Graph P artitioning a nd Clustering for Community Detection

GRAPH PARTITIONINGAND CLUSTERING FORCOMMUNITY DETECTION

Presented By: Group One

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Outline

Introduction: Hong Hande Graph Partitioning: Muthu Kumar C and Xie

Shudong Partitional Clustering: Agus Pratondo Spectral Clustering: Li Furong and Song

Chonggang Summary and Applications of Community

Detection: Aleksandr Farseev

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INTRODUCTION-BY HONG HANDE

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Facebook Group

https://www.facebook.com/thebeatles?rf=111113312246958

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Flickr group

http://www.flickr.com/groups/49246928@N00/pool/with/417646359/#photo_417646359

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CS6234 Advanced Algorithms

Sub-communityWhole class as a community

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Graph construction from web data(1)

Webpage www.x.comhref = “www.y.com”href = “www.z.com”

x

zy

ba

Webpage www.y.comhref = “www.x.com”href = “www.a.com”href = “www.b.com”

Webpage www.z.comhref = “www.a.com”

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Graph construction from web data(2) 8

Web pages as a graph

Cnn.com

Lots of links, lots of images. (1316 tags)

http://www.aharef.info/2006/05/websites_as_graphs.htm

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Internet as a graph

nodes = service providers edges = connections

hierarchical structure

S. Carmi,S. Havlin, S. Kirkpatrick, Y. Shavitt, E. Shir. A model of Internet topology using k-shell decomposition. PNAS 104 (27), pp. 11150-11154, 2007

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Emerging structures

Graph (from web, daily life) present certain structural characteristics

Group of nodes interacting with each other Dense inter-connections

functional/topical associations

Communitya.k.a. group, subgroup, module, cluster

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Community Types

Explicit The result of conscious human decision

Implicit Emerging from the interactions & activities of users Need special methods to be discovered

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Defining Communities

Often communities are defined with respect to a graph, G = (V,E) representing a set of objects (V) and their relations (E).

Even if such graph is not explicit in the raw data, it is usually possible to construct, e.g. feature vectors distances graph

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Communities and graphs

Internal edge

External edge

Given a graph, a community is defined as a set of nodes that are more densely connected to each other than to the rest of the network nodes

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Graph cuts

A cut is a partition of the vertices of a graph into two disjoint subsets.

The cut-set of the cut is the set of edges whose end points are in different subsets of the partition.

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Community detection methods

Graph partitioning

Node clustering K-means clustering Spectral clustering

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GRAPH PARTITIONING

MUTHU KUMAR C

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Graph Partitioning

Dividing vertices into groups of predefined size.

Given a graph G = (V, E, WE), with vertices V, edges E and edge weights WE.

Choose a partition such that: V = V1 U V2 U … U VP

V1∩ V2 …. ∩ Vp = Ø

Bisectioning: Partitioning into two equal sized groups of vertices.

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How many partitions?

There exists many possible partitioning to search. Just to divide into 2 partitions there are: which is exponential in n. Choosing optimal partitioning is NP-complete.

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Kernighan/Lin Algorithm1

An iterative, 2-way, balanced partitioning (bi-sectioning) heuristic.

The algorithm can also be extended to solve more general partitioning problems.

Given and find a partition such that: Cutsize T between A and B is minimized.

where

1. Kernighan, B. W., & Lin, S. (1970). An efficient heuristic procedure for partitioning graphs. Bell system technical journal, 49(2), 291-307.

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Let and be two vertices. External Cost Internal Cost

Moving a node from A to B increases T by and decreases T by

This is measured as , and are defined analogously for

b in B.

Kernighan-Lin: Definitions

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K/L Algorithm: Swap

A B

a

b

Cutsize

A Bb

a

A B

b

a

Cutsize

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Kernighan-Lin Algorithm

// KERNIGHAN-LIN Page 1 of 2COMPUTE T = COST(A,B) FOR INITIAL A, B

REPEAT // SWEEP BEGINS Compute costs D(v) for all v in V Unmark all vertices in V

While there are unmarked nodes Find an unmarked pair (ai,bi) with maximal gai,bi(i)

Mark ‘a’ and ‘b’ Update D(v) for all unmarked v Endwhile

Each sweep greedily computes |V|/2 possible X A, Y B to swap, picks a sequence of best such swaps.

but do not swap them.

as though ‘a’ and ‘b’ had been swapped.

(1)

(2)

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Kernighan-Lin Algorithm

// KERNIGHAN-LIN Page 2 of 2We have now computed:

*) a sequence of pairs(a1,b1), … , (ak,bk) and*) gains g(1),…., g(k) where k = |V|/2,

numbered in the order in which we marked themPick m ≤ k, which maximizes gain.

GAIN =

If Gain > 0 then // it is worth swapping Update newA = A - { a1,…,am } U { b1,…,bm } Update newB = B - { b1,…,bm } U { a1,…,am } Update T = T – Gain endifUNTIL GAIN <= 0 // SWEEP ENDS

Gain is reduction in cost from swapping (a1,b1) through (am,bm)

Kernighan-Lin Example

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Cut cost: 9Unmarked: 1,2,3,4,5,6,7,8

Edges are unweighted in this example

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Cut cost: 9Unmarked : 1,2,3,4,5,6,7,8

D(1) = 1 D(5) = 1 D(2) = 1 D(6) = 2 D(3) = 2 D(7) = 1 D(4) = 1 D(8) = 1

Costs D(v) of each node:

Nodes that lead to maximum gain

Kernighan/Lin Example31

Calculate D values to find best pair

Kernighan/Lin Example

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Cut cost: 9Unmarked : 1,2,3,4,5,6,7,8

D(1) = 1 D(5) = 1 D(2) = 1 D(6) = 2 D(3) = 2 D(7) = 1 D(4) = 1 D(8) = 1

g1 = 2+1-0 = 3 Swap (3,5) G1 = g1 =3


Gain in the current pass

Costs D(v) of each node:

Gain after node swapping

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Mark the identified pair as a candidate swap.



Cut cost: 6Unmarked: 1,2,4,6,7,8

D(1) = 1 D(5) = 1 D(2) = 1 D(6) = 2 D(3) = 2 D(7) = 1 D(4) = 1 D(8) = 1 g1 = 2+1-0 = 3 Swap (3,5) G1 = g1 =3

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New partitions and cut cost





D(1) = -1 D(6) = 2 D(2) = -1 D(7)=-1 D(4) = 3 D(8)=-1

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D(1) = -1 D(6) = 2 D(2) = -1 D(7)=-1 D(4) = 3 D(8)=-1 g2 = 3+2-0 = 5 Swap (4,6) G2 = G1+g2 =8




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Cut cost: 1Unmarked: 1,2,7,8

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Cut cost: 7Unmarked: 2,8


D(1) = -1 D(6) = 2 D(2) = -1 D(7)=-1 D(4) = 3 D(8)=-1 g2 = 3+2-0 = 5 Swap (4,6) G2 = G1+g2 =8

D(1) = -3D(7)=-3 D(2) = -3 D(8)=-3

g3 = -3-3-0 = -6 Swap (1,7) G3= G2 +g3 = 2




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Cut cost: 9Unmarked: –


Cut cost: 1Unmarked: 1,2,7,8

Cut cost: 7Unmarked: 2,8


D(1) = -1 D(6) = 2 D(2) = -1 D(7)=-1 D(4) = 3 D(8)=-1 g2 = 3+2-0 = 5 Swap (4,6) G2 = G1+g2 =8

D(1) = -3D(7)=-3 D(2) = -3 D(8)=-3

g3 = -3-3-0 = -6 Swap (1,7) G3= G2 +g3 = 2

D(2) = -1 D(8)=-1

g4 = -1-1-0 = -2 Swap (2,8) G4 = G3 +g4 = 0

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D(1) = -1 D(6) = 2 D(2) = -1 D(7)=-1 D(4) = 3 D(8)=-1 g2 = 3+2-0 = 5

Swap (4,6) G2 = G1+g2 =8

D(1) = -3D(7)=-3 D(2) = -3 D(8)=-3

g3 = -3-3-0 = -6 Swap (1,7) G3= G2 +g3 = 2

D(2) = -1 D(8)=-1

g4 = -1-1-0 = -2 Swap (2,8) G4 = G3 +g4 = 0

Since Gm > 0, the first m = 2 swaps (3,5) and (4,6) are executed. 2

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Since Gm > 0, more passes are needed until Gm 0.

Maximum positive gain Gm = 8 with m = 2.

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Escaping Local minima

Non monotonically increasing gains, that is, in the sequence of m swaps chosen, some may be negative.

Possibly escape “local minima”.

But there is no guarantee of optimal solution.

Demerits40

Bi-sectioning does not generalize well to k-way partitioning.

Partition to predefined sizes limits utility to niche applications.

ANALYSIS OF K/L ALGORITHM

XIE SHUDONG

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REPEATCompute costs D(v) for all v in VUnmark all vertices in VWhile there are unmarked nodes

Find an unmarked pair (a, b) with maximal g(a, b) Mark ‘a’ and ‘b’ (but do not swap them)Update D(v) for all unmarked v, as though ‘a’ and ‘b’ had been

swappedEndwhilePick m maximizing If Gain > 0 then … it is worth swapping

Update newA = A - { a1, …, am } { b∪ 1, …, bm } Update newB = B - { b1, …, bm } { a∪ 1, …, am }Update T = T – G

endifUNTIL GAIN <= 0

K/L Algorithm: AnalysisCOMPUTE T = COST(A,B) FOR INITIAL A, B

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swappedEndwhilePick m maximizing If Gain > 0 then … it is worth swapping



K/L Algorithm: AnalysisCOMPUTE T = COST(A,B) FOR INITIAL A, B

A B

O(|V|²)

Edges |V|/2

Nodes |V|/2

**

==

All Ext Edges |V|²/4

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swappedEndwhilePick m maximizing If Gain > 0 then



A B

a

b

For one node a: D(a) = E(a) – I(a) O(|V|)

For all |V| nodes

O(|V|²)

O(|V|²)

K/L Algorithm: AnalysisCOMPUTE T = COST(A,B) FOR INITIAL A, BO(|V|²)

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O(|V|²)

O(|V|)

K/L Algorithm: AnalysisCOMPUTE T = COST(A,B) FOR INITIAL A, BO(|V|²)

COMPUTE T = COST(A,B) FOR INITIAL A, B46






O(|V|²)

O(|V|)

O(|V|²)

K/L Algorithm: AnalysisO(|V|²) The (i+1)-th Candidate Pair

A B Pairs

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O(|V|²)

O(|V|)

O(1)

K/L Algorithm: Analysis

O(|V|²)

COMPUTE T = COST(A,B) FOR INITIAL A, BO(|V|²)

COMPUTE T = COST(A,B) FOR INITIAL A, B48






O(|V|²)

O(|V|)

O(1)

newD(a’) = D(a’) + 2*w(a’, a) - 2*w(a’, b) O(1)

O(|V|)


O(|V|²)

O(|V|²)

(i+1)-th loop: |V|-2i Unmarked Nodes

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O(|V|²)

O(|V|)

O(1)O(|V|)

|V|/2 pairs to be found

O(|V|³)


O(|V|²)







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O(|V|²)

O(|V|)

O(1)O(|V|)

O(|V|³)

g(1)g(1) + g(2)…

…g(1) + g(2) + … + g(m) + … + g(|V|/2)

g(1) + g(2) + … + g(m) → G O(|V|)

O(|V|)


O(|V|²)


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O(|V|²)O(|V|)

O(1)O(|V|)

O(|V|)

A

O(|V|)


O(|V|²)

O(|V|³)


O(|V|)

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Update newA = A - { a1, …, am } { b1, …, bm } ∪ Update newB = B - { b1, …, bm } { a1, …, am }∪Update T = T – G


O(|V|²)

O(|V|)

O(1)O(|V|)

O(|V|)

O(|V|)

O(|V|)


O(|V|²)

O(|V|³)


O(1)

O(|V|)

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Update newA = A - { a1, …, am } { b1, …, bm } ∪ Update newB = B - { b1, …, bm } { a1, …, am }∪Update T = T – G


O(|V|²)

O(|V|)

O(1)O(|V|)

O(|V|)

O(|V|)

O(|V|)

O(|V|)

(p iterations) O(p |V|³)


O(|V|²)

O(|V|³)


O(1)

How many Iterations?

Empirical testing by Kernighan and Lin on small graphs (|V|<=360) showed convergence after 2 to 4 passes

K-MEANS CLUSTERING

by Agus Pratondo

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Graph in Rn

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x ey

d

ba

c

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3

2

1 1

1

Graph in Rn

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x ey

d

ba

c

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3

2

1 1

1

a b c d e x ya 0 2 0 0 0 0 0b 2 0 0 0 0 1 5c 0 0 0 0 0 2 0d 0 0 0 0 1 3 0e 0 0 0 1 0 0 1x 0 1 2 3 0 0 0y 0 5 0 0 1 0 0

Adjacency matrix

Graph in Rn

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x ey

d

ba

c

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3

2

1 1

1

a b c d e x ya 0 2 0 0 0 0 0b 2 0 0 0 0 1 5c 0 0 0 0 0 2 0d 0 0 0 0 1 3 0e 0 0 0 1 0 0 1x 0 1 2 3 0 0 0y 0 5 0 0 1 0 0

Adjacency matrix

pa (0,2,0,0,0,0,0)b (2,0,0,0,0,1,5)c (0,0,0,0,0,2,0)d (0,0,0,0,1,3,0)e (0,0,0,1,0,0,1)x (0,1,2,3,0,0,0)y (0,5,0,0,1,0,0)

points in R7 :

Algorithm

Algorithm: Basic K-means1: Select K points as the initial centroids2: repeat3: Form K clusters by assigning all points to the closest centroids4: Re-compute the centroids of each cluster5: until the centroids do not change

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K-means example, step 159

X

Y

Let K = 3


k1

k2

k3

X

YPick 3 initialclustercenters(randomly)

Let K = 3

Algorithm


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K-means example, step 2-362

k1

k2

k3

X

Y

Assigneach pointto the closestclustercenter

Algorithm


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X

Y

Moveeach cluster centerto the meanof each cluster

k1

k2

k2

k1

k3

k3

`

``

Algorithm


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K-means example, step 3-4(repeat)66

X

YReassignpoints closest to a different new cluster center

k1

k2

k3


X

Y

Three points change

k1

k3k2


X

Y

re-compute cluster means

k1

k3k2

K-means example, step 3-4 (repeat)69

X

Y

move cluster centers to cluster means

k2

k1

k3

The centers change, repeat step 3-4


X

Y

No cluster center changes

convergedk2

k1

k3

Time Complexity

Algorithm: Basic K-means1: Select K points as the initial centroids2: repeat 3: Form K clusters by assigning all points to the closest centroids4: Re-compute the centroids of each cluster5: until the centroids do not change

Loop is repeated i times. Step 3: There are n points. Each points, the distance to k clusters is evaluated. Step 4: There are k cluster center that should be re-computed.

If there are k clusters with n points , and the iteration is repeated i times thenthe time complexity will be O(kni)

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Discussion72

Result can vary significantly depending on the initial choice of seeds (number and position) To increase chance of finding global optimum:

restart with different random seeds.

Problem with initializations 73

4 most top points 4 most bottom points

Problem with initializations 74

4 most left points 4 most right points

Discussion75

Result can vary significantly depending on initial choice of seeds (number and position) To increase chance of finding global optimum:

restart with different random seeds.

SPECTRAL CLUSTERING-BY LI FURONG

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Motivation77

Motivation

Two kinds of clusters

convex shaped non-convex shaped

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Motivation

Two kinds of clusters convex shaped, compact


k-means

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Motivation

Two kinds of clusters convex shaped, compact non-convex shaped, connected


k-means

spectral clustering

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Key Idea

Project the data points into a new space Clusters can be trivially detected in the new

space

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Key Idea

Project the data points into a new space Clusters can be trivially detected in the new

space

Next, we will cover How to find the new space How to represent data points in the space

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Matrix Representations of Graphs84

Matrix Representations of Graphs

Adjacency matrix W

Degree di of a node i

Degree matrix D

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Matrix Representations of Graphs

Adjacency matrix W

Degree di of a node i

Degree matrix D

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Graph Laplacian

Graph Laplacian87

Graph Laplacian

Graph Laplacian

Next, we will see some properties of L, which would be used for spectral clustering

We will work closely with linear algebra, especially eigenvalues and eigenvectors

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Properties of Graph Laplacian (1)

(1)(2)

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(1)(2)apply Equation 2

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Proof:


(1)(2)apply Equation 2

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Proof:


(1)(2)

apply Equation 1

apply Equation 2

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Proof:


(1)(2)

apply Equation 1

apply Equation 2

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Proof:


The smallest eigenvalue of L is 0, the corresponding eigenvector is the constant one vector . (1)(2)

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97



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Proof:



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Proof:

We Have Done So Many Works…101

We Have Done So Many Works…

Transform the graph to Laplacian L102


Transform the graph to Laplacian L

Study the properties of L, basically the eigenvalues and eigenvectors

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Transform the graph to Laplacian L

Study the properties of L, basically the eigenvalues and eigenvectors

Finally, we can see the relationship between the graph and the eigenvalues!

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Number of Connected Components & Eigenvalues of L

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a connected component of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices in the supergraph

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If an eigenvalue v has multiplicity k, then there are k linear independent eigenvectors corresponding to v


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indicator vector:




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indicator vector:




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indicator vector:




eigenvectors correspondingto eigenvalue 0

1⋮1

1⋮1

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Proof of Proposition 2111


1 connected component

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i j

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i jnm

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𝑓 =(1⋮1)


i jnm

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several connected components

1⋮1

1⋮1


i jnm 𝑓 =(1⋮1)

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several connected components

1⋮1

1⋮1


i jnm 𝑓 =(1⋮1)

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SPECTRAL CLUSTERING-BY SONG CHONGGANG

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Spectral Clustering Algorithm

Input: Graph , number k of clusters to form Compute adjacency matrix W and degree matrix D Laplacian L = D – W

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Input: Graph , number k of clusters to form Compute adjacency matrix W and degree matrix D Laplacian L = D – W Compute the first k eigenvectors of L Let contain the vectors as columns

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Input: Graph , number k of clusters to form Compute adjacency matrix W and degree matrix D Laplacian L = D – W Compute the first k eigenvectors of L Let contain the vectors as columns

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New space found!


Input: Graph , number k of clusters to form Compute adjacency matrix W and degree matrix D Laplacian L = D – W Compute the first k eigenvectors of L Let contain the vectors as columns Let be the vector corresponding to the i-th row of U Cluster the points into k clusters using k-means

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Representing data in the new space!



Time Complexity: O(n3)

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Example(1)

Now let’s go through an example. n = 6, k=2

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Example(2)

Step 1: Weighted adjacency matrix W and degree matrix D

Adjacency Matrix W Degree Matrix D

0 0 0

00

00

0

0 0

0

0

0

0

0 00

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Example(3)

Step 2: Laplacian matrix L=D-W

Laplacian Matrix L

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Example(4)

Step 3: Eigen-decomposition Eigenvalues =

Eigenvectors=

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Laplacian Matrix L

Example(5)

Step 3: Eigen-decomposition Eigenvalues =

Eigenvectors=

U

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Example(6)

Step 4: Embedding U=

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Example(6)


Each row represents a data point

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Example(7)


Map it to a two-dimensional space

0

0.25

0.5

-0.25

-0.5

0 0.25 0.5-0.25-0.5

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Example(8)

Step 5: Clustering K-means clustering

0

0.25

0.5

-0.25

-0.5

0 0.25 0.5-0.25-0.5

0

0.25

0.5

-0.25

-0.5

0 0.25 0.5-0.25-0.5

Cluster A

Cluster B

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Example(8)

Step 5: Clustering K-means clustering

0

0.25

0.5

-0.25

-0.5

0 0.25 0.5-0.25-0.5

0

0.25

0.5

-0.25

-0.5

0 0.25 0.5-0.25-0.5

Cluster A

Cluster B

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Why Spectral Clustering Works?(1)

Consider an ideal case There are no similarities between any nodes in

different connected components This conforms to Proposition 2:

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different connected components Compute the weighted adjacency matrix W and

degree matrix D. L = D - W; compute L’s 3 eigenvectors of eigenvalue

0.

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1⋮1

1⋮1

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different connected components Compute the weighted adjacency matrix W and

degree matrix D. L = D - W; compute L’s 3 eigenvectors of eigenvalue

0.


Consider an ideal case Let the three eigenvectors be three columns of

matrix U.

U=1⋮1

1⋮1

1⋮1

1⋮1

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Consider an ideal case Let the three eigenvectors be three columns of a

matrix U. Project the rows in U to a 3-dimensional space.

U=1⋮1

1⋮1

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Consider an ideal case Now we use K-Means in this space, we can have

very good results. # of 0 eigenvalues = # of connected components

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What if not the ideal case? We need to introduce Perturbation Theory.

Ideal Case

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What if not the ideal case? We need to introduce Perturbation Theory.

Perturbation is like noise.

Ideal Case Nearly ideal Case

Perturbation

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What if not the ideal case? Perturbation Theory will not be formally discussed

here. References will be offered on IVLE.

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What if not the ideal case? Perturbation Theory will not be formally discussed

here. What you need to know is:

For ideal case, the between-cluster similarity is 0. The first k eigenvectors of Laplacian matrix L are

indicators of clusters. For real case, L’ = L + H, where H is the perturbation. Perturbation theory tells us the eigenvectors

generated from L’ will be very close to the ideal vectors from L, bounded by a small value.

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APPLICATIONS AND SUMMARY-BY ALEKSANDR FARSEEV

Applications: VLSI

Very-large-scale integration (VLSI) - the process of creating integrated circuits by combining thousands of transistors into a single chip.

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Applications: VLSI design

ENTITY test isport a: in bit;

end ENTITY test;

DRCLVSERC

Circuit Design

Functional Designand Logic Design

Physical Design

Physical Verificationand Signoff

Fabrication

System Specification

Architectural Design

Chip

Packaging and Testing

Chip Planning

Placement

Signal Routing

Partitioning

Timing Closure

Clock Tree Synthesis

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Circuit:

Cut ca: four external connections

1

2

4

5

3

6

7 8

5

6

48

7 23

1

56

48

7 2

3 1

Cut ca

Block A Block B Block A Block B

Cut cb: two external connections

Cut cb

Applications: VLSI design(2)149

Applications: Social Media

One-modality (one-mode) network - type of networks where all vertices are of the same kind.

Multi-modality or multi-mode network - type of networks where vertices are of different kinds.

Hypergraph - is a generalization of a graph, where an edge (hyperedge) can connect any number of vertices.

(k, k) - (hyper) network - a network with k modalities and hyper edges involving exactly k vertices with each vertex from one unique modality. (3, 3) – network

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Applications: Social Media (2)

(3, 3) – network

Graphrepresentation

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(2, 2) User - Venue network

1 2 3 4

1 2 3 4 5

12 3

4 5

(1,2) User – User similarity network

UsersUsers

Venues

152


Reduction

(3, 3) User - Venue - Photo network(1, 2) User – User network

153


Detection

154


Understanding

Identification

155

Applications: Social Media (7) 156

http://next.comp.nus.edu.sg

http://next.comp.nus.edu.sg/

http://next.comp.nus.edu.sg/

Other applications

Parallel processing

Parallel Graph Computations

Complex Networks

Power Grids

Geographically Embedded Networks

Road Networks

Image Processing

157

Summary: KL Graph Partitioning

Able to perform only Bipartition.

Can not detect overlapping communities.

Time complexity – O

158

Summary: K-Means

Fast: O, where is # objects, is # clusters, and is #

iterations.

Easy to implement.

Need to specify k, the number of clusters, in advance

Not suitable to discover clusters with non-convex shapes

Can not detect overlapping communities

159

Summary: Spectral clustering

Need to specify k, the number of clusters, in advance

Can not detect overlapping communities.

Time complexity – O, where is # objects, is #

iterations.

Able to discover clusters with non-convex shapes

160

Sources162

1. Kernighan, B. W., & Lin, S. (1970). An efficient heuristic procedure for partitioning graphs. Bell system technical journal, 49(2), 291-307

2. James Demmel, CS 267: Applications of Parallel Computers , Graph Partitioning, http://www.cs.berkeley.edu/~demmel/cs267_Spr09

3. Andrew B. Kahng, Jens Lienig, Igor L. Markov, Jin Hu, VLSI Physical Design: From Graph Partitioning to Timing Closure

4. Sadiq M. Sait & Habib Youssef King, Chapter 2: Partitioning, Fahd University of Petroleum & Minerals College of Computer Sciences & Engineering Department of Computer Engineering September 2003.

5. http://shabal.in/visuals/kmeans/2.html6. www.cs.ucr.edu/~eamonn/205/MachineLearning3.ppt7. info.psu.edu.sa/psu/cis/asameh/cs-500/dm13-clustering.ppt

http://www.cs.berkeley.edu/~demmel/cs267_Spr09

http://shabal.in/visuals/kmeans/2.html

http://shabal.in/visuals/kmeans/2.html

Graph P artitioning a nd Clustering for Community Detection

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