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CSE 522 – Algorithmic and Economic Aspects of the InternetPowerPoint Presentation

CSE 522 – Algorithmic and Economic Aspects of the Internet

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### CSE 522 – Algorithmic and Economic Aspects of the Internet

Instructors:

Nicole Immorlica

Mohammad Mahdian

News Break: Nobel Prize in Economics

Robert Aumann

Thomas Schelling

…for having enhanced our understanding of conflict and cooperation through game-theory analysis.

Small-World Networks, recap

- Milgram’s Experiment (Psychology Today, 1967)
- Social networks have short paths

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

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!

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

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)

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.

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.)

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)

- Define phases based on how close m is to t:

Small-World Networks

- Milgram’s Experiment (Psychology Today, 1967)
- Social networks have short paths
- Strangers can find these paths

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]

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