introduction to data miningukang/courses/19s-dm/l20-graphs2.pdf · spectral graph partitioning ......
TRANSCRIPT
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1U Kang
Introduction to Data Mining
Lecture #20: Analysis of Large Graphs-2
U KangSeoul National University
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2U Kang
In This Lecture
Learn the min-cut problem in graphs, and its solutions
Learn an example of spectral graph theory (how linear algebra and graph problems interact)
Learn how to find small communities in graphs
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3U Kang
Outline
Spectral Clustering
Small Communities
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4U Kang
Graph Partitioning
Undirected graph 𝑮(𝑽, 𝑬):
Bi-partitioning task: Divide vertices into two disjoint groups 𝑨,𝑩
Questions: How can we define a “good” partition of 𝑮? How can we efficiently identify such a partition?
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A B
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5U Kang
Graph Partitioning
What makes a good partition?
Maximize the number of within-group connections
Minimize the number of between-group connections
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A B
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6U Kang
A B
Graph Cuts
Express partitioning objectives as a function of the “edge cut” of the partition
Cut: Set of edges with only one vertex in a group:
cut(A,B) = 21
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7U Kang
Graph Cut Criterion
Criterion: Minimum-cut Minimize weight of connections between groups
Degenerate case:
Problem: Only considers external cluster connections Does not consider internal cluster connectivity
arg minA,B cut(A,B)
“Optimal cut”
Minimum cut
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8U Kang
Graph Cut Criteria
Criterion: Normalized-cut [Shi-Malik, ’97]
Connectivity between groups relative to the density of each group
𝒗𝒐𝒍(𝑨): total weight of the edges with at least one endpoint in 𝑨: 𝒗𝒐𝒍 𝑨 = σ𝒊∈𝑨𝒌𝒊
Why use this criterion?
Produces more balanced partitions
How do we efficiently find a good partition?
Problem: Computing optimal normalized cut is NP-hard
[Shi-Malik]
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9U Kang
Spectral Graph Partitioning
A: adjacency matrix of undirected G Aij =1 if (𝒊, 𝒋) is an edge, else 0
x: a vector in n with components (𝒙𝟏, … , 𝒙𝒏) Think of it as a label/value of each node of 𝑮
What is the meaning of A x?
Entry yi is a sum of labels xj of neighbors of i
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10U Kang
What is the meaning of Ax?
ith coordinate of A x :
Sum of the x-values of neighbors of i
Make this a new value at node i
Spectral Graph Theory:
Analyze the “spectrum” of matrix representing 𝑮
Spectrum: Eigenvectors 𝒙𝒊 of a graph, ordered by the magnitude (strength) of their corresponding eigenvalues 𝝀𝒊:
𝑨 ⋅ 𝒙 = 𝝀 ⋅ 𝒙
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11U Kang
Example: d-regular graph
Suppose all nodes in 𝑮 have degree 𝒅and 𝑮 is connected
What are some eigenvalues/vectors of 𝑮?
𝑨 𝒙 = 𝝀 ⋅ 𝒙 What is ? What x?
Let’s try: 𝒙 = (𝟏, 𝟏,… , 𝟏)
Then: 𝑨 ⋅ 𝒙 = 𝒅, 𝒅,… , 𝒅 = 𝝀 ⋅ 𝒙. So: 𝝀 = 𝒅
We found eigenpair of 𝑮: 𝒙 = (𝟏, 𝟏,… , 𝟏), 𝝀 = 𝒅
Remember the meaning of 𝒚 = 𝑨 𝒙:
definition of
d-regular graph
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12U Kang
Matrix Representations
Adjacency matrix (A): n n matrix
A=[aij], aij=1 if edge between node i and j
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1 0 1 1 0 1 0
2 1 0 1 0 0 0
3 1 1 0 1 0 0
4 0 0 1 0 1 1
5 1 0 0 1 0 1
6 0 0 0 1 1 0
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13U Kang
Matrix Representations
Degree matrix (D): n n diagonal matrix
D=[dii], dii = degree of node i
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1 3 0 0 0 0 0
2 0 2 0 0 0 0
3 0 0 3 0 0 0
4 0 0 0 3 0 0
5 0 0 0 0 3 0
6 0 0 0 0 0 2
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14U Kang
Matrix Representations
Laplacian matrix (L): n n symmetric matrix
What is trivial eigenpair? 𝒙 = (𝟏,… , 𝟏) then 𝑳 ⋅ 𝒙 = 𝟎 and so 𝝀 = 𝝀𝟏 = 𝟎
Important properties of L: Eigenvalues are non-negative real numbers
Eigenvectors are real and orthogonal
𝑳 = 𝑫 − 𝑨
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1 3 -1 -1 0 -1 0
2 -1 2 -1 0 0 0
3 -1 -1 3 -1 0 0
4 0 0 -1 3 -1 -1
5 -1 0 0 -1 3 -1
6 0 0 0 -1 -1 2
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15U Kang
Our Goal
Recall: our goal is to find the minimum cut (or normalized cut)
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16U Kang
New Formulation of Min Cut [Fiedler’73]
Express partition (A,B) as a vector
𝒚𝒊 = ቊ+𝟏−𝟏
𝒊𝒇 𝒊 ∈ 𝑨𝒊𝒇 𝒊 ∈ 𝑩
Problem 1: Find a non-trivial vector y that minimizes f(y):
Problem 1 is equivalent to finding the min-cut (A,B) Why?
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17U Kang
Connection of min-cut problem and Laplacian L
Why?
yTL y = σ𝑖,𝑗=1𝑛 𝐿𝑖𝑗 𝑦𝑖𝑦𝑗 = σ𝑖,𝑗=1
𝑛 𝐷𝑖𝑗 − 𝐴𝑖𝑗 𝑦𝑖𝑦𝑗
= σ𝑖𝐷𝑖𝑖𝑦𝑖2 −σ 𝑖,𝑗 ∈𝐸 2𝑦𝑖𝑦𝑗
= σ 𝑖,𝑗 ∈𝐸(𝑦𝑖2 + 𝑦𝑗
2 − 2𝑦𝑖𝑦𝑗) = σ 𝒊,𝒋 ∈𝑬 𝒚𝒊 − 𝒚𝒋𝟐
New Formulation of Min Cut [Fiedler’73]
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18U Kang
Until now: finding the min-cut is equiv. to solving the following problem
But, the problem is NP-hard
So, we relax the constraint to make it tractable 𝒚𝒊 can be any number
Surprisingly, the solution of the problem is tightly connected with the eigenvector of L Detail: next slide
New Formulation of Min Cut [Fiedler’73]
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19U Kang
Rayleigh Theorem
𝝀𝟐 = 𝐦𝐢𝐧𝐲 𝒇 𝒚 : The minimum value of 𝒇(𝒚) is
given by the 2nd smallest eigenvalue λ2 of the Laplacian matrix L
𝐱 = 𝐚𝐫𝐠𝐦𝐢𝐧𝐲 𝒇 𝒚 : The optimal solution for y is
given by the corresponding eigenvector 𝒙, referred as the Fiedler vector
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20U Kang
Rayleigh Theorem
In fact, the minimum solution is given by y = 1 vector (the smallest eigenvector w/ eigenvalue 0); however, this does not say anything about the partition
Thus, we find the next best solution which is the Fiedler vector
Let the Fiedler vector = (𝛼1, … 𝛼𝑛). Then,σ𝛼𝑖
2 = 1 (unit length), σ𝛼𝑖 = 0 (orthogonal to the first eigenvector).
x
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21U Kang
So far…
How to define a “good” partition of a graph?
Minimize a given graph cut criterion
How to efficiently identify such a partition?
Approximate using information provided by the eigenvalues and eigenvectors of Laplacian
Spectral Clustering
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22U Kang
Spectral Clustering Algorithms
Three basic stages:
1) Pre-processing
Construct a matrix representation of the graph
2) Decomposition
Compute eigenvalues and eigenvectors of Laplacian
Map each point to a lower-dimensional representation based on one or more eigenvectors
3) Grouping
Assign points to two or more clusters, based on the new representation
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23U Kang
Spectral Partitioning Algorithm
1) Pre-processing: Build Laplacian
matrix L of the graph
2) Decomposition: Find eigenvalues
and eigenvectors xof the matrix L
Map vertices to corresponding components of x
0.0-0.4-0.40.4-0.60.4
0.50.4-0.2-0.5-0.30.4
-0.50.40.60.1-0.30.4
0.5-0.40.60.10.30.4
0.00.4-0.40.40.60.4
-0.5-0.4-0.2-0.50.30.4
5.0
4.0
3.0
3.0
1.0
0.0
= X =
-
0.66
-
0.35
-
0.34
0.33
0.62
0.31
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1 3 -1 -1 0 -1 0
2 -1 2 -1 0 0 0
3 -1 -1 3 -1 0 0
4 0 0 -1 3 -1 -1
5 -1 0 0 -1 3 -1
6 0 0 0 -1 -1 2
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24U Kang
Spectral Partitioning
3) Grouping: Sort components of reduced 1-dimensional vector
Identify clusters by splitting the sorted vector in two
How to choose a splitting point? Naïve approaches:
Split at 0 or median value
More expensive approaches: Attempt to minimize normalized cut in 1-dimension
(sweep over ordering of nodes induced by the eigenvector)
-0.66
-0.35
-0.34
0.33
0.62
0.31 Split at 0:
Cluster A: Positive points
Cluster B: Negative points
0.33
0.62
0.31
-0.66
-0.35
-0.34
A B
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25U Kang
Example: Spectral Partitioning
Rank in x2
Valu
e o
f x
2
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26U Kang
Example: Spectral Partitioning
Rank in x2
Valu
e o
f x
2
Components of x2
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27U Kang
Example: Spectral partitioningComponents of x1
Components of x3
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28U Kang
k-Way Spectral Clustering
How do we partition a graph into k clusters?
Two basic approaches:
Recursive bi-partitioning [Hagen et al., ’92]
Recursively apply bi-partitioning algorithm in a hierarchical divisive manner
Disadvantages: Inefficient, unstable
Cluster multiple eigenvectors [Shi-Malik, ’00]
Build a reduced space from multiple eigenvectors
Commonly used in recent papers
A preferable approach…
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29U Kang
Outline
Spectral Clustering
Small Communities
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30U Kang
Small Communities in Web
What is the signature of a community / discussion in a Web graph?
[Kumar et al. ‘99]
Dense 2-layer graph
Intuition: Many people all talking about the same things
… …
Use this to define “topics”:
What the same people on
the left talk about on the right
Remember HITS!
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31U Kang
Searching for Small Communities
A more well-defined problem:Enumerate complete bipartite subgraphs Ks,t Where Ks,t : s nodes on the “left” where each links to
the same t other nodes on the “right”
K3,4
|X| = s = 3
|Y| = t = 4X Y
Fully connected
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32U Kang
Frequent Itemset Enumeration
Market basket analysis. Setting:
Market: Universe U of n items
Baskets: m subsets of U: S1, S2, …, Sm U
(Si is a set of items one person bought)
Support: minimum number of occurrence f
Goal:
Find all subsets T appearing in at least f sets Si
(items in T were bought together at least f times)
What’s the connection between the itemsets and complete bipartite graphs?
[Agrawal-Srikant ‘99]
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33U Kang
From Itemsets to Bipartite Ks,t
Frequent itemsets = complete bipartite graphs!
How?
View each node i as a set Si of nodes i points to
Ks,t = a set Y of size tthat occurs in s sets Si
Looking for Ks,t findingfrequent sets of size t
with support s
[Kumar et al. ‘99]
ib
c
d
a
Si={a,b,c,d}
j
i
k
b
c
d
a
X Y
s … minimum support (|X|=s)
t … itemset size (|Y|=t)
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34U Kang
From Itemsets to Bipartite Ks,t
[Kumar et al. ‘99]
ib
c
d
a
Si={a,b,c,d}
x
y
z
b
c
a
X Y
Find frequent itemsets:
s … minimum support
t … itemset size
xb
c
a
We found Ks,t! Ks,t = a set Y of size tthat occurs in s sets Si
View each node i as a set Si of nodes i points to
Say we find a frequent itemset Y={a,b,c} of supp sSo, there are s nodes that link to all of {a,b,c}:
za
b
c
yb
c
a
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35U Kang
Example (1)
Minimum support s=2
{b,d}: support 3
{e,f}: support 2
And we just found 2 bipartite subgraphs:
c
a b
d
f
Itemsets:
a = {b,c,d}
b = {d}
c = {b,d,e,f}
d = {e,f}
e = {b,d}
f = {}
e
c
a b
d
e
c
d
fe
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36U Kang
Example (2)
Example of a community from a web graph
Nodes on the right Nodes on the left
[Kumar, Raghavan, Rajagopalan, Tomkins: Trawling the Web for emerging cyber-communities 1999]
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37U Kang
What You Need to Know
Min-cut problem
NP-hard graph theoretic problem
Can be formulated with an equivalent matrix problem which is also NP-hard
Relaxing constraints on the matrix problem leads to spectral clustering algorithm
Finding small communities in graphs
Can be solved by frequent itemset mining
![Page 38: Introduction to Data Miningukang/courses/19S-DM/L20-graphs2.pdf · Spectral Graph Partitioning ... ith coordinate of A x : Sum of the x-values of neighbors of i Make this a new value](https://reader034.vdocuments.us/reader034/viewer/2022042807/5f7789647768735b522f2c16/html5/thumbnails/38.jpg)
38U Kang
Questions?