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
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)
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
