1. Small-world networks

2. What is it?
Everyone talks about the small world phenomenon, but truly what is it? There are three landmark papers:
Stanley Milgram (1967)
Duncan Watts & Steve Strogatz (1998)
Jon Kleinberg (2001 )

3. 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.

4. Milgram’s experiment
Over many trials, the average number of intermediate steps in a successful chain was found to lie between 5 and 6.

5. 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

6. 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
Power grid of western US
Collaboration graph of movie actors
Neural network of worm C-elegans

7. 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.

8. 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.

9. A characterization of graphs
Completely regular
Small-world graphs (N >> k >> ln (N) >>1)
Completely random

10. 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
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

11. Completely random LOW clustering coefficient and LOW diameter.
C ~ k/n, L ~ log N

12. Small world graphs With probability p rewire each link
in a regular graph to a randomly selected node

13. Small world graphs Such a rewiring results in a a graph that has a
a high clustering coefficient but low diameter …

14. Small world graphs L= typical distance between two nodes
C= Clustering coefficient
N >> k >> ln(N) >> 1 C
Guarantees that the graph
Is connected L 1
0.01 p

15. 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!)

16. 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

17. Results
Theorem 1.
When r = 0, no decentralized algorithm can find the short chains (even if they exist).

18. 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].