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1The 2nd KIAS Conference on Statistical Physics (2006)
Yup KimKyung Hee University
Conserved Mass Aggregation and Lamb-lion Problem on complex
networks
CollaborationsSoon-Hyung Yook, Sungchul Kwon, Sungmin Lee
ReferencesS. Kwon, S. Lee and Y. Kim, PRE 73, 056102 (2006)S. Lee, S. Yook and Y. Kim, Submitted to PRE, cond-mat/0603647 S. Lee, S. Yook and Y. Kim, Submitted to PRL
2The 2nd KIAS Conference on Statistical Physics (2006)
Outline• Condensation phase transition on complex networks
– Symmetric Conserved mass aggregation (SCMA) model– SCMA model on complex networks– Mass distribution of a node with degree k, m(k)– Existence of infinite aggregation– Finite sized results for random walks (RWs) on scale-free netw
orks (SFNs)• Lamb-lion problem on complex networks• Application
– Peer-to-Peer network – Propose an efficient algorithm
• Conclusions
3The 2nd KIAS Conference on Statistical Physics (2006)
ijji mmm,0m
1mm,1mm jjii
Condensation phase transition
Diffusion
Chipping
Diffusion with unit rate :
Chipping with rate :
fluid phase Condensed phase
Examples : clouds, colloidal suspensions,polymer gels, aerosols, river networks
- Symmetric Conserved-mass aggregation (SCMA) model
S. N. Majumdar et al, J. Stat. Phys. 99, 1 (2000)
Diffusion tends to aggregate masses.
Chipping tends to split masses.
(j is one of nns to i)
4The 2nd KIAS Conference on Statistical Physics (2006)
For , competition between diffusion and chipping
→ phase transitions from condensed phase into fluid phase.
0
0
diffusion-dominant ( ) : aggregation on a site
chipping-dominant ( ) : masses scattered over entire lattice.
(zero-range process : ZRP)
Zero Range Process (ZRP)
A particle jumps out of the site at the rate
and hops to a site with the Probability .
A condensed phase, which a finite fraction of
total particles condenses on a single site,
arises or not according to , .
Braz. J. Phys. 30, 42 (2000)
Hopping
Jumping
5The 2nd KIAS Conference on Statistical Physics (2006)
Phase diagramOrder parameter : P(m) = mass distribution of a single site
= Probability that a site has mass m in the steady state.
Fluid phase
Condensed
phase
condensatem
ycriticalitatm
phasefluidinemP mm
~
~
~)(*/
2/5
11
c
theoryfieldMeanm
P(m)
6The 2nd KIAS Conference on Statistical Physics (2006)
Diffusion with unit rate :Chipping with rate ω :
ω= 0 : complete condensation on a node ω = ∞ : Zero-range process with constant chipping rate → Condensation always exists on scale free networks with ; J. D. Noh et al, Phys. Rev. Lett. 94, 198701 (2005).
ijji mmm,0m 1mm,1mm jjii
2
- SCMA model on complex networks
Degree distribution
scale-free networks (SFNs)
What about 0 < ω < ∞ case on SFNs ?What is the effect of underlying topology like SFNs on condensation transitions ?
7The 2nd KIAS Conference on Statistical Physics (2006)
(1) Random and scale free networks ofN = # of nodes = 10000, K = # of links = <k>N/2 = 20000
<k> = average degree = 4 in our simulations (a) = Random net. (RN) (b) = SFN of
3
3.4
8The 2nd KIAS Conference on Statistical Physics (2006)
Phase diagram : RNs (a) and SFN of (b)3.4
3.4:)2(33.2
RN:)3(38.2
The same type of condensation transition as those on regular lattices.(SFN with )
But the critical line depends on network structures.
3
9The 2nd KIAS Conference on Statistical Physics (2006)
(2) SFNs of(a) (b) (c) expected phase diagram
In limit, it is practically impossible to show the existence of the condensation. (Consideration of Diffusive Capture Process or Lamb-Lion Problem on the networks).
30.3 4.2
0
10The 2nd KIAS Conference on Statistical Physics (2006)
Total mass of nodes with degree k = kM
In a certain run of simulation,
)4.2(3
1
By diffusion, the aggregate diffuses around networks and the dominant hub is not the node at which the condensate is located unlike ZRP.
11The 2nd KIAS Conference on Statistical Physics (2006)
- Average mass of a node with degree k ,
At , it was shown that complete condensation always takes place on SFNs. What about on RNs ?
For , the behavior in the condensed phase ?
km
0
3
1
PRL 94, 198701
(2005)
ZRP on SFN
12The 2nd KIAS Conference on Statistical Physics (2006)
0
),(m
k kmPmM
kPmPkmP )(),(
= the probability of finding a random walker on degree k in k-space
kP
13The 2nd KIAS Conference on Statistical Physics (2006)
T
Condensedphase
Fluidphase
Lamb-lion problem
- For the existence of an infinite condensate, the two masses should aggregate again in finite time interval.
- If not, unit mass continuously chips off from the infinite aggregation, which will finally disappear.
Fluid phase
Condensed phase
Condensed phase
(no Fluid phase)
Or
: survival probability)(tS
: average life time
- Existence of infinite aggregation
finite
14The 2nd KIAS Conference on Statistical Physics (2006)
10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 10610-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
=2.4
N
v(t)/
N
t/N
N=1000 N=10000 N=100000 N=1000000
10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 10510-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
=3.0
Nv(t
)/N
t/N
N=1000 N=10000 N=100000 N=1000000
NtNT vsat ~~~For any
vN : The number of visited distinct sites of a random walker
satT : The saturation time
T : The average life time
- Finite sized results for RWs on SFNs
10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 10510-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
=4.3
N
v(t)/
N
t/N
N=1000 N=10000 N=100000 N=1000000
15The 2nd KIAS Conference on Statistical Physics (2006)
10-1 100 101 102 10310-1
100
N=1000 N=10000 N=100000 N=1000000
R/ (
log(
N))
0.97
t / (log(N))0.97
=4.3
10-1 100 101 102 103 104
1
N=1000 N=10000 N=100000 N=1000000
R /
(log(
N))
0.6
5
t / (log(N))0.65
=3.0
100 101 102 103 104 105
1
2
3
4
5
6
N=1000 N=10000 N=100000 N=1000000
R /
(log(
log(
N))
)0.4
5
t / (log(log(N)))0.45
=2.4
R: the distance between two random walkers (the shortest path)
N: the number of nodes
16The 2nd KIAS Conference on Statistical Physics (2006)
Static trap
Hub effect
random walker to random walker random walker
No!!
On networks
On regular lattice
Yes
?
Diffusive capture process = lamb-lion problem
17The 2nd KIAS Conference on Statistical Physics (2006)
Lamb-lion problemThe diffusion-controlled reactions, in which diffusing
particlesare immediately converted to a product if a pair of
them meetstogether, have many physical applications.Examples : electron trapping and recombination, wetting, melting,exciton fusion, and commensurate-incommensurate transitions
Among these examples, dynamic properties of wetting, melting, and commensurate-
incommensurate transition are known to be related to the diffusive capture process,
whose kinetics can be simplified by lamb-lion problem (diffusing preys-predators model).
P.L.Krapivsky and S.Redner J.Phys.A 29, 5347 (1996)
What is the survival probability of a diffusing lamb which is hunted by hungry lions?
On regular lattice
PRB 39, 889 (1989), JSP 34, 667 (1984), PRB 29, 239 (1984), JPA 21, L89 (1988)
Diffusion-controlled reaction
First passagephenomena of RWs
Survival probability of a diffusing lamb
18The 2nd KIAS Conference on Statistical Physics (2006)
The major searching engines, such as Google, use general
random walking robots along the links between hyper-texts
to collect information of each web page. The searching algorithm can be mapped to the
system of a diffusing particle to find an immobile
absorbing particle.
One of interesting applications can be found in searching information over the Internet.
If the lamb meets the lion, the lamb is captured.
At each time step, a lamb and lions
take random walks simultaneously.
Initially a lamb and lions are randomly
distributed to the nodes on the networks.
Our model
Korean Phys. Soc. 48, S202 (2006)
Degree distribution
19The 2nd KIAS Conference on Statistical Physics (2006)
We measure and on LSFNs with various and network size .
20The 2nd KIAS Conference on Statistical Physics (2006)
We measure the average life time and the survival probability on TSFNs.
21The 2nd KIAS Conference on Statistical Physics (2006)
Origin of long-living tail of for
The data explicitly shows that lamb-lion
with corresponds to the long
surviving tail. In the used networks, the
explicitly demonstrates that the
lamb and the lion are in different
branches.
22The 2nd KIAS Conference on Statistical Physics (2006)
Relation between degrees and capture events
We measure the number of captures
occurring at a node with degree .
PRL 92, 118701 (2004).
Assume ( : the model dependent parameter satisfying )
increases as for . Determined 's from the data in (a) and (b) are for LSFN and
for TSFN.
23The 2nd KIAS Conference on Statistical Physics (2006)
Relation between degrees and capture events
Assume ( : the model dependent parameter satisfying )
increases as for . Determined 's from the data in (a) and (b) are for LSFN and
for TSFN. provides the topological origin of the gathering behavior of
randomwalks at hubs. This implies that the lamb and the lion have a strong
tendency to move intobig hubs.
24The 2nd KIAS Conference on Statistical Physics (2006)
Complex Network
Lamb
Lion
Information packet
Query packet
Implementing an efficient searching algorithm is the key to a better performance of P2P protocol design.
P2P systems are distributed systems in which nodes exchange files directly with each other.
We apply results of our study on diffusive capture process to the searching algorithm to find file in the Peer-to-Peer (P2P) file-sharing networks.
Application
25The 2nd KIAS Conference on Statistical Physics (2006)
Each node forwards the received query packets to all of its nearest neighbors until the pre-assigned time-to-live (TTL) becomes 0.
Flooding based algorithm (FB) n-random walker algorithm (n-RW)
The node who want to search a file produces n query packets. Each querying packet takes random walks along the P2P connections until the pre-assigned TTL becomes 0.
FB causes significant traffic congestion. For example, the P2P traffic consumes 60-70% of European Operators’ bandwidth.
n-RW algorithm can cause long waiting time if there are a few requested files in the network and they are located at the node with small number of connections.
(http://www.theregister.co.uk/2003/10/14 /edonkey_rides_like_the_wind/)
s
T
s
query packet
T
26The 2nd KIAS Conference on Statistical Physics (2006)
In general, the degree distribution of P2P networks follows the power law, with , or highly skewed fat-tailed distributions.(=> Expect attracting hubs)
=> exists effective attractor (Hubs)
Degree distribution of P2P network
L.A.Adam, R.M.Lukose, B.Huberman, & A.R.Puniyani, PRE 64, 46135 (2001)M.Ripeanu, I.Foster & A.Iamnitchi, IEEE Internet Computing Journal 6, 50 (2002)Stutzbach, D. & Rejaie, R. In Global Internet Symposium, 127 Mar. (2005)
We expect two main benefits by using new algorithm.1) the amount of traffic is constant and much less than
FB algorithm2) provide more scalable searching time than n-RW
algorithm
27The 2nd KIAS Conference on Statistical Physics (2006)
ii) Independently, a randomly chosen node sends out one query packet to find
a specific file. (The query packet also takes random walks.) iii) If the query packet meets an information packet which has the requested file
name in its list, then the IP information in the information packet is sent back
to the requesting node. iv) And then the query packet is discarded but the information
packets continue random walks for the next query.
i) Each node sends out an information packet (names of files + IP address). (Each of these packets takes random walks along the P2P
connections.)
One query event
s
query packetinformation packet
n-lion and lamb algorithm (NLL)
Propose an efficient algorithm
28The 2nd KIAS Conference on Statistical Physics (2006)
The inset shows the time evolution of obtained from a single run of simulation of FB.
The local maximum exceeds which is 2,000 times larger than the traffic generated
by NLL.
The average traffic of each algorithm.
FB generates around 50 times more traffic than NLL on the average.
At the moment of occurring such large amount of traffic, FB would consume the most of the bandwidth of the Internet and cause severe traffic congestion over the network. However, NLL always guarantees a constant level of traffic, which is much less than that of FB and comparable to that of n-RW.
29The 2nd KIAS Conference on Statistical Physics (2006)
: the number of available files on a network : the average searching time
(More scalable searching time than n-RW)
The average searching time of NLL on SFNs is, at least, 10 times faster than n-RW on SFNs.
The average searching times satisfy
The difference between NLL and 2-RW, , increases as .
Since the probability that two random walks visits a node with degree is proportional to , the hub can collect all information packets.
increases almost linearly for n-RW
. However, NLLgrows as for small , but
seems to be less than 0.5 or
becomes saturated to a fixed finite value
for large .
hub
30The 2nd KIAS Conference on Statistical Physics (2006)
ConclusionOn RN and SFNs with , CMA model undergoes the same typeof condensation transitions as one dimensional lattice.(The critical line depends on the underlying network structures.)
On SFNs with , an infinite aggregation with exponentiallydecaying background mass distribution always takes place for anynonzero density. (no phase transitions)
On LSFNs,
On TSFNs,
31The 2nd KIAS Conference on Statistical Physics (2006)
The lamb and the lion have a strong tendency to move into big hubs. By numerical simulations, we verify that our new searching algorithm can
drastically decrease the traffic congestion compared to FB algorithm and
can provide more scalable average searching time than n-RW algorithm
and comparable with FB algorithm.The hubs spontaneously play a very similar role of the
directory serversin structured P2P networks. However, we expect the one of
the advantagesof using NLL algorithm can be in reducing large amount of
systemresources for the directory server to store and handle the
huge centralizedinformation packet, because most of information packets
stay on thedominant hubs and their nearest neighbors for a
considerable amount oftime. Therefore, the information on the nearest neighbors of
the hubs at agiven time is easily accessed through the hubs by random
walks withoutstoring all information at the hubs.