Small-world networks

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 )

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

Watts and Strogatz 1998 Research originally inspired by Watts' 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 ( rewiring with probability p=0) Small-world graphs ( p = small, close to 0) Completely random ( p=1) What is rewiring?

Completely regular A ring lattice N=20 K= 4 (each node has k neighbors ) High clustering coefficient and high diameter. C = 3/4, L ~ N/k

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

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

Clustering Coefficient and Characteristic Path Length p in log-scale

Milgram’s experiment revisited Milgram’s experiment showed that (a) There exist short paths in large networks that connect individuals (b) People are able to find these short paths using a simple, possibly greedy, decentralized algorithm Small world models only take care of (a) What about (b)? Watts-Strogatz model is of little help.

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!)

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

Results Theorem 2. When r=0, no decentralized algorithm can find the short chains (even if they exist). The expected no of hops to connect is O(n 2/3 ). [ Uniform distribution prevents a decentralized algorithm from using any clue from the geometry of the grid].

Kleinberg’s model Consider a directed 2-dimensional lattice where each node u has four neighbors at lattice distance = 1 For each vertex u add q shortcuts choose vertex v as the destination of the shortcut with probability proportional to [d(u,v)] -r when r = 0, we have uniform probabilities