1. CSE 522 – Algorithmic and Economic Aspects of the Internet Instructors: Nicole Immorlica Mohammad Mahdian

2. News Break: Nobel Prize in Economics Robert AumannThomas Schelling …for having enhanced our understanding of conflict and cooperation through game-theory analysis.

3. This lecture How to find short paths in small-world networks.

4. Small-World Networks, recap Milgram’s Experiment (Psychology Today, 1967)  Social networks have short paths

5. Short Paths Why should short paths exist? Watts and Strogatz (Nature, 1998)  People know their neighbors – “local” contacts  and a few others – “long-range” contacts regular grapha few random edgeslow diameter+=

6. Short Paths Why should strangers be able to find them? Kleinberg (STOC, 2000): Suppose long- range contacts are drawn from a distribution which favors closer nodes  Gives navigational cues to message-passers  Increases path length There is a value for the tradeoff where strangers can find the paths!

7. Generative Model Start with an n £ n grid  Local contacts: connect each node to all nodes within lattice distance p  Long-range contacts: connect each node u to q random nodes v chosen independently with probability proportional to d(u,v) -r Generalizes Watts-Strogatz for r = 0 Biases long-range contacts towards closer neighbors when r > 0

8. Tradeoff Distribution uniform highly local Guaranteed path length

9. Decentralized Algorithm Node s must send message m to node t At any moment, current message holder u must pass m to a neighbor given only:  Set of local contacts of all nodes (grid structure)  Location on grid of destination node t  Location and long-range contacts of all nodes that have seen m (but not long-range contacts of nodes that have not seen m)

10. Delivery Time Definition: Expected delivery time is the expectation, over the choice of long-range contacts and a uniformly random source and destination, of the number of steps taken to deliver message.

11. Results [Kleinberg, 2000] Theorem 1: There is a decentralized algorithm A so that when r = 2 and p = q = 1, the expected delivery time of A is O(log 2 n). Theorem 2: (a) For 0 · r 2, the expected delivery time of any decentralized algorithm is  (n (r – 2)/(r – 1) ). (Constants depend on p, q, and r.)

12. Proof of Theorem 1 Algorithm: In each step, u sends m to his neighbor v which is closest (in lattice distance) to t. Proof Sketch:  Define phases based on how close m is to t: algorithm is in phase j if 2 j · dist(m,t) · 2 (j+1)  Prove we don’t spend much time any phase: expected time in phase j is at most log n for all j  Conclude since at most log n + 1 phases, and so expected delivery time is O(log 2 n)

13. Small-World Networks Milgram’s Experiment (Psychology Today, 1967)  Social networks have short paths  Strangers can find these paths

14. Discussion Generalizations of underlying structure  Higher dimensional lattices [Kleinberg]  Hierarchical network models [Kleinberg] Finding shorter paths  Greedy is  (log 2 n) [Barriere, Fraigniaud, Kranakis, Krizanc]  NoN greedy routing is  (log n / loglog n) in other models [Manku, Naor, Wieder]