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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. - PowerPoint PPT Presentation
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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
How to find short paths in
small-world networks.
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
Tradeoff
Distribution
uniform highly local
Gu
ara
nte
ed
pat
h le
ngt
h
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)
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]