Transcript Chord: A Scalable P2P Lookup protocol for Internet

Small-world networks
What is it?
Everyone talks about the small world phenomenon,
but truly what is it? There are three landmark
papers:
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Stanley Milgram (1967)
Duncan Watts & Steve Strogatz (1998)
Jon Kleinberg (2001 )
Milgram’s experiment
A person P in Nebraska was given a letter to
deliver to another person Q in Massachusetts. P
was told about Q’s address and occupation, and
instructed to send the letter to someone she knew
on a first-name basis in order to transmit the
letter to the destination as fast as possible.
Milgram’s experiment
Over many trials, the average number of
intermediate steps in a successful chain was found
to lie between 5 and 6.
Milgram’s experiment
Initial success rate was very low (5%). The follow-up
experiments made some modifications of the original
experiment.
The outcome of the experiment led to the term:
six degrees of separation
Other example of small world
Small world graphs are highly clustered like
regular lattices, yet paths of short length exist
between random peers. Example of such graphs
are
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Power grid of western US
Collaboration graph of movie actors
Neural network of worm C-elegans
Watts and Strogatz 1998
Research originally inspired by Watt’s efforts to
understand the synchronization of cricket chirps,
which show a high degree of coordination over long
ranges, as though the insects are being guided by
an invisible conductor.
Disease spreads faster over a small-world network.
Questions not answered
Why six degrees of separation? Any scientific
reason? What properties do these social
graphs have? Are there other situations in
which this model is applicable?
Time to reverse engineer this.
A characterization of graphs
Completely regular
Small-world graphs (N >> k >> ln (N) >>1)
Completely random
N = number of nodes, k = degree of each node
Completely regular
N=20
K= 4 (each node has k neighbors)
High clustering coefficient
and high diameter.
C = 3/6 = 1/2, L ~ N/k
L= diameter
The clustering coefficient for a vertex
is given by the proportion of links between
the vertices within its neighborhood
divided by the number of links that could
possibly exist between them.
A ring lattice
Completely random
LOW clustering coefficient
and LOW diameter.
C ~ k/n, L ~ log N
Small world graphs
With probability p rewire each link
in a regular graph to a randomly selected node
Small world graphs
Such a rewiring results in a graph that has a
high clustering coefficient but low diameter …
Small world graphs
L= diameter
C= Clustering coefficient
N >> k >> ln(N) >> 1
C
Guarantees that the graph
Is connected
L
0
0.01
1
p
Kleinberg’s question
Watts and Strogatz’s research only showed the
existence of short paths between arbitrary pair
of nodes. What is the guarantee that one can find
such a path for communication?
(If there is no algorithm for finding the short
path, then it is not of much value!)
Kleinberg’s Small-World Model
Embed the graph into a grid. Each node has links to every node at lattice distance p
(short range neighbors) & q long range links. Choose long-range links s.t. the
prob. to have a long range contact is proportional to 1/dr (r is a new parameter)
p = 1, q = 2
r = 2
Results
Theorem 1.
When r = 0, no decentralized algorithm can
find the short chains (even if they exist).
Results
Theorem 2.
When r=0, the expected no of hops needed
to connect with a peer is O(n2/3).
[Uniform distribution prevents a decentralized
algorithm from using any clue from the geometry
of the grid].