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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. This lecture.

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cse 522 algorithmic and economic aspects of the internet

CSE 522 – Algorithmic and Economic Aspects of the Internet

Instructors:

Nicole Immorlica

Mohammad Mahdian

news break nobel prize in economics
News Break: Nobel Prize in Economics

Robert Aumann

Thomas Schelling

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

this lecture
This lecture

How to find short paths in

small-world networks.

small world networks recap
Small-World Networks, recap
  • Milgram’s Experiment (Psychology Today, 1967)
    • Social networks have short paths
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 paths1
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
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
tradeoff
Tradeoff

Guaranteed path length

highly local

uniform

Distribution

decentralized algorithm
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
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
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
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)
small world networks
Small-World Networks
  • Milgram’s Experiment (Psychology Today, 1967)
    • Social networks have short paths
    • Strangers can find these paths
discussion
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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