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Approximates pp the optimal p cut [ [Shi‐Malik, , ’00] ] Can be used to approximate the optimal k‐way normalized cut Emphasizes cohesive clusters Increases the h unevenness in i the h distribution di ib i of f the h data d Associations between similar points are amplified, associations between dissimilar points are attenuated Th The data dt bbegins i t to “ “approximate i t a clustering” l t i ” Well‐separated space Transforms data to a new “embedded embedded space”, space , consisting of k orthogonal basis vectors NB: Multiple eigenvectors prevent instability due to information f loss l 11/8/2010 Jure Leskovec, <strong>Stanford</strong> CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 42
Eigengap: The difference between two consecutive eigenvalues Most stable clustering is generally given by the value k that maximises eigengap: p Example: Eigenvalue 50 45 40 35 30 25 20 15 10 5 0 λ 1 λ 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 k k k k1 max k Choose kk=22 11/8/2010 Jure Leskovec, <strong>Stanford</strong> CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu 43 2 1
Page 1 and 2: CS224W: Social and Information Netw
Page 3 and 4: Today: 3 methods (3) Trawling: Comm
Page 5 and 6: (2) A more well‐defined well defi
Page 7 and 8: Marketbasket analysis: What items
Page 9 and 10: U={1,2,3,4,5}, U={12345} f=3 S1={1
Page 11 and 12: [Kumar et al. ‘99] Claim: (3) (i
Page 13 and 14: For the proof we will need the foll
Page 15 and 16: Note: As soon as s nodes appear in
Page 17 and 18: We have: Total height of all bucket
Page 19 and 20: Undirected graph G(V,E): G(V E): B
Page 21 and 22: A Express partitioning objectives
Page 23 and 24: Criterion: Normalized‐cut [ [Shi
Page 25 and 26: jth j coordinate of Ax: Sum of th
Page 27 and 28: What if G is not connected? Say G
Page 29 and 30: Degree matrix (D): n n diagonal ma
Page 31 and 32: For symmetric matrix M: 2 min x T
Page 33 and 34: Express p p partition ( (A,B) , ) a
Page 35 and 36: How to define a “good” good par
Page 37 and 38: p g Pre‐processing: Build Lapla
Page 39 and 40: 11/8/2010 Jure Leskovec, Stanford C
Page 41: k‐eigenvector k eigenvector Algor
Page 45: Non‐overlapping Non overlapping v

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