr + e xz + e yr + e yz ≤ v x + v y (1b) e yr + e xy + e zr + e xz ≤ v y + v z (1c) e xr + e yr + e zr > v x + v z (1a) + (1b) and use (1c) gives e xr +2e yr +e zr +e xy +e yz +2e xz ≤ v x +2v y +v z < e xr +e yr +e zr +v y  e yr + e xy + e yz < v y

 NAE-3-SAT: The problem of determining whether there exists a truth assignment for a 3-CNF Boolean formula such that each clause has at least one true literal and at least one false literal. Fact: NAE-3-SAT is NP-Complete [1]  WHISKER: Given an unweighted undirected graph, determine whether there exists a whisker or not. WHISKER is NP-Complete (of course, from a reduction from NAE-3-SAT)

 Road map ◦ 1. Construct a special graph G of 2n vertices and show that G admits 2 n whiskers and no more. ◦ 2. Construct a G-like graph for the 3- SAT problem. ◦ 3. Make a reduction from NAE-3-SAT problem to WHISKER

 WHISKER is in NP  Reduction from NAE-3-SAT to WHISKER ◦ Consider the following graph (constructed in poly time)  At each row, pick only one vertex (i.e., either x i or ¬x i )  The resulted graph G of n vertices is a whisker  Total number of whiskers is 2 n …………  And no more than that

 2 n whiskers and no more than that!!! Why???  Suppose there is a whisker W of 2k+j vertices  Cut size of W  By definition of suitable cut size, we have which implies !!!!

 NAE-3-SAT ≤ P WHISKER  Consider an instance of NAE-3-SAT with n variables and c clauses.  Construct G 1, G 2, …, G c as follow

 NAE-3-SAT ≤ P WHISKER  Now, combine all G i ’s and add up all edge weights to get G’.  Next GG G’ G* 3CNF has a satisfied assignment  contains a whisker update

 Update G ( )  Update G’ ◦ Amplify all edge weights of G’ by a small amount δ where cn 2 δ << 1  All whiskers in new G are the same as in old G.

 G* = G + G’  Goal: If the 3CNF instance has a satisfied truth assignment, then selecting true literal from each row of G* gives us a whisker of size n, and vice versa.  For any truth assignment of 3SAT, rearrange the literals in to TRUE and FALSE columns.  If there is a satisfied not-all-equal assignment for 3SAT ◦ Each clause must have one TRUE and one FALSE literals. ◦ Not all the literals in each clause can be in the same column. ◦ For each i th clause, G i contains n 2 -2 edges connecting its two columns ◦ Total cut size is required to satisfied

 If there is NO satisfied not-all-equal assignment for 3SAT ◦ At least one clause i has its literals located in the same column  n 2 edges between the two columns of G i. ◦ For the other (c-1) clauses, there are at most (n 2 -2) edges connecting the their two columns. Total number of edges: (c-1)(n 2 -2)+n 2 = cn 2 –2c+2. ◦ Of course, we don’t want selecting the true literal in each row give us a whisker, thus Combining the two inequalities, if ℇ and δ is chosen such that Then If the 3CNF instance has a satisfied truth assignment, then selecting true literal from each row of G* gives us a whisker of size n, and vice versa. ◦ Hence, NAE-3-CNF ≤ P WHISKER □

 On random graph ◦ Alg 2 can positively find an approximate core ◦ Alg 3 fails to find approximate core ◦ The size of core growing linearly with d = np (fixed n) and logarithmically with n (fixed d) ◦ ??? G(n,p) displays core structure with high probability when p > 1/n ???

 Textual graph ◦ Vertices and Edges: Words and their semantic Correlations ◦ Data is crawled from 10K scientific papers of KDD conf. ( ) ◦ Pointwise mutual information ◦ Total: 685 vertices and edges

 Both alg 2 and 3 successfully find approximate cores.  Higher values of λ indicate smaller core sizes.  Fig (b), the best community of the textual graph has a large conductance of.3  best community has as many internal edges as cut edges.  Alg 3 is believed to be more useful.

 Is a “whisker” make sense?

 [1] Schaefer, T. J. The complexity of satisfiability problems. In Proc. 10th Ann. ACM Symp. on Theory of Computing (1978), Association for Computing Machinery, pp