Four graph partitioning algorithms. Fan Chung University of California, San Diego

Transcription

1 Four graph partitioning algorithms Fan Chung University of California, San Diego

2 History of graph partitioning NP-hard approximation algorithms Spectral method, Fiedler 73, Folklore Multicommunity flow, Leighton+Rao 88 Semidefinite programming, xxxxxxxxxxxxxarora+rao+vazirani 04 Expander flow, Arora+Hazan+Kale 04 Single commodity flows, xxxxxxxxxxkhandekar+rao+vazirani 06

3 traditional applications of graph partition algorithms: Divide-and-conquer algorithms Circuit layout & designs Parallel computing Hierarchical clusterings Bioinformatics

4 Applications of partitioning algorithms for massive graphs Web search identify communities locate hot spots trace targets combat link spam epidemics

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6 Motivations Outline of the talk Conductance and Cheeger s inequality Four graph paritioning algorithms xxby using: eigenvectors random walks PageRank heat kernel Local graph algorithms Future directions

7 Two types of cuts: Vertex cut edge cut How good is the cut?

8 e(s,v-s) Vol S e(s,v-s) S Vol S = Σ v ε S deg(v) S = Σ 1 v ε S V-S S

9 The Cheeger constant for graphs The Cheeger constant h Φ G G = ess (, ) min S min( vol S, vol S ) The volume of S is vol( S) = dx h G and its variations are sometimes called Φ G x S conductance, isoperimetric number,

10 The Cheeger inequality The Cheeger constant Φh G = ess (, ) min S min( volsvols, ) The Cheeger inequality 2Φ λ G λ : the first nontrivial eigenvalue of the xx(normalized) Laplacian. Φ 2 G 2

11 The spectrum of a graph Adjacency matrix Many ways to define the spectrum of a graph. How are the eigenvalues related to properties of graphs?

12 The spectrum of a graph Adjacency matrix Combinatorial Laplacian L= D A diagonal degree matrix Normalized Laplacian Random walks Rate of convergence adjacency matrix

13 The spectrum of a graph Discrete Laplace operator 1 Δ f ( x) = ( f ( x) f( y)) d Lxy (, ) = not symmetric in general x y x 1 if x = y { 1 if x y and x y d x Normalized Laplacian symmetric normalized with eigenvalues L( x, y ) = { 1 if x = y 1 if x y and x y d d x 0= λ0 λ1 λn 1 2 = λ y

14 Can you hear the shape of a network? λ dictates many properties of a graph. connectivity diameter isoperimetry z(bottlenecks) How good is the cut by using the eigenvalue λ?

15 Finding a cut by a sweep Using a sweep by the eigenvector, can reduce the exponential number of choices of subsets to a linear number.

16 Finding a cut by a sweep Using a sweep by the eigenvector, can reduce the exponential number of choices of subsets to a linear number. Still, there is a lower bound guarantee by using the Cheeger inequality. 2Φ λ Φ 2 2

17 Partitioning algorithm The Cheeger inequality Using eigenvector f, the Cheeger inequality can be stated as α Φ 2Φ λ where λ is the first non-trivial eigenvalue of the Laplacian and α is the minimum Cheeger ratio in a sweep using the eigenvector. f

18 Eigenvalue problem for n x n matrix:. n 30 billion websites Hard to compute eigenvalues Even harder to compute eigenvectors

19 In the old days, compute for a given (whole) graph. In reality, can only afford to compute locally. (Access to a (huge) graph, e.g., for a vertex v, find its neighbors. Bounded number of access.)

20 Finding a cut by a sweep Using a sweep by the eigenvector can reduce the exponential number of choices of subsets to a linear number. Using a local sweep by random walks, PageRank and its variations can further reduce the a linear number of choices to a specified finite number of sizes.

21 Four one-sweep graph partitioning algorithms graph spectral method spectral partition algorithm random walks PageRank local partition algorithms heat kernel

22 4 Partitioning algorithm 4 Cheeger inequalities graph spectral method random walks PageRank Fiedler 73, Cheeger, 60 s Mihail 89 Lovasz, Simonovits, 90, 93 Spielman, Teng, 04 Andersen, Chung, Lang, 06 heat kernel Chung, PNAS, 08.

23 Graph partitioning Local graph partitioning Courtesy of Reid Andersen

24 What is a local graph partitioning algorithm? A local graph partitioning algorithm finds a small cut near the given seed(s) with running time depending only on the size of the output.

25 The definition of PageRank given by Brin and Page is based on random walks.

26 Partitioning Computing PageRank History of computing Pagerank Brin+Page 98 Personalized PageRank, Haveliwala 03 Computing personalized PageRank, xxxxxxxxxxxxxxxxxxxxxxxxjeh+widom 03 xxxxxxxxxxxxxxxxxxxxxxxx Berkhin 06

27 Random walks in a graph. G : a graph P : transition probability matrix Puv (, ) = 1 d u if u v, 0 otherwise. d u := the degree of u. A lazy walk: W = I + P 2

28 Original definition of PageRank A (bored) surfer either surf a random webpage with probability α or surf a linked webpage with probability 1- α α : the jumping constant p = α (,,..., ) (1 α n n n ) pw

29 Definition of personalized PageRank Two equivalent ways to define PageRank pr(α,s) (1) p = αs+ (1 α) pw s: the seed as a row vector α : the jumping constant s

30 Definition of PageRank Two equivalent ways to define PageRank p=pr(α,s) (1) p = αs+ (1 α) pw (2) p s = ( 1, 1,..., 1 n n n) t t = α (1 α) ( sw ) t= 0 s = some seed, e.g., (1,0,...,0) the (original) PageRank personalized PageRank (Organize the random walks by a scalar α.)

31 Partitioning algorithm using random walks Mihail 89, Lovász+Simonovits, 90, 93 k vol( S) W ( u, S) π ( S) 1 du β 2 k 8 k Leads to a Cheeger inequality: β 2Φ λ 8logn β G Φ 8logn 2 2 G where is the minimum Cheeger ratio over sweeps by using a lazy walk of k steps from every vertex for an appropriate range of k.

32 Algorithmic aspects of PageRank Fast approximation algorithm for x personalized PageRank greedy type algorithm, linear complexity Can use the jumping constant to approximate PageRank with a support of the desired size. Errors can be effectively bounded.

33 Approximate the pagerank vector : pr( α, s) = p+ pr( α, r) Approximate pagerank Residue vector

34 Partitioning algorithm using PageRank Using the PageRank vector with seed as a subset S vol( S) vol( G)/4, and a Cheeger inequality can be obtained : γ ΦS 8logs γ u Φ 2 2 u u 8log where is the minimum Cheeger ratio over sweeps by using personalized PageRank with a random seed in S. The volume of the set of such u is > vol(s)/4. s

35 A partitioning algorithm using PageRank Algorithm(φ,s,b): Compute ε-approximate Pagerank p=pr(α,s) with α=0.1/(φ 2 b), ε=2 -b /b. One sweep algorithm using p for finding cuts with conductance < φ. Performance analysis: If s is in a set S with conductance Φ>φ 2 log s, with constant probability, the algorithm outputs a cut C with condutance < φ, of size 1 order s and vol( C S) > vol( S). (Improving previous bounds by a factor of φlog s. ) 4

36 Courtesy of Reid Andersen.

37 Courtesy of Reid Andersen

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39 Kevin Lang 2007

40 4 Partitioning algorithm 4 Cheeger inequalities graph spectral method random walks PageRank Fiedler 73, Cheeger, 60 s Mihail 89 Lovasz, Simonovits, 90, 93 Spielman, Teng, 04 Andersen, Chung, Lang, 06 heat kernel Chung, PNAS, 08.

41 PageRank versus heat kernel, s k k (1 ) ( sw ) ρts, k = 0 pα = α α t = e k = 0 s ( tw ) k! Geometric sum Exponential sum k

42 PageRank versus heat kernel p k k α, s = α (1 α) ( sw ) ρts, k = 0 t = e k = 0 s ( tw ) k! Geometric sum Exponential sum k p = α + (1 α) pw ρ = ρ( I W) t recurrence Heat equation

43 Definition of heat kernel t t t Ht = e I + tw + W + + W + 2 k! t( I W) = e = e tl 2 k 2 k (......) 2 k t 2 k t k = I tl+ L ( 1) L k! Ht = ( I W) Ht t ρ = sh ts, t

44 Partitioning algorithm using the heat kernel Theorem: vol( S) 2 tκtu, /4 ρtu, ( S) π( S) e du where κ tu, is the minimum Cheeger ratio over sweeps by using heat kernel pagerank over all u in S.

45 Partitioning algorithm using the heat kernel Theorem: vol( S) 2 tκtu, /4 ρtu, ( S) π( S) e du where κ tu, is the minimum Cheeger ratio over sweeps by using heat kernel pagerank over all u in S. Theorem: For vol S 2/3 ( ) vol( G), ρ, ( S ) π th ( S ) e S ts. (Improving the previous PageRank lower bound 1-t h S.)

46 Theorem: ( S), ( S ) ( S S ts ) (1 ( S )) e π ρ π π ht/1 ( ) Sketch of a proof: Consider Show Then Ft ( ) = log( ρ ( S) π ( S)) 2 Ft () 0 2 t ts, ΦS Ft () F(0) = t t 1 π S ( ) Solve and get /1 ( ( )), ( S ) ( S ht S S ts ) (1 ( S )) e π ρ π π

47 Random walks versus heat kernel How fast is the convergence to the stationary distribution? For what k, can one have fw k π? Choose t to satisfy the required property.

48 Partitioning algorithm using the heat kernel Using the upper and lower bounds, a Cheeger inequality can be obtained : 2 2 κ S ΦS ΦS λs 8 8 where λ S is the Dirichlet eigenvalue of the Laplacian, and κ S is the minimum Cheeger ratio over sweeps by using heat kernel with seeds S for appropriate t.

49 Dirichlet eigenvalues for a subset S V λ S = inf f u v ( f( u) f( v)) w f( w) 2 d w 2 S V over all f satisfying the Dirichlet boundary condition: f() v = 0 for all v S.

50 Partitioning algorithm using the heat kernel Using the upper and lower bounds, a Cheeger inequality can be obtained : 2 2 κ S ΦS ΦS λs 8 8 where λ S is the Dirichlet eigenvalue of the Laplacian, and κ S is the minimum Cheeger ratio over sweeps by using heat kernel with seeds S for appropriate t.

51 Partitioning algorithm using the heat kernel Using the upper and lower bounds, a Cheeger inequality can be obtained : 2 2 κ S ΦS ΦS λs 8 8 where λ S is the Dirichlet eigenvalue of the Laplacian, and κ S is the minimum Cheeger ratio over sweeps by using heat kernel with seeds S for appropriate t.

52 Partitioning algorithm using the heat kernel Using the upper and lower bounds, a Cheeger inequality can be obtained : 2 2 κu Φu ΦS λs 8logs 8logs where λ S is the Dirichlet eigenvalue of the Laplacian, and κ u is the minimum Cheeger ratio over sweeps by using heat kernel with a random seed in S. The volume of the set of such u is > vol(s)/4.

53 Courtesy of Reid Andersen

54 Future directions: Use PageRank and the heat kernel xxpagerank to shed light on: The geometry of graphs? Solving combinatorial problems, such xxas covering, packing, matching, etc. Graph drawing, visualization Metric embedding

55 Courtesy of Reid Andersen