Title: Variation on preferential-attachment
Abstract
In this talk, I will describe how preferential attachment arises from the first principle using game theory. Next, I will extend the model of preferential attachment into a general model, which allows for the incorporation of Homophily ties in the network. This talk is based on joint works with Prof. Chen Avin, Avi Cohen, Yinon Nahum, Prof. Pierre Fraigniaud, and Prof. David Peleg.
3. 3
Power Law Distributions
Observed in
both network
and non-
network
structures
โEmergence of
Scaling in
Random
Networksโ
(Barabรกsi and
Albert, 1999)
Curabitur a nisl facilisis lectus
posuere pharetra porta sed
neque. Fusce porttitor venenatis
ipsum at ullamcorper.
Suspendisse sodales leo vehicula
libero pharetra, ac porttitor
metus mollis.
Object Photography
Pr ๐ฅ๐ฅ > ๐ก๐ก ~๐ฟ๐ฟ ๐ก๐ก ๐ก๐กโ๐ฝ๐ฝ+1,
lim
๐๐โ โ
L r t /๐ฟ๐ฟ[๐ก๐ก] = 1
4. 4
History
๐ฝ๐ฝ=3 ๐ฝ๐ฝ=3 ๐ฝ๐ฝ=3 ๐ฝ๐ฝ=3 ๐ฝ๐ฝ=3๐ฝ๐ฝ=3 ๐ฝ๐ฝ โ(2,3]๐ฝ๐ฝ โ(2,3]
1925,
1925,
1976
1976
1999
1999
Udny Yule Price Barabรกsi
Pr[ ๐ฃ๐ฃ๐ก๐กconnects to ๐ฃ๐ฃ๐๐ ] =
๐๐๐๐
โ๐๐ ๐๐๐๐
2006
2006
Chung and Lu
5. 05
Preferential-Attachment
โข Evolutionary model of networks
โข There are two operations: node event,
edge event.
โข In the node event node arrives
โข connected to net with only one edge
โข Node arriving connects according to
degree.
โข In edge event
โข we select two nodes according to the
degree.
6. 6
OUR Preferential-Attachment
An edge
event
happens
with
probability
๐๐๐ก๐ก
Edge
event
A component
event
happens with
probability
๐๐๐ก๐ก
Component
event
Can change in
time.
But
๐๐๐ก๐ก+๐๐๐ก๐ก + ๐๐๐ก๐ก = 1
Time
varies
In fact, it is also
possible to
prove the
results on
Hyper graph
Hyper
graph
A node
event
happens
with
probability
๐๐๐ก๐ก
Node
event
The basic model can be expanded simply in many
directions.
7. Theorem
r=1-1/log(t), p=1/log(t)
E=๐๐ log ๐๐, sub-linear core
Consider
p=1/2,r=0,q=1/2
Consider
p=0,r=0.25,q=0.75
Giant component
Push ๐ฝ๐ฝ to be
Full domain [2,โ)
p=ฮต,r=1/2,q=1- ฮต
Full domain (1,2]
r=1-1/t^a, p=1/t^a
โข PA follows a power law with exponent
โ ๐ฝ๐ฝ = 1 +
2
๐๐+2๐๐
8. 8
Example
P=1/2 r=1/3 q=1/6
๐๐๐๐๐๐(๐ก๐ก)
๐๐๐ก๐ก
= ๐๐๐๐(๐ก๐ก)/2t
Node E
๐๐๐๐๐๐(๐ก๐ก)
๐๐๐๐
= 2๐๐๐๐(๐ก๐ก)/2t
Edge E
๐๐๐๐๐๐(๐ก๐ก)
๐๐๐๐
= 0
Comp E
p=1/2,r=1/3,q=1/6
Let ๐๐๐๐(๐ก๐ก) denote the Degree of vertices of i at time t
๐๐๐๐๐๐(๐ก๐ก)
๐๐๐ก๐ก
= 1/2๐๐๐๐(๐ก๐ก)/2t+1/3๐๐๐๐(๐ก๐ก)/t
๐๐๐๐(๐ก๐ก)=(t/i)^(7/12)
๐ฝ๐ฝ = 1 + 12/7
10. 10
Game Theory in one slide
๏
๏
๏
๏
There are
players who
can choose
strategies
A profile is an
assignment
strategy for
each
The players
wish to
maximize their
(expected)
payoff
A profile
determines an
outcome, and
an outcome
determines a
payoff for each
player
11. 11
Game Theory, cont. ANALYSIS
Nash Equilibrium is a
profile such that no single player
can gain by changing her strategy
unilaterally.
example
A B
A (10,10) (5,0)
B (0,5) (0,0)
12. 12
Network Formation Game [Fabrikant et al, 2003]
Player = node
Strategy =
which nodes
to connect to
.
Goal:
minimize
their average
distance
Resulting
graph is not
Power Law
13. With probability ๐ผ๐ผ,
๐ฃ๐ฃ๐ก๐ก connects, and
with
probability 1โ๐ผ๐ผ, ๐ฃ๐ฃ๐ก๐ก
connects to a
random neighbour
of the host.
Start with one
node ๐ฃ๐ฃ1.
Wealth Based
Recommendation.
At time ๐ก๐ก, node
๐ฃ๐ฃ๐ก๐ก arrives.
๐ฃ๐ฃ๐ก๐ก proposes to
an (existing)
host node.
Wealth&Recommendation
(W&R) Game
Utility = (expected) degree
14. 14
How to play W&R
Each time a node arrives, it has to choose a single
node to connect to or receive a recommendation.
After selecting a node, either the node connects to
that node, or gets a proposal and then connects to
that node.
W&R game
End of
The game
๐๐
Wealth
๐ผ๐ผ
Partial
Information
deg seq
Utility
Max
degree
Question
what is
Nash
15. 15
Strategy
Player ๐๐๐๐ Strategy ๐ ๐ ๐๐.
is a probability distribution on existing
nodes.
๐ ๐ ๐๐ ๐ ๐ ๐๐, ๐ซ๐ซ
prob. of choosing the node of degree ๐๐_๐๐ in
the degree sequence ๐ท๐ท=(๐๐_1,โฆ,๐๐_(๐ก๐กโ1)).
.
Strategy Profile -๐ท๐ท = ๐ ๐ ๐๐ , (๐๐โฅ๐๐)
16. 016
Examples
โข What to do if every one plays
uniform
โข Better to connect to small deg
nodes?
โข if ๐ผ๐ผ=1?
โข Is it Nash Equilibrium?
5
1
1
1
1
1
1
1
1
3
3
3
1/n
1/n
17. The Preferential Attachment Strategy
The Preferential Attachment (PA) strategy at time ๐ก๐ก
over the degree sequence ๐ท๐ท = (๐๐1, โฆ , ๐๐๐ก๐กโ1) is:
=
๐๐๐๐
2(๐ก๐ก โ 2)
๐๐๐ก๐ก ๐๐๐๐, ๐ท๐ท =
๐๐๐๐
โ๐๐ ๐๐๐๐
The Preferential Attachment strategy profile is the
strategy profile where all players ๐ฃ๐ฃ๐ก๐ก for ๐ก๐ก โฅ 5 play the PA strategy.
18. 1
9 RUNDO
The stationary distribution of a
simple random
Walk is a probability of PA
PA and
Random
Walks
Pr ๐ฃ๐ฃ๐ก๐กconnects to ๐ฃ๐ฃ๐๐ =Pr[RW to visit ๐ฃ๐ฃ๐๐]=
๐๐๐๐
โ๐๐ ๐๐๐๐
19. 4 FOUR
2
0
TheoremThe Preferential Attachment
strategy profile is the only
universal Nash equilibrium.
Provides a possible explanation why preferential
attachment occurs in social networks.
20. 21
PA is a universal Nash Equilibrium
Suppose PA profile is played ๐ฃ๐ฃ๐๐ changes its strategy to ๐๐๐๐โฒ At step ๐ก๐ก>๐๐, No deviating from PAPROCESS
๐ฃ๐ฃ๐๐ starts at
degree ๐๐ = 1.
๐๐ โ ๐๐ + 1
w.p.
๐๐
2(๐ก๐กโ2)
.
Hence, ๐๐๐๐โฒ doesnโt matter