k-clique percolation and clustering in directed and weighted networks

Transcripción

Introduction
Directed communities
Weighted communities
k -clique percolation and clustering in directed
and weighted networks
Gergely Palla1 Dániel Ábel2 Imre Derényi2
Illés Farkas1 Péter Pollner1 Tamás Vicsek1,2
1 Statistical
and Biological Physics Research Group,
Hungarian Acedmy of Sciences,
2 Department
of Biological Physics,
Eötvös University, Hungary
March 2008, Louvain-la-Neuve
Vicsek group
Directed and weighted communities
Introduction
Outline
Introduction
The Clique Percolation Method (CPM)
Phase transition in the Erdős-Rényi graph
Relative in- and out degree
Directed CPM
Results
Weights in the original CPM
Weighted CPM
Results
Vicsek group
Introduction
CPM
Definitions
k -clique: a complete (fully connected) subgraph of k
vertices.
k -clique adjacency: two k -cliques are adjacent if they
share k − 1 vertices, i.e., if they differ only in a single node.
Vicsek group
Introduction
CPM
Definitions
vertices.
Vicsek group
Introduction
CPM
Definitions
vertices.
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
k -clique community: the union of k -cliques that can be
reached from one to the other through a sequence of
adjacent k -cliques.
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
Vicsek group
Introduction
CPM
CPM
(continued)
Definition
Illustration:
same at k = 4:
Vicsek group
Introduction
CPM
Advantages of the CPM
The main advantages of the CPM:
Allows overlaps between the communities.
The definition is based on the density of the links.
It is local. (No resolution limit).
Vicsek group
Introduction
CPM
The order parameters
The Erdős-Rényi graph:
N nodes,
every pair is independently linked with probability p.
A giant k -clique percolation cluster can be found if p ≥ pc (k ).
The order parameter of the phase transition is the size of the
giant cluster:
The number of nodes, N ∗
−→ Φ ≡ N ∗ /N,
∗
The number of k -cliques, N
−→ Ψ ≡ N ∗ /N .
Vicsek group
Introduction
CPM
N nodes,
giant cluster:
−→ Φ ≡ N ∗ /N,
∗
−→ Ψ ≡ N ∗ /N .
Vicsek group
Introduction
CPM
N nodes,
giant cluster:
−→ Φ ≡ N ∗ /N,
∗
−→ Ψ ≡ N ∗ /N .
Vicsek group
Introduction
CPM
Results
Numerical results:
1
1
0.8
Φ
0.6
0.4
0.2
0
0.2
k=4
k=4
0.8
N=100
N=200
N=500
N=1000
N=2000
N=5000
0.4
0.6
Ψ
N=100
N=200
N=500
N=1000
N=2000
N=5000
0.6
0.4
0.2
0.8
1
p/pc(k )
1.2
1.4
1.6
0
0.2
1.8
1
pc (k ) =
1
0.6
.
[N(k − 1)] k −1
Vicsek group
1
p/pc(k )
1.4
1.8
Introduction
Relative in- and out-degree
Directed CPM
Results
Directed links
Direction of the links:
Direction of some kind of flow (e.g. information, energy).
Asymmetrical relation (e.g. superior-inferior).
Out-hubs in communities represent “sources”, whereas in-hubs
correspond to “drains”:
Vicsek group
Introduction
Directed CPM
Results
Directed links
source/top
drain/bottom
Vicsek group
Introduction
Directed CPM
Results
Directed links
source/top
interface
Vicsek group
Introduction
Directed CPM
Results
We define the relative in-degree and relative out-degree of
node i in community α as
α
Di,in
≡
α
Di,out
≡
α
di,in
α
di,in
,
α
+ di,out
α
di,out
,
α
α
di,in
+ di,out
For weighted networks these can be replaced by the relative
in-strength and relative out-strength:
α
Wi,in
≡
α
wi,in
,
α
α
wi,in
+ wi,out
α
Wi,out
≡
α
wi,out
,
α
α
wi,in
+ wi,out
Vicsek group
Introduction
Directed CPM
Results
We define the relative in-degree and relative out-degree of
node i in community α as
α
Di,in
≡
α
Di,out
≡
α
di,in
α
di,in
,
α
+ di,out
α
di,out
,
α
α
di,in
+ di,out
For weighted networks these can be replaced by the relative
in-strength and relative out-strength:
α
Wi,in
≡
α
wi,in
,
α
α
wi,in
+ wi,out
α
Wi,out
≡
α
wi,out
,
α
α
wi,in
+ wi,out
Vicsek group
Introduction
Directed CPM
Results
Directed k -cliques?
Comparing undirected and directed connections:
A
B
A
B
A
B
A
B
In case of k -cliques:
k (k − 1)/2 links −→ 3k (k −1)/2 possible configurations.
However, we would like the k -clique to have some kind of
directionality as a whole as well.
Vicsek group
Introduction
Directed CPM
Results
Directed k -cliques?
Comparing undirected and directed connections:
A
B
A
B
A
B
A
B
In case of k -cliques:
k (k − 1)/2 links −→ 3k (k −1)/2 possible configurations.
However, we would like the k -clique to have some kind of
directionality as a whole as well.
Vicsek group
Introduction
Directed CPM
Results
Definition
A directed k -clique has to fulfil the following conditions:
In the absence of double links:
Any directed link in the k -clique points from a node with a
higher order (larger restricted out-degree) to a node with a
lower order.
The k -clique contains no directed loops.
The restricted out-degree of each node in the k -clique is
different.
If double links are present:
It is possible to eliminate the double links in such a way that the
single links fulfil the above conditions.
Vicsek group
Introduction
Directed CPM
Results
Definition
lower order.
different.
Vicsek group
Introduction
Directed CPM
Results
Definition
lower order.
different.
Vicsek group
Introduction
Directed CPM
Results
Definition
lower order.
different.
Vicsek group
Introduction
Directed CPM
Results
Definition
lower order.
different.
Vicsek group
Introduction
Directed CPM
Results
Illustration
a)
3
b)
NO
1
0
c)
YES
contains double links?
2
3
d)
1 (or 2)
2 (or 1)
0
YES
NO
directed k−clique?
Vicsek group
Introduction
Directed CPM
Results
Phase transition in the directed E-R graph
The directed E-R graph:
N nodes,
The N(N − 1) possible “places” for the directed links are
filled independently with probability p.
Theoretical prediction of the critical point for the appearance of
a giant directed k -clique percolation cluster:
pctheor =
1
1
.
[Nk (k − 1)] k −1
Order parameters: Φ, Ψ (same as in the undirected case).
Vicsek group
Introduction
Directed CPM
Results
Phase transition in the directed E-R graph
The directed E-R graph:
N nodes,
The N(N − 1) possible “places” for the directed links are
filled independently with probability p.
Theoretical prediction of the critical point for the appearance of
a giant directed k -clique percolation cluster:
pctheor =
1
1
.
[Nk (k − 1)] k −1
Order parameters: Φ, Ψ (same as in the undirected case).
Vicsek group
Introduction
Directed CPM
Results
Numerical results
1
1
(a)
(b)
0.8
0.8
k=4
0.6
0.4
N = 25
50
100
200
400
800
1600
Ψ(p)
Φ(p)
k=4
0.2
0.6
0.4
N = 50
100
200
400
800
1600
0.2
0
0
0.6
0.8
1
1.2
1.4
1.6
1.8
0.6
0.8
theor
1.7
(c)
(d)
1.6
k=3
4
5
6
1.6
N = 25
50
100
200
400
800
1600
theor
k=4
num
/ pC
pCnum
pC
χ(p)
1.4
1.8
0.3
0.1
1.2
p / pC
0.4
0.2
1
theor
p / pC
1.5
1.4
1.3
1+cN-α fit: α = 0.49(2)
0.49(1)
0.48(2)
0.45(2)
1.2
1.1
1
0
0.6
0.8
1
1.2
1.4
1.6
1.8
p / pCtheor
Vicsek group
1e-04
0.001
1/N
0.01
1.8
Introduction
Directed CPM
Results
Word association network
Local picture of the communities:
PRESTIGE (1.0)
SUCCESS (0.46)
BRONZE
(0.95), (1.0)
BRASS
(0.87)
TARNISH
(0.93)
FAME
(0.89)
PROSPER (0.84)
WEALTH (0.60) POVERTY (1.0)
FORTUNE
(0.56)
OLYMPICS (0.79)
METAL (0.22.)
POOR (0.28)
GOLD (0.29), (0.0),
(0.73), (0.15)
LUXURY (0.91)
MANSION (1.0)
TREASURE (1.0)
SILVER (0.39)
COPPER
(0.63)
RICH (0.31)
MEDAL (0.54)
SUCCEED (1.0)
MONEY
(0.0),(0.06)
VALUABLE (0.80)
DIAMOND (0.25)
BRACELET
(0.80)
LIMOUSINE (1.0)
EXPENSIVE (0.14)
NECKLACE
(0.55)
JEWELRY
(0.43)
PRICELESS (0.93)
JEWEL (0.67)
RING (0.13)
EARRING
(0.87)
STONE (0.0)
PEARL
(0.81)
SAPPHIRE
(1.0)
GEM
(0.71)
α
Wout
1
0.8
0.6
0.4
0.2
0
EMERALD (0.86)
RUBY (0.60)
Vicsek group
Introduction
Directed CPM
Results
Relative out-degree and number of hits
α :
The number of hits in Google as a function of Wi,out
1e+10
number of hits
1e+09
1e+08
1e+07
1e+06
1e+05
0
0.2
Vicsek group
0.4
Wi,αout
0.6
0.8
1
Introduction
Directed CPM
Results
Google’s on web pages
Local picture of the communities:
enterprise/maps (0.83)
enterprise/government (1.0)
enterprise
addurl (0.83)
enterprise/whygoogle.html (0.50)
enterprise/news_events.html (0.43)
enterprise/gsa/onebox.html (0.64)
enterprise/support.html (0.43)
enterprise/gep (0.43)
services/websearch.html (1.0)
enterprise/apps (0.91)
enterprise (0.52)
enterprise/customers.html (0.43)
enterprise/gsa (0.43)
psearch (0.17)
privacy.html (0.17)
+
accounts/Login (0.83)
help/faq_accounts.html (0.54)
accounts
jobs
newsalerts (0.56)
jobs/working.html (0.50)
jobs/international.html (0.67)
(https) accounts (0.60)
alerts (0.80)
accounts/ServiceLogin (0.83.)
accounts (0.89)
about.html (0.33), (0.06)
sitemap.html (1.0), (0.83)
jobs (0.50)
terms_of_service.html (0.18)
accounts/NewAccount (0.67)
enterprise/mini (0.39)
+ www.google.com (0.0),(0.08),(0.0)
accounts/ForgotPasswd (0.67)
Vicsek group
Introduction
Directed CPM
Results
Comparing overlaps
α :
Membership number in function of Di,out
Google’s own pages
word associations
email network
yeast transc. reg.
9
avg. # of modules of a node
8
3
7
6
2
5
4
1
3
0
0.5
1
2
1
0
0.2
0.4
0.6
dout / ( din + dout )
0.8
1
Community overlaps:
word association net, Google’s web pages −→ in-hubs,
e-mail net, transcription regulatory network −→ out-hubs.
Vicsek group
Introduction
Weighted CPM
Results
Link weights in the original CPM
In the original CPM we can take into account the weights by
ignoring links weaker than a certain threshold w ∗ .
Changing w ∗ and k is similar to changing the resolution in a
microscope.
Optimal k -clique size and w ∗
Where the community structure is as highly structured as
possible: just below the critical point of the appearance of a
giant k -clique community.
Vicsek group
Introduction
Weighted CPM
Results
microscope.
Vicsek group
Introduction
Weighted CPM
Results
microscope.
Vicsek group
Introduction
Weighted CPM
Results
k -clique intensity
The intensity I of a sub-graph is defined as the geometrical
mean of its link weights.
0
12/k /(k −1)
BY C
wij C
For a k -clique C: I(C) = B
A
@
i<j
i,j∈C
Weighted k -clique
A k -clique with an intensity greater or equal to a given intensity
threshold I ∗ .
Vicsek group
Introduction
Weighted CPM
Results
Percolation transition in the E-R graph
A weighted E-R graph:
N nodes,
every pair is linked independently with uniform probability
p,
each link is assigned a weight chosen randomly from a
uniform distribution on the (0, 1] interval.
The critical linking probability is a function of the intensity
threshold. At I = 0 we recover pc (I = 0) = [N(k − 1)]−1/(k −1) .
Vicsek group
Introduction
Weighted CPM
Results
Percolation transition in the E-R graph
A weighted E-R graph:
N nodes,
every pair is linked independently with uniform probability
p,
each link is assigned a weight chosen randomly from a
uniform distribution on the (0, 1] interval.
The critical linking probability is a function of the intensity
threshold. At I = 0 we recover pc (I = 0) = [N(k − 1)]−1/(k −1) .
Vicsek group
Introduction
Weighted CPM
Results
Results
Vicsek group
Introduction
Weighted CPM
Results
NYSE graph
New York Stock Exchange graph:
We studied the pre-computed stock correlation matrix
containing the averaged correlation between the daily
logarithmic returns.
The correlation coefficients can be used as link weights.
We kept only the strongest 3%.
Vicsek group
Introduction
Weighted CPM
Results
Summary
Directed communities:
Relative in- and out-degree,
Directed k -cliques.
Weighted communities:
k -clique intensity.
Publications:
New Journal of Physics 9, 180 (2007),
New Journal of Physics 9, 186 (2007).
Downloadable community finding software:
http://cfinder.org
Vicsek group

k-clique percolation and clustering in directed and weighted networks

Transcripción

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