DMTM Lecture 18 Graph mining

Prof. Pier Luca Lanzi
Graph Mining
Data Mining andText Mining (UIC 583 @ Politecnico di Milano)

References
• Jure Leskovec, Anand Rajaraman, Jeff Ullman. Mining of Massive
Datasets, Chapter 5 & Chapter 10
• Book and slides are available from http://www.mmds.org
2

Facebook social graph
4-degrees of separation [Backstrom-Boldi-Rosa-Ugander-Vigna, 2011]

Connections between political blogs
Polarization of the network [Adamic-Glance, 2005]

Citation networks and Maps of science
[Börner et al., 2012]

Web as a graph: pages are nodes, edges are links

How is the Web Organized?
• Initial approaches
§Human curated Web directories
§Yahoo, DMOZ, LookSmart
• Then, Web search
§Information Retrieval investigates:
Find relevant docs in a small
and trusted set
§Newspaper articles, Patents, etc.
8
Web is huge, full of untrusted documents,
random things, web spam, etc.

Web Search Challenges
• Web contains many sources of information
§Who should we “trust”?
§Trick: Trustworthy pages may point to each other!
• What is the “best” answer to query “newspaper”?
§No single right answer
§Trick: Pages that actually know about newspapers
might all be pointing to many newspapers
9

Page Rank
10

https://www.youtube.com/watch?v=wPMZr9RDmVk

Page Rank Algorithm
• The underlying idea is to look at links as votes
• A page is more important if it has more links
§In-coming links? Out-going links?
• Intuition
§www.stanford.edu has 23,400 in-links
§www.joe-schmoe.com has one in-link
• Are all in-links are equal?
§Links from important pages count more
§Recursive question!
12

B
38.4
C
34.3
E
8.1
F
3.9
D
3.9
A
3.3
1.6
1.6 1.6 1.6 1.6

Simple Recursive Formulation
• Each link’s vote is proportional
to the importance of its source page
• If page j with importance rj has n
out-links, each link gets rj/n votes
• Page j’s own importance is
the sum of the votes on its in-links
14
j
ki
rj/3
rj/3rj/3
rj = ri/3+rk/4
ri/3 rk/4

The “Flow” Model
• A “vote” from an important page is worth more
• A page is important if it is pointed to by other important pages
• Define a “rank” rj for page j
where di is the out-degree of node i
• “Flow” equations
ry = ry /2 + ra /2
ra = ry /2 + rm
rm = ra /2
15
å®
=
ji
i
j
r
r
id y
ma
a/2
y/2
a/2
m
y/2

Solving the Flow Equations
• The equations, three unknowns variables, no constant
§No unique solution
§All solutions equivalent modulo the scale factor
• An additional constraint (ry+ra+rm=1) forces uniqueness
• Gaussian elimination method works for small examples, but we
need a better method for large web-size graphs
16
We need a different formulation that scales up!

The Matrix Formulation
• Represent the graph as a transition matrix M
§Suppose page i has di out-links
§If page i is linked to page j Mji is set to 1/di else Mji=0
§M is a “column stochastic matrix” since
the columns sum up to 1
• Given the rank vector r with an entry per page, where ri is the
importance of page i and the ri sum up to one
17
å®
=
ji
i
j
r
r
id
The flow equation can be written as r = Mr

The Eigenvector Formulation
• Since the flow equation can be written as r = Mr, the rank vector
r is also an eigenvector of M
• Thus, we can solve for r using a simple iterative scheme
(“power iteration”)
• Power iteration: a simple iterative scheme
§Suppose there are N web pages
§Initialize: r(0) = [1/N,….,1/N]T
§Iterate: r(t+1) = M · r(t)
§Stop when |r(t+1) – r(t)|1 < e
18

The Random Walk Formulation
• Suppose that a random surfer that at time t is on page i and will
continue it navigation by following one of the out-link at random
• At time t+1, will end up on page j and from there it will continue
the random surfing indefinitely
• Let p(t) the vector of probabilities pi(t) that the surfer is on
page i at time t (p(t) is the probability distribution over pages)
• Then, p(t+1) = Mp(t) so that
19
p(t) is the stationary distribution for the random walk

Existence and Uniqueness
For graphs that satisfy certain conditions,
the stationary distribution is unique and
eventually will be reached no matter
what the initial probability distribution is

Hubs and Authorities (HITS)

Hubs and Authorities
• HITS (Hypertext-Induced Topic Selection)
§Is a measure of importance of pages and
documents, similar to PageRank
§Proposed at around same time as PageRank (1998)
• Goal: Say we want to find good newspapers
§Don’t just find newspapers. Find “experts”, that is,
people who link in a coordinated way to good newspapers
• The idea is similar, links are viewed as votes
§Page is more important if it has more links
§In-coming links? Out-going links?
22

• Each page has 2 scores
• Quality as an expert (hub)
§Total sum of votes of authorities pointed to
• Quality as a content (authority)
§Total sum of votes coming from experts
• Principle of repeated improvement
23

• Authorities are pages containing
useful information
§Newspaper home pages
§Course home pages
§Home pages of auto manufacturers
• Hubs are pages that link to authorities
§List of newspapers
§Course bulletin
§List of US auto manufacturers
24

Counting in-links: Authority 25
(Note this is idealized example. In reality graph is not bipartite and each page
has both the hub and authority score)
Each page starts with hub score
1. Authorities collect their
votes

Counting in-links: Authority
26
Sum of hub
scores of nodes
pointing to NYT.
Each page starts with hub score
1. Authorities collect their
votes
26

Expert Quality: Hub
27
Hubs collect authority scores
Sum of authority
scores of nodes that
the node points to.
27

Reweighting
28
Authorities again collect
the hub scores
28

Mutually Recursive Definition
• A good hub links to many good authorities
• A good authority is linked from many good hubs
• Model using two scores for each node:
§Hub score and Authority score
§Represented as vectors and
29
29

The HITS Algorithm
• Initialize scores
• Iterate until convergence:
§Update authority scores
§Update hub scores
§Normalize
• Two vectors a = (a1, …, an) and h=(h1, …, hn) and
the adjacency matrix A, with Aij=1 is 1 if i connects to j
are connected, 0 otherwise
30
•
•
•
•

The HITS Algorithm (vector notation)
• Set ai = hi = 1/√n
• Repeat until convergence
§h = Aa
§a = ATh
• Convergence criteria
• Under reasonable assumptions about A, HITS converges to
vectors h* and a* where
§h* is the principal eigenvector of matrix A AT
§a* is the principal eigenvector of matrix AT A
31

PageRank vs HITS
• PageRank and HITS are two solutions to the same problem
§What is the value of an in-link from u to v?
§In the PageRank model, the value of the link
depends on the links into u
§In the HITS model, it depends on
the value of the other links out of u
• The destinies of PageRank and HITS
after 1998 were very different
32

Community Detection
33

We often think of networks being organized into modules, cluster, communities:

The goal is typically to find densely linked clusters

advertiser
query
Micro-Markets in Sponsored Search: find micro-markets by partitioning the
query-to-advertiser graph (Andersen, Lang: Communities from seed sets, 2006)

Clusters in Movies-to-Actors graph
(Andersen, Lang: Communities from seed sets, 2006)

Discovering social circles, circles of trust
(McAuley, Leskovec: Discovering social circles in ego networks, 2012)

how can we identify communities?

Girvan-Newman Method
• Define edge betweenness as the number of shortest paths
passing over the edge
• Divisive hierarchical clustering based on the notion of edge
betweenness
• The Algorithm
§Start with an undirected graph
§Repeat until no edges are left
Calculate betweenness of edges
Remove edges with highest betweenness
• Connected components are communities
• Gives a hierarchical decomposition of the network
40

Need to re-compute
betweenness at every step
49
33
121

Step 1: Step 2:
Step 3: Hierarchical network
decomposition

Communities in physics collaborations

how to select the number of clusters?

Network Communities
• Communities are viewed as sets of tightly connected nodes
• We define modularity as a measure
of how well a network is partitioned
into communities
• Given a partitioning of the network
into a set of groups S we define the
modularity Q as
45
Need a null model!

Modularity is useful for selecting the number of clusters:
Q

Example #1 – Compute betweenness
# First we load the ipgrah package
library(igraph)
# let's generate two networks and merge them into one graph.
g2 <- barabasi.game(50, p=2, directed=F)
g1 <- watts.strogatz.game(1, size=100, nei=5, p=0.05)
g <- graph.union(g1,g2)
# let's remove multi-edges and loops
g <- simplify(g)
plot(g)
# compute betweenness
ebc <- edge.betweenness.community(g, directed=F)
48

Example #1 – Build Dendrogram and
Compute Modularity
mods <- sapply(0:ecount(g), function(i){
g2 <- delete.edges(g, ebc$removed.edges[seq(length=i)])
cl <- clusters(g2)$membership
# March 13, 2014 - compute modularity on the original graph g
# (Thank you to Augustin Luna for detecting this typo) and not on the
induced one g2.
modularity(g,cl)
})
# we can now plot all modularities
plot(mods, pch=20)
50

Example #1 – Select the cut
# Now, let's color the nodes according to their membership
g2<-delete.edges(g, ebc$removed.edges[seq(length=which.max(mods)-1)])
V(g)$color=clusters(g2)$membership
# Let's choose a layout for the graph
g$layout <- layout.fruchterman.reingold
# plot it
plot(g, vertex.label=NA)
52

Spectral Clustering

What Makes a Good Cluster?
• Undirected graph G(V,E)
• Partitioning task
§Divide the vertices into two disjoint
groups A, B=VA
• Questions
§How can we define a “good partition” of G?
§How can we efficiently identify such a partition?
55
1
3
2
5
4
6
1
3
2
5
4
6
A B

1
3
2
5
4
6
What makes a good partition?
Maximize the number of within-group connections
Minimize the number of between-group connections

Graph Cuts
• Express partitioning objectives as a function of
the “edge cut” of the partition
• Cut is defined as the set of edges with only one vertex in a group
• The cut of the set A, B is cut(A,B) = 2 or in more general
57
1
3
2
5
4
6
A B

Graph Cut Criterion
• Partition quality
§Minimize weight of connections between groups,
i.e., arg minA,B cut(A,B)
• Degenerate case:
• Problems
§Only considers external cluster connections
§Does not consider internal cluster connectivity
58
“Optimal cut”
Minimum cut

Graph Partitioning Criteria:
Normalized cut (Conductance)
• Connectivity of the group to the rest of the network should be
relative to the density of the group
• Where vol(A) is the total weight of the edges that have at least
one endpoint in A
59

Spectral Graph Partitioning
• Let A be the adjacent matrix of the graph G with n nodes
§Aij is 1 if there is an edge between i and j, 0 otherwise
§x a vector of n components (x1, …, xn) that represents
labels/values assigned to each node of G
§Ax returns a vector in which each component j is the sum of
the labels of the neighbors of node j
• Spectral Graph Theory
§Analyze the spectrum of G, that is, the eigenvectors xi of the
graph corresponding to the eigenvalues Λ of G sorted in
increasing order
§Λ = { λ1, …, λn} such that λ1≤λ2 ≤… ≤λn
62

Example: d-regular Graph
• Suppose that all the nodes in G have degree d and G is
connected
• What are the eigenvalues/eigenvectors of G? Ax=λx
§Ax returns the sum of the labels of each node’s neighbors and
since each node has exactly d neighbors, x = (1, …, 1) is an
eigenvector and d is an eigenvalue
• What if G is not connected but still d-regular
• A vector with all the ones is A and all the zeros in B (or
viceversa) is still an eigenvector of A with eigenvalue d
63
A B

Example: d-regular Graph (not
connected)
• What if G has two separate components
but it is still d-regular
• A vector with all the ones is A and all the
zeros in B (or viceversa) is still an eigenvector
of A with eigenvalue d
• Underlying intuition
64
A B
A B A B
λ1=λ2
λ1≈λ2

Spectral Graph Partitioning
• Adjacency matrix A (nxn)
§Symmetric
§Real and orthogonal
eigenvectors
• Degree Matrix
§nxn diagonal matrix
§Dii = degree of node i
65
1
3
2
5
4
6
1 2 3 4 5 6
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
1 2 3 4 5 6
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

Graph Laplacian Matrix
• Computed as L = D-A
§nxn symmetric matrix
§x=(1,…,1) is a trivial eigenvector since Lx=0 so λ1=0
• Important properties of L
§Eigenvalues are non-negative
real numbers
§Eigenvectors are real
and orthogonal
66
1 2 3 4 5 6
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

λ2 as optimization problem
• For symmetric matrix M,
• What is the meaning of xTLx on G? We can show that,
• So that, considering that the second eigenvector x is the unit
vector, and x is orthogonal to the unit vector (1, …, 1)
67

λ2 as optimization problem
• So that, considering that the second eigenvector x is the unit
vector, and x is orthogonal to the unit vector (1, …, 1)
• Such that,
68

0 x
λ2 and its eigenvector x balance to minimize
xi xj

Finding the Optimal Cut
• Express the partition (A,B) as a vector y where,
§yi = +1 if node i belongs to A
§yi = -1 if node i belongs to B
• We can minimize the cut of the partition by finding a non-trivial
vector that minimizes
70
Can’t solve exactly! Let’s relax y and
allow y to take any real value.

Rayleigh Theorem
• We know that,
• The minimum value of f(y) is given by the second smallest
eigenvalue λ2 of the Laplacian matrix L
• Thus, the optimal solution for y is given by the corresponding
eigenvector x, referred as the Fiedler vector
71

Spectral Clustering Algorithms
1. Pre-processing
§Construct a matrix representation of the graph
2. Decomposition
§Compute eigenvalues and eigenvectors of the matrix
§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
72

Spectral Partitioning Algorithm
• Pre-processing:
§Build Laplacian
matrix L of the
graph
• Decomposition:
§Find eigenvalues l
and eigenvectors x
of the matrix L
§Map vertices to
corresponding
components of l2
73
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
l= X =
-0.66
-0.35
-0.34
0.33
0.62
0.31
1 2 3 4 5 6
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

Spectral Partitioning Algorithm
• 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)
74
74
-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

Example: Spectral Partitioning 75
Rank in x2
Valueofx2

Example: Spectral Partitioning
76
Rank in x2
Valueofx2
Components of x2
76

Example: Spectral partitioning
77
Components of x1
Components of x3
77

Example using R

Example #1
ComputeDegreeMatrix <- function(v) {
n <- max(dim(v))
m <- diag(n)
d = v %*% rep(1,n)
for(i in 1:n ) {
m[i,i] = d[i,1]
}
return(m)
}
ComputeLaplacianMatrix <- function(v) {
D = ComputeDegreeMatrix(v)
L = D-v
return(L)
}
79

Example #1
#### 2 ----6
#### / |
#### 1 4 |
#### / |
#### 3 5
G = rbind(
c(0, 1, 1, 0, 0, 0),
c(1, 0, 0, 1, 0, 1),
c(1, 0, 0, 1, 0, 0),
c(0, 1, 1, 0, 1, 0),
c(0, 0, 0, 1, 0, 1),
c(0, 1, 0, 0, 1, 0)
)
L = ComputeLaplacianMatrix(G)
E = eigen(L)
second_eigen_value = (E$value)[max(dim(L))-1]
second_eigen_vector = (E$vectors)[,max(dim(L))-1]
5.000000e-01 1.165734e-15 5.000000e-01 4.718448e-16 -5.000000e-01 -
5.000000e-01
80

Example #2
G = rbind(
c(0, 0, 1, 0, 0, 1, 0, 0),
c(0, 0, 0, 0, 1, 0, 0, 1),
c(1, 0, 0, 1, 0, 1, 0, 0),
c(0, 0, 1, 0, 1, 0, 1, 0),
c(0, 1, 0, 1, 0, 0, 0, 1),
c(1, 0, 1, 0, 0, 0, 1, 0),
c(0, 0, 0, 1, 0, 1, 0, 1),
c(0, 1, 0, 0, 1, 0, 1, 0)
)
L = ComputeLaplacianMatrix(G)
E = eigen(L)
second_eigen_value = (E$value)[max(dim(L))-1]
second_eigen_vector = (E$vectors)[,max(dim(L))-1]
5.000000e-01 -5.000000e-01 3.535534e-01 -3.885781e-16 -3.535534e-01
3.535534e-01 -4.996004e-16 -3.535534e-01
81

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…
82
82

Why Use Multiple Eigenvectors?
• Approximates the optimal cut [Shi-Malik, ’00]
§Can be used to approximate optimal k-way normalized cut
• Emphasizes cohesive clusters
§Increases the unevenness in the distribution of the data
§Associations between similar points are amplified, associations
between dissimilar points are attenuated
§The data begins to “approximate a clustering”
• Well-separated space
§Transforms data to a new “embedded space”,
consisting of k orthogonal basis vectors
• Multiple eigenvectors prevent instability due to information loss
83

Searching for Small Communities
(Trawling)

Searching for small communities in
the Web graph (Trawling)
• Trawling
§ What is the signature of a community in a Web graph?
§ The underlying intuition, that small communities involve
many people talking about the same things
§ Use this to define “topics”: what the same people on
the left talk about on the right?
• More formally
§ Enumerate complete bipartite subgraphs Ks,t
§ Ks,t has s nodes on the “left” and t nodes on the “right”
§ The left nodes link to the same node of on the right,
forming a fully connected bipartite graph
85
[Kumar et al. ‘99]
Dense 2-layer
…
…
K3,4
X Y

Mining Bipartite Ks,t using Frequent
Itemsets
• Searching for such complete bipartite graphs can be viewed as a
frequent itemset mining problem
• View each node i as a
set Si of nodes i points to
• Ks,t = a set Y of size t
that occurs in s sets Si
• Looking for Ks,t is equivalento to
settting the frequency threshold
to s and look at layer t
(i.e., all frequent sets of size t)
86
[Kumar et al. ‘99]
i
b
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)

i
b
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
We found Ks,t!
Ks,t = a set Y of size t
that 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 s;
so, there are s nodes that link to
all of {a,b,c}:
x
b
c
a
z
a
b
c
y
b
c
a

Example
• Support threshold s=2
§{b,d}: support 3
§{e,f}: support 2
• And we just found 2 bipartite subgraphs:
88
c
a b
d
f
e
c
a b
d
e
c
d
f
e
• Itemsets
§ a = {b,c,d}
§ b = {d}
§ c = {b,d,e,f}
§ d = {e,f}
§ e = {b,d}
§ f = {}

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DMTM Lecture 18 Graph mining

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