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Community Detection:Overlapping Communities

CS224W: Social and Information Network AnalysisJure Leskovec, Stanford Universityhttp://cs224w.stanford.edu

Slide2Overlapping Communities

Non-overlapping vs. overlapping communities11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

2

Slide3Overlaps of Social Circles

A node belongs to many social circles11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

3

[

Palla

et al., ‘05]

Slide411/17/2011

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu4

Slide5Clique Percolation Method (CPM)

Two nodes belong to the same community if they can be connected through adjacent k-cliques:

k

-clique:

Fully connected

graph on

k

nodes

Adjacent k

-cliques:

overlap in

k-1

nodes

k

-clique community

Set of nodes that can

be reached through a

sequence of adjacent

k

-cliques

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

5

3-clique

adjacent

3-cliques

[

Palla

et al., ‘05]

Give an example of two non-

overallping

3-cliques!

Slide6Clique Percolation Method (CPM)

Two nodes belong to the same community if they can be connected through adjacent k-cliques:

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

6

4-clique

adjacent

4-cliques

Communities for k=4

[

Palla

et al., ‘05]

Give an example of two non-

overallping

4-cliques!

Slide7CPM: Steps

Clique Percolation Method:

Find maximal-cliques

(not

k

-cliques!)

Clique overlap graph:

Each clique is a node

Connect two cliques if they

overlap in at least

k-1

nodes

Communities:

Connected components of

the clique overlap matrix

How to set

k

?

Set

k

so that we get the “richest” (most widely distributed cluster sizes) community structure

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

7

A

C

D

B

A

C

D

B

Cliques

Communities

k=3

On clique overlap graph show the communities – circles of connected components.

Emphasize that this is for parameter k=3.

Define maximal clique!

Slide8CPM method: Example

Start with graphFind maximal cliques Create clique overlap matrixThreshold the matrix at value k-1If

a

ij

<k-1

set 0

Communities are the connected components of the

thresholded

matrix

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

8

(1) Graph

(2) Clique overlap

matrix

(3)

Thresholded

matrix at 3

(4) Communities

(connected components)

Slide9Example: Phone-Call Network11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

9

Communities in a “tiny” part of a phone call network of 4 million users

[

Palla

et al., ‘07]

[

Palla

et al., ‘07]

Slide10Example: Website

11/17/2011Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

10

[

Farkas

et. al

.

07]

Slide11How to Find Maximal Cliques?

No nice way, hard combinatorial problem

Maximal clique:

clique

that can’t be extended

{

a,b,c

} is a clique but not maximal clique{

a,b,c,d} is maximal cliqueAlgorithm: Sketch

Start with a seed node

Expand the clique around the seed

Once the clique cannot be further

expanded we found the maximal clique

Note:

This will generate the same clique multiple times

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

11

a

b

d

c

Slide12How to Find Maximal Cliques?

Start with a seed vertex “a”Goal: Find the maximal clique Q “a” belongs toObservation: If some “x” belongs to Q then it is a

neighbor of

“a”

Why?

If

a,x

Q but not a–x, then Q is not a clique!Recursive algorithm:

Q … current cliqueR … candidate vertices to expand the clique toExample: Start with “a” and expand around it

11/17/2011

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

12

Q= {a} {

a,b

} {

a,b,c

}

bktrack

{

a,b,d

}

R= {

b

,c,d

} {

b,c,d

}

{d}

(c)={}

{c}(d)={}

(b)={c

,d}

Steps of the recursive algorithm

(u)…neighbor set of u

d

a

b

c

Slide13How to Find Maximal Cliques?

Start with a seed vertex “a”Goal: Find the maximal clique Q “a” belongs toObservation: If some “x” belongs to Q then it is a member of “a”

Why?

If

a,x

Q but not a–x, then Q is not a clique!

Recursive algorithm:Q … current cliqueR … candidate vertices

to expand the clique toExample: Start with “a” and expand around it

11/17/2011

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

13

Q= {a} {

a,b

} {

a,b,c

}

bktrack

{

a,b,d

}

R= {

b

,c,d

} {

b,c,d

}

{d}

(c)={}

{c}

(d)={}

(b)={c,d

}

Steps of the recursive algorithm

(u)…neighbor set of u

d

a

b

c

Slide14How to Find Maximal Cliques?

Q … current cliqueR … candidate verticesExpand(R,Q)

while

R ≠ {}

p = vertex in R

Q

p

= Q

{p}

R

p

= R

(p)

if

R

p

≠ {}: Expand(

R

p,

Q

p

)

else:

output

Q

p

R = R – {p}

11/17/2011

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

14

a

e

b

c

f

d

Start: Expand(V, {})

R={a,…f}, Q={}

p = {a}

Q

p

= {a}

R

p

= {

b,d

}

Expand(

Rp

, Q):

R = {b,d}, Q={a}

p = {b}

Qp = {

a,b} R

p = {d}

Expand(Rp, Q

): R = {d}, Q={

a,b} p = {d}

Qp = {

a,b,d}

Rp = {} : output {a,b,d} p = {d} Q

p = {a,d} Rp = {b} Expand(R

p

, Q):

R =

{b},

Q={

a,d

}

p =

{b}

Q

p

= {

a,d

}

R

p

= {} :

output {

a,d,b

}

Have an animation about R and Q on the example graph.

Slide15How to Find Maximal Cliques?

Q … current cliqueR … candidate verticesExpand(R,Q)

while

R ≠ {}

p = vertex in R

Q

p

= Q

{p}

R

p

= R

(p)

if

R

p

≠ {}: Expand(

R

p,

Q

p

)

else:

output

Q

p

R = R – {p}

11/17/2011

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

15

a

e

b

c

f

d

Start: Expand(V, {})

R={a,…f}, Q={}

p = {b}

Q

p

= {b}

R

p

= {

a,c,d

}

Expand(

Rp, Q):

R = {a,c,d}, Q={b}

p = {a}

Qp = {b,a

} R

p = {d} Expand(

Rp, Q

): R = {d}, Q={b,a

} p = {d}

Qp = {

b,a,d}

Rp = {} : output {b,a,d} p = {c} Q

p = {b,c} Rp = {d} Expand(

R

p

, Q):

R =

{d},

Q

={

b,c

}

p =

{d}

Q

p

=

{

b,c,d

}

R

p

= {} :

output

{

b,c,d

}

Slide16How to Find Maximal Cliques?

How to prevent maximal cliques to be generated multiple times?Only output cliques that are lexicographically minimum{a,b,c} < {b,a,c

}

Even better:

Only expand to

the nodes higher in the lexicographical order

11/17/2011

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

16

a

e

b

c

f

d

Start: Expand(V, {})

R={a,…f}, Q={}

p = {a}

Q

p

= {a}

R

p

= {

b,d

}

Expand(

R

p

, Q):

R = {

b,d

}, Q={a}

p = {b}

Q

p

= {

a,b

}

R

p

= {d}

Expand(

Rp, Q):

R = {d}, Q={a,b

} p = {d}

Q

p = {a,b,d}

R

p = {} : output {a,b,d

} p =

{d}

Qp = {a,d

}

Rp =

{b}

Don’t expand d >

bBetter explain the lexicographical ordering and why cascades are generated multiple times.

Slide17How to Model Networks with Communities?

11/17/2011Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

17

Slide18Reflections: Finding Communities

Let’s rethink what we

are doing…

Given a network

Want to find communities!

Need to:

Formalize the notion

of a community

Need

an algorithm that will find

sets of nodes

that are “good” communities

More generally:

How to think about clusters in large networks?

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

18

Better motivate what we want to do. And how we want to do that – why NCP and what will we get out of it?

Slide19Community Score

How community like is a set of nodes?A good cluster S hasMany edges internally

Few edges pointing outside

Simplest objective function:

Conductance

Small

conductance

corresponds to good clusters

19

S

S’

Slide20Network Community Profile Plot

Define: Network community profile (NCP) plot

Plot the score of

best

community of size

k

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

20

Community size, log k

log

Φ

(k)

k=5

k=7

[WWW ‘08]

k=10

Slide21How to (Really) Compute NCP?11/12/2009

Jure Leskovec, Stanford CS322: Network Analysis

21

Run the favorite clustering method

Each dot represents a cluster

For each size find “best” cluster

Cluster size, log k

Cluster score, log

Φ

(k)

Spectral

Graclus

Metis

Slide22NCP Plot: Meshes

Meshes, grids, dense random graphs:Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

22

d-dimensional meshes

California road network

11/10/2010

[WWW ‘08]

Slide23NCP plot: Network Science

Collaborations between scientists in networks

[Newman, 2005]

23

Community size, log k

Conductance, log

Φ

(k)

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

11/10/2010

[WWW ‘08]

Slide24Natural Hypothesis

Natural hypothesis about NCP:

NCP of real networks slopes

downward

Slope

of the NCP corresponds to the “

dimensionality

“ of the network

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

24

What about large networks?

[Internet Mathematics ‘09]

Slide25Large Networks: Very Different

Typical example:

General Relativity collaborations

(

n=4,158, m=13,422

)

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

25

[Internet Mathematics ‘09]

Slide26More NCP Plots of Networks11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

26

[Internet Mathematics ‘09]

Slide27

Φ(k), (score)

k, (cluster size)

NCP:

LiveJournal

(

n=5m, m=42m

)

27

Better and better clusters

Clusters get worse and worse

Best cluster has ~100 nodes

Slide28Explanation: The Upward PartAs clusters grow the number of edges

inside grows slower that the number crossing

28

Φ

=2/10 = 0.2

Each node has twice as many children

Φ

=1/7=0.14

Φ

=8/20 = 0.4

Φ

=64/92 = 0.69

Slide29Explanation: Downward Part

Empirically we note that best clusters

are

barely connected

to the network

29

NCP plot

Core-periphery structure

Make this slide first, before the infinite tree.

Slide30What If We Remove Good Clusters?

30

Nothing happens!

Nestedness

of the core-periphery structure

Slide31Suggested Network Structure

Nested Core-Periphery (jellyfish, octopus)

Whiskers are responsible for good communities

Denser and denser core of the network

Core contains 60% node and 80% edges

31

Slide32******* END *********Good lecture

Overlapping part went wellPeople had questions – make it more interactiveMake more homeworks, quizes so that students do more work – now they just come to class and stare at the slides.

11/17/2011

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

32

Slide33Communities: Issues and Questions

11/17/2011

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

33

Slide34Communities: Issues and Questions

Some issues with community detection:Many different formalizations of clustering objective functions Objectives are NP-hard to optimize exactlyMethods can find clusters that are systematically “biased”

Questions:

How well do algorithms optimize objectives?

What clusters do different methods find?

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

34

Slide35Many Different Objective Functions

Single-criterion:Modularity: m-E(m)Edges cut:

c

Multi-criterion:

Conductance

:

c/(2m+c)

Expansion:

c/n

Density:

1-m/n

2

CutRatio

:

c/n

(N-n)

Normalized Cut:

c/(2m+c) + c/2(M-m)+c

Flake-ODF:

frac

. of nodes with more than ½ edges

pointing outside S

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

35

S

n

: nodes in S

m

: edges in S

c

: edges pointing

outside S

[WWW ‘09]

Slide36Many Classes of Algorithms

Many algorithms to that implicitly or explicitly optimize objectives and extract communities:Heuristics:Girvan-Newman,

Modularity optimization:

popular heuristics

Metis:

multi-resolution heuristic

[Karypis-Kumar ‘98]

Theoretical approximation algorithms:

Spectral partitioning

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

36

[WWW ‘09]

Slide37NCP: Live Journal

LiveJournal

Spectral

Metis

37

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

[WWW ‘09]

Slide38Properties of Clusters (1)

500

node communities from

Spectral

:

500

node communities from

Metis

:

38

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

[WWW ‘09]

Slide39Properties of Clusters (2)

Metis gives sets with better conductanceSpectral gives

tighter and more well-rounded sets

39

Conductance of bounding cut

Spectral

Disconnected

Metis

Connected

Metis

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

[WWW ‘09]

Diameter of the cluster

External / Internal

conductance

Lower is good

Expand this slide into 3 different slides to illustrate what each of

the figures plots

.

Slide40Multi-criterion Objectives

40

All qualitatively similar

Observations:

Conductance, Expansion, Norm-cut, Cut-ratio are similar

Flake-ODF

prefers larger clusters

Density

is bad

Cut-ratio

has high variance

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

[WWW ‘09]

Slide41Single-criterion Objectives

41Observations:All measures are monotonic

Modularity

prefers large clusters

Ignores small clusters

11/10/2010

Jure Leskovec, Stanford CS224W: Social and Information Network Analysis, http://cs224w.stanford.edu

[WWW ‘09]

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