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Some work looking at the theoretical implications of the use of supernodes in peer-to-peer networks.

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- 1. Necessity of Supernodes David Hadaller, Kevin Regan and Tyrel Russell
- 2. P2P Networks Internet application Application Layer Overlay Networks 2
- 3. Problems of P2P Networks Scalability Security Anonimity Fault Tolerant 3
- 4. P2P Structured Unstructured Networks Centralized Napster ??? Decentralized Chord/Tapestry Gnutella/Kazaa 4
- 5. The centralized and unstructured decentralized systems are not scalable in their vanilla form Structured decentralized networks are scalable But require the topology to be of a certain structure 5
- 6. Alternatives... The addition of supernodes seems to make unstructured decentralized networks scalable Why? 6
- 7. Three Directions Traditional Decentralized Unstructured (Flooding) P2P Networks Graph Theoretical Bounds Decentralized Structured P2P Networks 7
- 8. What is a Peer-to- Peer System? Most commonly used for ﬁle sharing Napster released 1999, shutdown 2001 Gnutella released 2000 Kazaa released 2001 Today: millions of users sharing petabytes of data
- 9. System Interaction User issues a keyword search Network returns list of peers contain matching ﬁles
- 10. Architecture A peer operates as both a client and a server Idea: Everyone is equal, everyone cooperates (both not true) File sharing: -FastTrack (Kazaa, Kazaa Lite) -Gnutella (Morpheus, LimeWire)
- 11. How do Unstructured Peer- to-Peer Systems Work?
- 12. Scalability Issues Gnutella not scalable
- 13. Making Gnutella Scalable Chawathe et al. obtained 3 to 5x improvement in system capacity Adding nodes of higher degree Kazaa
- 14. Host Characteristics 6% users very well connected 10% of sessions >5 hrs
- 15. Impact of File Sharing Study at the University of Washington P2P accounts for 43% Internet traﬃc Web accounts for 14%
- 16. Graph Theory Formulate the problem as a Graph Theory Problem Let the P2P network be a graph G where G is a set of vertices V and there exists an edge between two nodes u,v∈V when u is a neighbour of v in the overlay network 16
- 17. Some Deﬁnitions Maximum Degree We denote ∆ as the maximum over the degree of all vertices of a graph. Minimum (u,v)-path We denote the minimum path between two vertices u, v, u ≠ v, as d(u, v). Diameter We deﬁne diameter D as the length of the max d(u, v) for all u, v ∈ V 17
- 18. Moore Bound Upper bound on number of vertices in a graph with max degree Δ and diameter D D n ≤ 1 + ∆ 1 + (∆ − 1) + · · · + (∆ − 1) ∆(∆ − 1)D − 2 ≤ = n0 (D, ∆) ∆−2 n(∆−2)+2 log ∆ D≥ log(∆ − 1) 18
- 19. Moore Bound d(u,v) = D v d(u,v) = 2 v v d(u,v) = 1 v v !-1 v v !-1 v u ! v v n ≤ 1 + ∆ 1 + (∆ − 1) + · · · + (∆ − 1)D 19
- 20. Special Graphs Moore Graph Have equivilence in the Moore bound Diameter of 2: Moore graphs only exist with Δ = 3, 7, 57 Diameter more than 3: No Moore graphs exist de Bruijn Graph 8 4 Leland-Solomon Graph 6 3 9 2 7 1 5 0 20
- 21. Random Graphs We are interested in property Q, so for a graph of size n Enumerate the number of possible graphs with Q If proportion of graphs with Q → 1 as n → ∞ Then we say that almost every graph has property Q 21
- 22. Random Graphs Some properties: almost every graph has diameter 2 is k-connected for a ﬁxed k > 2 has no complete subgraph Hk where k > 2log2n 22
- 23. Random Graphs For a ﬁxed Δ ≥ 3 almost every Δ-regular graph has diameter 6∆ D ≥ log∆−1 n + log∆−1 logn − log∆−1 +1 ∆−2 23
- 24. Graph Theory Summary There is a theorectical limit to search in a P2P network As logn increases the time to search will increase by roughly O(logn) 24
- 25. Distributed Hash Tables Distributed Hash Tables (DHTs) are an implementation of a decentralized, structured peer-to-peer network The diameter of the network scales logarithmically with the size of the network Node degree varies from O(1) to O(log n) 25
- 26. Implementations Tapestry - Plaxton Mesh CAN - d-dimensional Torus Pastry - Logical Ring Chord - Logical Ring with Chords Koorde - Logical Ring with deBruijn Graphs HyperCuP - Hypercube 26
- 27. Number of Name Degree Diameter Nodes Tapestry N O(log N) O(log N) CAN N O(d) O(n1/d) Pastry N O(log N) O(log N) Chord N O(log N) O(log N) Koorde N O(1) O(log N) HyperCuP N O((b-1)l) O(log N)
- 28. Conclusion Graph theory gives us a way to bound almost any random graph with a given number of nodes and a maximum degree Flooding P2P networks have looked at scalability from an empirical perspective Distributed Hash Tables provide a scalable method for P2P at the cost of 28
- 29. Questions? 29

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