[PPT] - Proximity measures applied to community detection in complex PowerPoint Presentation

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Proximity measures applied to community detection in complex networks

Maximilien Danisch Thesis supervised by: Jean-Loup Guillaume and B´ en´ edicte Le Grand Complex networks team, LIP6-CNRS-UPMC CRI, Univ. Paris 1

December 15, 2015

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Complex networks

Definition

Network/Graph: a set of nodes linked by edges. networks nodes edges Facebook profiles friendship Internet computers connections Web web pages hyperlinks P2P peers file exchanges

Remark

increasingly more networks
increasingly larger networks
have common properties

⇒ need for fast and generic algorithms to understand them and extract knowledge.

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Why detecting groups of nodes?

Organization

Classification of documents (e.g. Wikipedia pages). Organization of friends’ lists on Facebook.

Recommendation

“People you may know” on Facebook or LinkedIn. “You may also like...” on Amazon.

Prediction

What is this unknown P2P file? What is the function of this unknown protein?

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Group of nodes = Community

Some definitions

a set of nodes that are similar,
a set of nodes that are close to one another,
a set of nodes highly connected inside, but poorly connected
utside.

Remark

Different visions of communities:

overlapping / partition
global / local

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Positioning

Survey articles

Partition: Fortunato’s survey 2010 (500 references)

⇒ A lot of work has been done.

Overlapping communities: Xi et al. 2013

⇒ More realistic than partition, but less work has been done and methods do not scale.

From local to global: Kanawati 2014

⇒ if you solve locally, you may be able to solve globally.

Position

⇒ Chose to work on overlapping communities with a local approach in order to

be realistic and
be able to treat large networks.

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Communities detection: how?

Observation

Local community detection is done mainly using greedy heuristics to optimize an ad hoc quality function.

Examples of quality function

rd(S) =

li li+lo

C(S) = △i(S)

(|S|

3 ) ×

△i(S) △i(S)+△o(S)

Friggeri et al. 2011

Problems:

1 design of the quality function is difficult, 2 optimization can be trapped in local minima.

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Let’s not use quality functions!

Idea

A community can be defined as a set of nodes close to each other. ⇒ Let’s try to use proximity measures instead of quality functions to find communities.

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Outline / Contributions

1 How to use proximity measures for community detection 2 Proposition of two proximity measures:

propagated opinion
Katz+

3 Framework to:

find the communities of a given node (local)
find overlapping communities (global)
complete a set of nodes into a community

Maximilien Danisch — Proximity measures and communities — December 15, 2015 8/56

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Networks and communities

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Ego-centered community

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Ego-centered community

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On a small visualizable network

50 100 150 200 250 300

RANK OF THE NODE

0.000 0.005 0.010 0.015 0.020 0.025

PROXIMITY

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Wikipedia category = community

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PROXIMITY TO THE NODE MAGNUS CALSEN

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RANK OF THE NODE

500 1000 1500 2000 2500 3000 3500 4000

NUMBER OF TOPK NODES IN THE CATEGORY CHESS

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Ego-centered community

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Ego-centered community

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On a small visualizable network

50 100 150 200 250 300 350 400

RANK OF THE NODE

0.0 0.2 0.4 0.6 0.8 1.0

PROXIMITY

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Several behaviors in Wikipedia:

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sharp transition smooth transition deformed power-law perfect power-law

RANK OF THE NODE PROXIMITY

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Ego-centered community

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Ego-centered community

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Ego-centered community

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Multi-ego-centered community

for each node: minimum

f the two proximities

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Multi-egocentered communities

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Bi-ego-centered communities

In Wikipedia: Folk wrestling + Torii school =

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Torii school Folk wrestling

PROXIMITY RANK OF THE NODE

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Bi-ego-centered communities

In Wikipedia: Folk wrestling + Torii school = Sumo

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Torii school Folk wrestling

PROXIMITY RANK OF THE NODE

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PROXIMITY

Sumo Minimum Rescaled minimum

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Which proximity measure?

Classical proximity measures

distance: number of hops between two nodes
number of common neighbors between two nodes
Katz index
commuting time
hitting time
rooted page-rank

Remark

need to have a discriminative measure
need to compute the proximity of all nodes to the node of

interest in a fast way

with or without parameters?

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Outline / Contributions

1 How to use proximity measures for community detection 2 Proposition of two proximity measures:

propagated opinion
Katz+

3 Framework to:

find the communities of a given node (local)
find overlapping communities (global)
complete a set of nodes into a community

Maximilien Danisch — Proximity measures and communities — December 15, 2015 25/56

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A parameter free proximity measure: the propagated opinion

50 100 150 200 250 300 350 400

RANK OF THE NODE

0.0 0.2 0.4 0.6 0.8 1.0

PROXIMITY

Xt = MXt−1 AVERAGING X i

t

= 1 RESETING

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A parameter free proximity measure: the propagated opinion

50 100 150 200 250 300 350 400

RANK OF THE NODE

0.0 0.2 0.4 0.6 0.8 1.0

PROXIMITY

10 iterations 20 iterations 100 iterations 10000 iterations

Xt = MXt−1 AVERAGING Xt =

Xt−min(Xt) 1−min(Xt)

RESCALING X i

t

= 1 RESETING

Maximilien Danisch — Proximity measures and communities — December 15, 2015 26/56

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Problem

50 100 150 200 250

RANK OF THE NODE

0.00 0.01 0.02 0.03 0.04 0.05 0.06

PROXIMITY

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Outline / Contributions

1 How to use proximity measures for community detection 2 Proposition of two proximity measures:

propagated opinion
Katz+

3 Framework to:

find the communities of a given node (local)
find overlapping communities (global)
complete a set of nodes into a community

Maximilien Danisch — Proximity measures and communities — December 15, 2015 28/56

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A parametrized proximity measure: Katz+

n common neighbors 2 1 3

Which node is closer to 1: 2 or 3?

“distance VS redundancy” trade-off: Pα(i, j) =

λ

l=0

αlNpath

l

(i, j)

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A parametrized proximity measure: Katz+

2 1 3

Which node is closer to 1: 2 or 3?

“popularity VS intimacy” trade-off: Pα,β(i, j) = 1 dβ

j λ

l=0

propagated opinion
Katz+

3 Framework to:

find the communities of a given node (local)
find overlapping communities (global)
complete a set of nodes into a community

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1- Choosing a set of candidates

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2- Looking for bi-egocentered communities

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Chess Boxing Chessboard MIN: Chess notation

PROXIMITY RANK OF THE NODE

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Chess Boxing Achi, Nagano MIN: Morabaraba

PROXIMITY RANK OF THE NODE

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3- Cleaning and labeling the output

Jaccard matrix for the communities obtained before cleaning:

100 200 300 400 500 600 700 100 200 300 400 500 600 700 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

3000 candidates ⇒ 770 communities ⇒ 5 communities after cleaning.

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Outline / Contributions

1 How to use proximity measures for community detection 2 Proposition of two proximity measures:

propagated opinion
Katz+

3 Framework to:

find the communities of a given node (local)
find overlapping communities (global)
complete a set of nodes into a community

Maximilien Danisch — Proximity measures and communities — December 15, 2015 39/56

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Representative’s community

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Representatives in Wikipedia

Category candidate # nodes in the F-score representative found community Chess Magnus Carlsen 4259 0.965 (3850 nodes) Chess 4239 0.970 Queen’s Gambit 4542 0.977 World Chess Championship 4338 0.970 Chess Boxing 37 0.009 Sumo Sumo 443 0.951 (445 nodes) Taiho Koki 486 0.965 Yokozuna 504 0.962 Lyoto Machida 87 0.004 Boxing Boxing 6814 0.890 (7289 nodes) Cruiserweight 8739 0.924 Vitali Klitschko 9942 0.835 Chess Boxing 37 0.001

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Comparing found communities to ground truth communities

Comparing two communities

F1(a, b) = 2 |a∩b|

|a|+|b|

Comparing two sets of communities

F1(A, B) =

1 |A|

a∈A

max

b∈B F1(a, b)

⇒ how much A looks like B: not symmetric

S(A, B) = 2 F1(A,B) F1(B,A)

F1(A,B)+F1(B,A)

(symmetric)

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Comparison

Network k-clique BigClam link approche percolation partition proximity scholar 0.14 0.16 0.15 0.07 0.00 0.01 0.16 0.12 0.14 0.13 0.13 0.13 16773 26318 (k=3) 100 11133 18845 DBLP 0.00 0.03 0.00 0.05 0.10 0.07 0.00 0.03 0.00 0.32 0.19 0.24 24 3113 (k=5) 50 12871 15 wiki08 0.67 0.03 0.04 79365 355 wiki12 0.72 0.03 0.05 730752 2046

F1(FC, GTC), F1(GTC, FC), S(FC, GTC) and number of communities.

Performance equivalent or better than state of the art.
Scale to very large graphs contrary to state of the art.

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Outline / Contributions

1 How to use proximity measures for community detection 2 Proposition of two proximity measures:

propagated opinion
Katz+

3 Framework to:

find the communities of a given node (local)
find overlapping communities (global)
complete a set of nodes into a community

Maximilien Danisch — Proximity measures and communities — December 15, 2015 44/56

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Learning the parameters of a parametrized proximity measure

: in the input set of nodes : not in the input set of nodes

Learn the parameters by AUC grid-search optimization.

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Learning by AUC optimization

0.0001 0.001 0.01 0.1 1 alpha 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 beta 0.989 0.9892 0.9894 0.9896 0.9898 0.99 0.9902 0.9904

AUC of the ranking as a function of α and β.

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Validation

500 1000 1500 2000 2500 3000 3500 4000

k

0.0 0.2 0.4 0.6 0.8 1.0

PROPORTION KATZ+ KATZ T and F PROP-OP DISTANCE

Comparison to other proximity measures for the task of completing the Wikipedia category “graph theory”.

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Conclusion

We showed how to use proximity measures for community

detection.

We proposed two new proximity measures.
We proposed frameworks to solve three variations of the

community detection problem.

Our approach is a viable alternative to the classical approach
f quality function greedy optimization.
We suggested solutions to two community-related problems:

1 Social capitalists detection in Twitter, 2 Pedophile queries made on P2P networks.

We’ve underlined the lack of a realistic benchmark to test
verlapping community detection algorithms and tried to
pen a path towards one.

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Perspectives

Adapt the method to follow dynamic communities

Proximity approach is stable and communities labeled by representatives.

Take into account links’ orientation

Survey: Malliaros & Vazirgiannis 2013.
How to better take into account links’ orientation in community

detection algorithms?

The study of oriented triangles could lead to some insights!

Redefine the notion of community

Current classic definition “set of nodes highly connected inside but poorly

connected outside” is wrong. Other definitions are too broad.

Study ground truth communities and capture what makes them

communities.

Design a better model of network with communities and a community

detection framework out of the study.

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Following dynamic communities

Observation

Traditional approaches detect communities at the different timestamps and match them. Two problems with the quality function optimization:

1 the optimization is not stable to perturbations 2 it can be hard to match the communities

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Stability despite perturbations on Wikipedia

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PROXIMITY TO THE NODE MAGNUS CARLSEN 0% 5% 10% 20% 40%

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Use representatives as labels Wikip´ edia 2008 VS Wikip´ edia 2012

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PROXIMITY TO THE NODE MAGNUS CARLSEN IN 2008

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PROXIMITY TO THE NODE MAGNUS CARLSEN IN 2012

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Perspectives

Adapt the method to follow dynamic communities

Proximity approach is stable and communities labeled by representatives.

Take into account links’ orientation

Survey: Malliaros & Vazirgiannis 2013.
How to better take into account links’ orientation in community

detection algorithms?

The study of oriented triangles could lead to some insights!

Redefine the notion of community

Current classic definition “set of nodes highly connected inside but poorly

connected outside” is wrong. Other definitions are too broad.

Study ground truth communities and capture what makes them

communities.

Design a better model of network with communities and a community

detection framework out of the study.

Maximilien Danisch — Proximity measures and communities — December 15, 2015 53/56

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Oriented triangles and communities

1 2 3 4 5 6 7

Network Wikipedia Twitter triangle 1 0.011 0.021 triangle 2 0.448 0.266 triangle 3 0.028 0.172 triangle 4 1.539 0.733 triangle 5 1.418 0.436 triangle 6 0.797 2.224 triangle 7 12.608 45.9634 Number of triangles divided by the expected number of triangles.

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Perspectives

Adapt the method to follow dynamic communities

Proximity approach is stable and communities labeled by representatives.

Take into account links’ orientation

Survey: Malliaros & Vazirgiannis 2013.
How to better take into account links’ orientation in community

detection algorithms?

The study of oriented triangles could lead to some insights!

Redefine the notion of community

Current classic definition “set of nodes highly connected inside but poorly

connected outside” is wrong. Other definitions are too broad.

Study ground truth communities and capture what makes them

communities.

Design a better model of network with communities and a community

detection framework out of the study.

Maximilien Danisch — Proximity measures and communities — December 15, 2015 55/56

SLIDE 58

c n r s - u p m c l a b o r a t o i r e d ’ i n f o r m a t i q u e d e p a r i s 6

13 publications

2 international journals:
Towards multi-ego-centered communities: a node

similarity approach.

M. Danisch, J.-L. Guillaume and B. Le Grand.

IJWC 2013.

Multi-ego-centered communities in practice.
M. Danisch, J.-L. Guillaume, B. Le Grand.

SNAM 2013.

4 international conferences:
Mining bipartite graphs to improve semantic

pedophile activity detection.

R. Fournier and M. Danisch. RCIS 2014.
On the importance of considering social

capitalism when measuring influence on Twitter.

M. Danisch, N. Dugu´

e and A. Perez. BESC 2014.

Direct Generation of Random Graphs Exactly

Realising a Prescribed Degree Sequence.

D. Obradovic and M. Danisch. CASoN 2014.
Learning a Proximity Measure to Complete a

Community.

M. Danisch, J.-L. Guillaume and B. Le Grand.

DSAA 2014.

1 international workshop:
Unfolding ego-centered community structures

with ”a similarity approach”.

M. Danisch, J.-L. Guillaume and B. Le Grand.

CompleNet IV 2013.

1 international book chapter:
Multi-ego-centered communities.
M. Danisch, J.-L. Guillaume and B. Le Grand.

ComNetBook 2013.

4 national conferences:
Une approche `

a base de similarit´ e pour la d´ etection de communaut´ es egocentr´ ees.

M. Danisch, J.-L. Guillaume and B. Le Grand.

ALGOTEL 2013.

Compl´

etion de communaut´ es par l’apprentissage d’une mesure de proximit´ e.

M. Danisch, J.-L. Guillaume and B. Le Grand.

ALGOTEL 2014.

Prendre en compte le capitalisme social dans la

mesure de l’influence sur Twitter.

M. Danisch, N. Dugu´

e and A. Perez. MARAMI 2014.

Structures biparties et communaut´

es recouvrantes des graphes de terrains.

R. Tackx, M. Danisch and F. Tarissan.

MARAMI 2014.

1 national workshop:
D´

eplier les structures communautaires egocentr´ ees - une approche ` a base de similarit´ e.

M. Danisch, J.-L. Guillaume and B. Le Grand.

AFGG-EGC 2013.

Maximilien Danisch — Proximity measures and communities — December 15, 2015 56/56