CSE 522 – Algorithmic and Economic Aspects of the Internet

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

2. News Break: Nobel Prize in Economics Robert Aumann Thomas 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 graph + a few random edges = low 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 Guaranteed path length highly local uniform Distribution

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(log2n). • Theorem 2: (a) For 0 · r < 2, the expected delivery time of any decentralized algorithm is (n(2 – r)/3). (b) For 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 2j· 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(log2 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 (log2n) [Barriere, Fraigniaud, Kranakis, Krizanc] • NoN greedy routing is (log n / loglog n) in other models [Manku, Naor, Wieder]