1 The 2nd KIAS Conference on Statistical Physics (2006) Yup Kim Kyung Hee University Conserved Mass Aggregation and Lamb- lion Problem on complex networks Collaborations Soon-Hyung Yook, Sungchul Kwon, Sungmin Lee References S. Kwon, S. Lee and Y. Kim, PRE 73, (2006) S. Lee, S. Yook and Y. Kim, Submitted to PRE, cond-mat/ S. Lee, S. Yook and Y. Kim, Submitted to PRL

2 The 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 networks (SFNs) Lamb-lion problem on complex networks Application –Peer-to-Peer network –Propose an efficient algorithm Conclusions

3 The 2nd KIAS Conference on Statistical Physics (2006) Condensation phase transition Diffusion Chipping Diffusion with unit rate : Chipping with rate : fluid phaseCondensed 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)

4 The 2nd KIAS Conference on Statistical Physics (2006) For, competition between diffusion and chipping → phase transitions from condensed phase into fluid phase. 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

5 The 2nd KIAS Conference on Statistical Physics (2006) Phase diagram Order parameter : P(m) = mass distribution of a single site = Probability that a site has mass m in the steady state. Fluid phase Condensed phase m P(m)P(m)

6 The 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, (2005). - 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 ?

7 The 2nd KIAS Conference on Statistical Physics (2006) (1)Random and scale free networks of N = # of nodes = 10000, K = # of links = N/2 = = average degree = 4 in our simulations (a) = Random net. (RN) (b) = SFN of

8 The 2nd KIAS Conference on Statistical Physics (2006) Phase diagram : RNs (a) and SFN of (b) The same type of condensation transition as those on regular lattices. (SFN with ) But the critical line depends on network structures.

9 The 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).

10 The 2nd KIAS Conference on Statistical Physics (2006) Total mass of nodes with degree k = In a certain run of simulation, By diffusion, the aggregate diffuses around networks and the dominant hub is not the node at which the condensate is located unlike ZRP.

11 The 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 ? PRL 94, (2005) ZRP on SFN

12 The 2nd KIAS Conference on Statistical Physics (2006) = the probability of finding a random walker on degree k in k-space

13 The 2nd KIAS Conference on Statistical Physics (2006) Condensed phase Fluid phase 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 (no Fluid phase) Or : survival probability : average life time - Existence of infinite aggregation finite

14 The 2nd KIAS Conference on Statistical Physics (2006) For any : The number of visited distinct sites of a random walker : The saturation time : The average life time - Finite sized results for RWs on SFNs

15 The 2nd KIAS Conference on Statistical Physics (2006) : the distance between two random walkers (the shortest path) : the number of nodes

16 The 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

17 The 2nd KIAS Conference on Statistical Physics (2006) Lamb-lion problem The diffusion-controlled reactions, in which diffusing particles are immediately converted to a product if a pair of them meets together, 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 passage phenomena of RWs Survival probability of a diffusing lamb

18 The 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

19 The 2nd KIAS Conference on Statistical Physics (2006) We measure and on LSFNs with various and network size.

20 The 2nd KIAS Conference on Statistical Physics (2006) We measure the average life time and the survival probability on TSFNs.

21 The 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.

22 The 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, (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.

23 The 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 random walks at hubs. This implies that the lamb and the lion have a strong tendency to move into big hubs.

24 The 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

25 The 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. ( /edonkey_rides_like_the_wind/) s T s query packet T

26 The 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, (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 algorithm 2) provide more scalable searching time than n-RW algorithm

27 The 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 packet information packet n-lion and lamb algorithm (NLL) Propose an efficient algorithm

28 The 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.

29 The 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, NLL grows as for small, but seems to be less than 0.5 or becomes saturated to a fixed finite value for large. hub

30 The 2nd KIAS Conference on Statistical Physics (2006) Conclusion On RN and SFNs with, CMA model undergoes the same type of condensation transitions as one dimensional lattice. (The critical line depends on the underlying network structures.) On SFNs with, an infinite aggregation with exponentially decaying background mass distribution always takes place for any nonzero density. (no phase transitions) On LSFNs, On TSFNs,

31 The 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 servers in structured P2P networks. However, we expect the one of the advantages of using NLL algorithm can be in reducing large amount of system resources for the directory server to store and handle the huge centralized information packet, because most of information packets stay on the dominant hubs and their nearest neighbors for a considerable amount of time. Therefore, the information on the nearest neighbors of the hubs at a given time is easily accessed through the hubs by random walks without storing all information at the hubs.

32 The 2nd KIAS Conference on Statistical Physics (2006) Static trap Hub effect If Then is finite. SFNs random walker to random walker random walker No!! On networks On regular lattice Yes : survival probability