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

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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]