Yup Kim Kyung Hee University

1. Conserved Mass Aggregation and Lamb-lion Problem on complex networks Yup Kim Kyung Hee University References S. 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 Collaborations Soon-Hyung Yook, Sungchul Kwon, Sungmin Lee

2. 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 problemon complex networks Application Peer-to-Peer network Propose an efficient algorithm Conclusions Outline

3. Diffusion Chipping Condensation phase transition Examples : clouds, colloidal suspensions, polymer gels, aerosols, river networks fluid phase Condensed phase - Symmetric Conserved-mass aggregation (SCMA) model Diffusion with unit rate : Chipping with rate : (j is one of nns to i) Diffusion tends to aggregate masses. Chipping tends to split masses. S. N. Majumdar et al, J. Stat. Phys. 99, 1 (2000)

4. diffusion-dominant ( ) : aggregation on a site chipping-dominant ( ) : masses scattered over entire lattice. (zero-range process : ZRP) For , competition between diffusion and chipping → phase transitions from condensed phase into fluid phase. Braz. J. Phys. 30, 42 (2000) Zero Range Process (ZRP) Hopping A particle jumps out of the site at the rate and hops to a site with the Probability . Jumping A condensed phase, which a finite fraction of total particles condenses on a single site, arises or not according to , .

5. P(m) m Phase diagram Order parameter : P(m) = mass distribution of a single site = Probability that a site has mass m in the steady state. Condensed phase Fluid phase

6. - SCMA model on complex networks scale-free networks (SFNs) 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). Degree distribution What about 0 < ω < ∞ case on SFNs ? What is the effect of underlying topology like SFNs on condensation transitions ?

7. Random and scale free networks of N = # of nodes = 10000, K = # of links = <k>N/2 = 20000 <k> = average degree = 4 in our simulations (a) = Random net. (RN) (b) = SFN of

8. 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. (2) SFNs of • (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. 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. - Average mass of a node with degree k , At , it was shown that complete condensation always takes place on SFNs. What about on RNs ? ZRP on SFN PRL 94, 198701 (2005) For , the behavior in the condensed phase ?

12. = the probability of finding a random walker on degree k in k-space

13. Condensed phase Fluid phase Condensed phase (no Fluid phase) - Existence of infinite aggregation Condensed phase Or 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. : survival probability finite : average life time

14. For any - Finite sized results for RWs on SFNs : The number of visited distinct sites of a random walker : The saturation time : The average life time

15. : the distance between two random walkers (the shortest path) : the number of nodes

16. Diffusive capture process = lamb-lion problem Static trap On regular lattice Yes random walker to random walker random walker On networks No!! Hub effect

17. What is the survival probability of a diffusing lamb which is hunted by hungry lions? 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 PRB 39, 889 (1989), JSP 34, 667 (1984), PRB 29, 239 (1984), JPA 21, L89 (1988) 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). Diffusion-controlled reaction First passage phenomena of RWs Survival probability of a diffusing lamb P.L.Krapivsky and S.Redner J.Phys.A 29, 5347 (1996) On regular lattice

18. One of interesting applications can be found in searching information over the Internet. 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. Our model Korean Phys. Soc. 48, S202 (2006) Initially a lamband lions are randomly distributed to the nodes on the networks. At each time step, a lamb and lions take random walks simultaneously. If the lamb meets the lion, the lamb is captured. Degree distribution

19. We measure and on LSFNs with various and network size .

20. We measure the average life time and the survival probability on TSFNs.

21. 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. PRL 92, 118701 (2004). Relation between degrees and capture events We measure the number of captures occurring at a node with degree . 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. 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. Application Complex Network Lamb Information packet Lion Query packet 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. P2P systems are distributed systems in which nodes exchange files directly with each other. Implementing an efficient searching algorithm is the key to a better performance of P2P protocol design.

25. query packet s s T T 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. Each node forwards the received query packets to all of its nearest neighbors until the pre-assigned time-to-live (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/)

26. In general, the degree distribution of P2P networks follows the power law, with , or highly skewed fat-tailed distributions. Degree distribution of P2P network (=> Expect attracting hubs) 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) => exists effective attractor (Hubs) 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. Propose an efficient algorithm information packet query packet s n-lion and lamb algorithm (NLL) i) Each node sends out an information packet (names of files + IP address). (Each of these packets takes random walks along the P2P connections.) 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 packetis discarded but the information packets continue random walks for the next query. One query event

28. The average traffic of each algorithm. FB generates around 50 times more traffic than NLL on the average. 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. 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 average searching time : the number of available files on a network The average searching time of NLL on SFNs is, at least, 10 times faster than n-RW on SFNs. increases almost linearly for n-RW . However, NLL grows as for small , but The average searching times satisfy seems to be less than 0.5 or becomes saturated to a fixed finite value The difference between NLL and 2-RW, for large . , increases as . Since the probability that two random walks visits a node with degree is proportional to , the hub can collect all information packets. hub (More scalable searching time than n-RW)

30. 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 TSFNs, On LSFNs,

31. 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. SFNs : survival probability If Then is finite. Static trap On regular lattice Yes random walker to random walker random walker On networks No!! Hub effect