Replication Strategies in Unstructured Peer-to-Peer Networks

1. Replication Strategies in Unstructured Peer-to-Peer Networks Edith Cohen Scott Shenker This is a modified version of the original presentation by the authors

2. Search in Basic P2P Architectures • Centralized: central directory server. (Napster) • Decentralized: • Structured (DHTs): Only exact-match queries, tightly controlled overlay. • Unstructured: (Gnutella, FastTrack); search is “blind” - probed peers are unrelated to query.

3. Replication in P2P architectures • No proactive replication (Gnutella) • Hosts store and serve only what they requested • A copy can be found only by probing a host with a copy • Proactive replication of “keys” (= meta data + pointer) for search efficiency (FastTrack, DHTs) • Proactive replication of “copies” – for search and download efficiency, anonymity. (Freenet)

4. QUESTION How to use replication to improve search efficiency in unstructured networks with a proactive replication mechanism ?

5. Search and replication model Unstructured networks with replication of keys or copies. Peers probed (in the search and replication process) are unrelated to query/item - Probe success likelihood can not be better, on average, than random probes. • Replication: Each host can store up to r copies (or keys=metadata+pointer) of items. Goal: minimize average search size(number of probes till query is satisfied) Search: probe hosts, uniformly at random, until the query is satisfied (or the search max size is exceeded)

6. Search size • Query is soluble if there are sufficiently many copies of the item. • Query is insoluble if item is rare or non existent. What is the search size of a query ? • Insoluble queries: maximum search size • Soluble queries: number of nodes a query need to visit until the answer is found. We look at the Expected Search Size (ESS) of each item. The ESS is inversely proportional to the fraction of peers with a copy of the item.

7. Search Example 4 probes 2 probes

8. Notations • m items with relative query rates • n nodes (peers), each has a uniform capacity r • R = n ris the total available space • ri = number of copies of item i. Thus pi = ri/R is the fraction of the total space allocated to item i. • Si pi = 1 • qi = normalized query rate for item i. Thus • Si qi= 1

9. Notations • Allocation p = (r1/R, r2/R, …, rm/R) • Areplication strategy is a mapping from q to p. • Assumption R ≥ m ≥ r. (If m < r, then one can copy every item in all the nodes. If R < m then no allocation can store a copy of all m objects)

10. Expected Search Size (ESS) • m items with relative query rates • q1 > q2 > q3 > … > qm. Siqi = 1 • Search size for ith item is a geometric r.v. with mean Ai = 1/(r pi). • ESS is Si qiAi = (Si qi / pi)/r Allocation: p1, p2, p3,…, pmSi pi = 1 ri/n = r.piis the fraction of hosts storing a copy of i

11. Uniform and Proportional Replication • Two natural strategies: • Uniform Allocation: pi = 1/m • Simple, resources are divided equally • Proportional Allocation: pi = qi • “Fair”, resources per item proportional to demand • Reflects current P2P practices

12. Example: 3 items, q1=1/2, q2=1/3, q3=1/6 q1 > q2 > q3 Uniform Proportional Uniform and Proportional Replication

13. Basic Questions • How do Uniform and Proportional allocations perform/compare ? • Which strategy minimizes the Expected Search Size (ESS) ? • Is there a simple protocol that achieves optimal replication in decentralized unstructured networks ?

14. Insoluble queries What about the cost of soluble queries? Answer is more surprising … Search always extends to the maximum allowed search size. If we fix the available storage for copies, the query rate distribution, and the number if items that we wish to be “locatable”, then The maximum required search size depends on the smallest allocation of an item. Thus, Uniform allocation minimizes this maximum and thus the cost induced by insoluble queries.

15. Uniform and Proportional Allocations (ESS for soluble queries) • Lemma: The ESS under either Uniform or Proportional allocations is m/r • Independent of query rates (!!!) • Same ESS for Proportional and Uniform (!!!)

16. Proof outline Proportional: Average Search Size is (Siqi / pi)/r = (Siqi / qi)/r = m/r Uniform: Average Search Size is (Siqi / pi)/r = (Sim qi)/r = (m/r) Si qi = m/r

17. Proportional and Uniform are the worst “reasonable” strategies (!!!) Space of Possible Allocations • Theorem2: p is worse than Uniform/Proportional if • for all i, q i+1/q i> 1 (more popular gets less) OR • for all i, q i+1/q i> p i+1/p i(less popular gets less than “fair share”) • (These are unreasonable strategies) Definition: Allocation p1, p2, p3,…, pm is “in-between” Uniform and Proportional if for 1< i <m, q i+1/q i< p i+1/p i< 1 Theorem1: All (strictly) in-between strategies are (strictly) better than Uniform and Proportional

18. Worse than prop/uni More popular item gets less. SR Better than prop/uni Worse than prop/uni More popular gets more than its proportional share Space of allocations on 2 items Uniform Proportional p2/p1 q2/q1

19. So, what is the best strategy for soluble queries ?

20. Square-Root Allocation • Lies “In-between” Uniform and Proportional • Theorem: Square-Root allocation minimizes the ESS (on soluble queries) • Minimize Siqi / pi such thatSipi = 1 (pi)is proportional to square-root of(qi)

21. How much can we gain by using SR ? Zipf-like query rates

22. OPT is a hybrid of Uniform and SR Tuned to balance cost of soluble and insoluble queries. • OK • SR is best for soluble queries • Uniform minimizes cost of insoluble queries What is the optimal strategy?

23. 10^4 items, Zipf-like w=1.5 All Soluble 85% Soluble All Insoluble SR Uniform

24. We now know what we need. How do we get there?

25. Replication Algorithms • Uniform and Proportional are “easy” : • Uniform: When item is created, replicate its key in a fixed number of hosts. • Proportional: for each query, replicate the key in a fixed number of hosts Desired properties of algorithm: Fully distributed where peers communicate through random probes; minimal bookkeeping; and no more communication than what is needed for search. Converge to/obtain SR allocation when query rates remain steady.

26. Property of the process: <Ci> average value of C used to replicate ith item. Claim: If <Ci>/<Cj> remains fixed over time, and <Ci>, <Cj> >e, then pi/pjgqi <Ci>/qj <Cj> Model for Copy Creation/Deletion • Creation: after a successful search, C(s) new copies are created at random hosts. • Deletion: is independent of the identity of the item; copy survival chances are non-decreasing with creation time. (i.e., FIFO at each node)

27. Algorithm for square-root allocation needs to have <Ci> equal to or converge to a value inversely proportional to Creation/Deletion Process Corollary: then If

28. SR Replication Algorithms • Path replication: number of new copies C(s) is proportional to the size of the search (Freenet) • Converges to SR allocation (+reasonable conditions) • Convergence unstable with delayed creations • Sibling memory: each copy remembers the number of sibling copies, • Quickly “on target” • For “good estimates” need to find several copies. • Probe memory: each peer records number and combined search size of probes it sees for each item. C(S) is determined by collecting this info from number of peers proportional to search size. • Immediately “on target” • Extra communication (proportional to that needed for search).

29. Algorithm 1: Path Replication Number of new copies produced per query, <Ci>, is proportional to search size 1/pi Creation rate is proportional to qi <Ci> Steady state: creation rate proportional to allocation pi, thus

30. Simulation Delay = 0.25 * copy lifetime; 10000 hosts Path replication Sibling number Hosts with copy time

31. Summary • Random Search/replication Model: probes to “random” hosts • Proportional allocation – current practice • Uniform allocation – best for insoluble queries • Soluble queries: • Proportional and Uniform allocations are two extremes with same average performance • Square-Root allocation minimizes Average Search Size • OPT (all queries) lies between SR and Uniform • SR/OPT allocation can be realized by simple algorithms.