Lecture 1-5
Power Law Structure
Weili Wu Ding-Zhu Du
Univ of Texas at Dallas
[email protected]
“The small world network
is a type of mathematical graph in which
most nodes are not neighbors of one
another, but most nodes can be reached
from every other by a small number of
hops or steps.”
Why small distance and large
size can stay together?
2
Power Law
• During the evolution and
growth of a network, the
great majority of new
edges are to nodes with an
already high degree.
3
Power-law distribution
Log-log scale:
log f(x) ~ –αlog x
Power law distribution:
f(x) ~ x–α
4
Power Law
• Nodes with high degrees may have “butterfly
effect”.
• Small number
• Big influence
5
Important Facts on Power-law
• Many NP-hard network problems are still NPhard in power-law graphs.
• While they have no good approximation in
general, they have constant-approximation in
power-law graphs.
What is Power Law Graph?
G  P ( ,  )
 e 
G  P( ,  ) means | {v V (G) | deg( v)  k} |   
k 
The max degree is e /  . (e  k  )




(

)
e
if   1
e /  
e
 
# nodes  n     e
if   1
k 1 k
 e / 
if   1

1 
8
Warning
In study on Power-law Graph, a lot
of real numbers are treated as
integers!!!
9
The max degree is e
 /
.
1



(


1
)
e
if   2

2
 /

1e
e  1
# edges  m   k      e
if   2
2 k 1 k
 4 2 / 
1e
if   2
 2 2  

1
where  ( )   
i 1 i
10
Scientific collaborat ion networks, 2.1    2.45.
A.L. Barabasi, et al., Evolution of the social network of scientific
collaborations, Physica A, vol. 311, 2002.
World Wide Web,   2.1 (indegree) , 2.45 (outdegree ).
R. Albert, et al., Erro and attack tolerance of complex networks,
Nature, vol. 406,
Internet at router and intra - domain level,   2.48.
M. Faloutsos, et al., On power-law relationship of the internet topology,
SIGCOMM’99,
11
Why still NP-hard in Power-law?
12
Proof Techniques
• NP-hard in graph with constant degree, e.g.,
the Vertex-Cover is NP-hard in cubic graphs.
• Embedding a constant-degree graph into a
power-law graph.
13

in cubic graph
Vertex - Cover
p Vertex - Cover
m in power - law graphs
cubic graph
# nodes  e / 3
for any k  3, costruct a graph with # nodes  e / k 
and with an easily - known min vertex - cover.
14
Why approximate easily in Powerlaw?
15
• More nodes with low degree
• Less nodes with high degree
• Size of opt solution is often determined by #
of nodes with low degree.
for   1,
 e 
# nodes with degree k :   
k 
total # of nodes   ( )e
16
Modularity Maximization
Modularity Function (Newman 2006)
Consider a graph G  (V , E ) with adjacency matrix (aij ).
Given a partition (V1 , V2 ,..., Vk ) of V , define
 L(V ,V )  L(V ,V )  L(V ,V )  2 
s
s
s
s
s
s
 
Q  
 
L(V , V )
s 1  L (V , V )

 

where L(U , W )   aij .
k
iU , jW
This is the total difference of the fraction of the edges
within a community minus the expected number of
such fraction if edges were distribute d at random.
17
Modularity Maximization
 L(V ,V )  L(V ,V )  L(V ,V )  2 
s
s
s
s
s
s
 
Q  
 
L(V ,V )
s 1  L(V , V )

 

k
k

 L(V ,V )
s 1
s
L(V ,V )
s
1
18
Idea
• Lower-degree nodes
follow higher-degree
nodes.
19
Low-Degree Following Algorithm
T.N. Dinh & M.T. Thai, 2013
L  , M  , O  , pi  0 i  1..n;
for each i  V do
if (deg( i )  d 0 ) & (i  L  M )
i
j
i
t
then if N (i )  M
then select j  N (i ) \ M and
let M  M  {i}, L  L  { j}, pi  j
else selet t  N (i ) and set
O  O  {i}, pi  t.
L : set of leaders
M : set of members
O : set of orbiters
| M  O | 0.5 | {nodes with degree  d 0 } |
20
Low-Degree Following Algorithm
L  , M  , O  , pi  0 i  1..n;
for each i  V do
if (deg( i )  d 0 ) & (i  L  M )
then if N (i )  M then select j  N (i ) \ M and let
M  M  {i}, L  L  { j}, pi  j
else selet t  N (i ) and set
O  O  {i}, pi  t.
" i follows pi ."
i  M : pi  L. i  O : pi  M .
i  O : N (i )  M .
21
Low-Degree Following Algorithm
{C (i) | i V  ( M  O)}
C (i)  {i}  { j  M | p j  i}  {t  O | p pt  i}
Choice of d0
Choose d 0 from 1,2, .., n such that above community
partition reach maximun modularity .
22
Low-Degree Following Algorithm
{C (i) | i V  ( M  O)}
C (i)  {i}  { j  M | p j  i}  {t  O | p pi  t}
Theorem
For power - law graph with   2 and for any   0,
with LDF algorithm, we can construct a polynomial - time
algorithm A which generates community partition with
modularity at least
 ( )

 ( - 1)
 ( )
n

 (  1) m
23
Idea of Proof
{C (i) | i V  ( M  O)}  {V1 ,V2 ,...,Vh }
C (i)  {i}  { j  M | p j  i}  {t  O | p pt  i}
In algorithm, we choose a d 0 to reach largest modularity .
In the proof, we show that ther e exists a d 0 which can
give a community partition with modularity at least
 ( )
-.
 ( - 1)
24
Lower bound for positive part
{C (i) | i V  ( M  O)}  {V1 ,V2 ,...,Vh }
C (i)  {i}  { j  M | p j  i}  {t  O | p pt  i}
i  M  O : i and pi are in the same community.
k
d0
s 1
k 1
Hence,  L(Vs , Vs )  2 | M  O |  e  k 
 e ( ( )   )
for sufficient ly large d 0 , independen t from  .

(note :  k    ( ).)
k 1
25
Upper bound for negative part
{C (i) | i V  ( M  O)}  {V1 ,V2 ,...,Vh }
C (i)  {i}  { j  M | p j  i}  {t  O | p pt  i}
i  L with Vs  C (i ) :
L(Vs , Vs )  L(Vs , Vs )  deg( i )  deg( i )  d 0  deg( i )  d 02
 deg( i )  2d 02
h
2

))
V
,
V
(
L

)
V
,
V
(
L
(
 s s
s
s
s 1
2 2
d
2

)
i
(deg(

0)
iV  ( M O )
e / 
e
    (2kd02 ) 2  e /   e (2d 02 ) 2
k 1 k
26
h

L
(
V
,
V
)

2
|
M

O
|

e
( ( )   )
 s s
s 1
h
2
2 2
(
L
(
V
,
V
)

L
(
V
,
V
))

(deg(
i
)

2
d
 s s

s
s
0)
s 1
iL
e / 
e
    (2kd02 ) 2  e /   e (2d 02 ) 2
k 1 k
 L(V , V )  L(V , V )  L(V , V )  2 
s
s
s
s
s
s
 
Q  
 
L(V , V )
s 1  L (V , V )

 

(2d 02 ) 2
 ( )  
 ( )




 (11 /  )
 (  1)  (  1)e
 (  1)
for sufficient ly large  .
h
(note : d 0 depends on  ,  is independen t from  .)
27
Warning
In study on Power-law Graph, a lot
of real numbers are treated as
integers!!!
Can we get same results if not do
so?
28
References
1. Thang N. Dinh, My T. Thai : Community detection
in scale - free networks : approximat ion algorithms for
maximizing modularity , IEEE Journal on Sele cted
Areas in C ommunications 31(6) : 997 - 1006 (2013)
29
THANK YOU!