Minimizing Seed Set Selection with Probabilistic Guarantee ... fileKleinberg’s Small-World...

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Transcript of Minimizing Seed Set Selection with Probabilistic Guarantee ... fileKleinberg’s Small-World...

Page 1: Minimizing Seed Set Selection with Probabilistic Guarantee ... fileKleinberg’s Small-World Model •Put ๐‘›๐‘˜people on a ๐‘˜-dimensional grid •Connect each to its immediate

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Page 2: Minimizing Seed Set Selection with Probabilistic Guarantee ... fileKleinberg’s Small-World Model •Put ๐‘›๐‘˜people on a ๐‘˜-dimensional grid •Connect each to its immediate

Milgramโ€™67: Six Degrees of Separation

โ€ข 296 People in Omaha, NE, were given a

letter, asked to try to reach a stockbroker in

Boston, MA, via personal acquaintances

โ€ข 20% reached target

โ€ข average number of โ€œhopsโ€ in the completed

chains = 6.5

โ€ข Why are chains so short?

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Watts & Strogatzโ€™98: Small-World Model

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โ€ข Propose two important features of

the small-world networks

โ€“ Low diameter

โ€“ High clustering

โ€ข Propose a random rewiring model

โ€ข But one feature of the Milgram

experiment is missing!

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Kleinbergโ€™00: Navigable Small World

โ€ข Notice the feature of efficient

decentralized navigation in Milgramโ€™s

experiment --- navigability

โ€“ Subjects only use local information to

navigate the network

โ€ข Adjust the rewiring model of

Watts&Strogatz

โ€ข Prove the navigability of the model at a

critical parameter setting

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Kleinbergโ€™s Small-World Model

โ€ข Put ๐‘›๐‘˜ people on a ๐‘˜-dimensional grid

โ€ข Connect each to its immediate grid neighbors

โ€ข Add one directed long-range link per node

โ€“ Node ๐‘ข connects to ๐‘ฃ with probability

Pr ๐‘ข โ†’ ๐‘ฃ 1

๐‘‘ ๐‘ข, ๐‘ฃ ๐‘Ÿ

โ€“ ๐‘Ÿ โˆˆ [0,+โˆž) is connection preference: โ€ข ๐‘Ÿ close to โˆž: prefers to connect to nodes in the vicinity

โ€ข ๐‘Ÿ close to 0: prefer to connect to faraway nodes equally as neighboring nodes

โ€ข ๐‘Ÿ = 0: reduces to the Watts & Strogatz model (random network)

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Decentralized Routing in Kleinbergโ€™s Model

โ€ข Decentralized greedy routing:โ€“ given a target ๐‘ก, every node ๐‘ข routes the message for ๐‘ก to ๐‘ขโ€™s

neighbor (local or long-range contact) closest to ๐‘ก in grid distance

โ€ข Main result:โ€“ ๐‘Ÿ = ๐‘˜: routing is efficient ๐‘‚(log2 ๐‘›) --- navigable network

โ€“ ๐‘Ÿ < ๐‘˜ or ๐‘Ÿ > ๐‘˜: routing is not efficient ฮฉ ๐‘›๐‘ for some ๐‘ related to ๐‘Ÿ.

โ€ข Intuition:โ€“ ๐‘Ÿ > ๐‘˜: long-range links are too close to move towards the target

โ€“ ๐‘Ÿ < ๐‘˜: long-range links are too random to zoom into the target

โ€“ ๐‘Ÿ = ๐‘˜: right balance between fast moving towards and zooming into the target

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๐‘Ÿ > ๐‘˜

๐‘Ÿ < ๐‘˜

๐‘Ÿ = ๐‘˜What is the parameter in real networks?

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Empirical Validation

โ€ข Liben-Nowell et al. โ€˜05:

โ€“ fractional dimension ๐›ผ for non-uniform population distribution

๐‘ค: ๐‘‘ ๐‘ข,๐‘ค โ‰ค ๐‘‘ ๐‘ข, ๐‘ฃ = ๐‘ โ‹… ๐‘‘ ๐‘ข, ๐‘ฃ ๐›ผ

โ€“ When ๐‘Ÿ = ๐›ผ, the network is navigable

โ€“ LiveJournal dataset (495K nodes): ๐›ผ โ‰ˆ 0.8, ๐‘Ÿ = 1.2

โ€ข Ours:

โ€“ Renren dataset (10 mil nodes)

โ€“ ๐›ผ โ‰ˆ 1.0, ๐‘Ÿ = 0.9

โ€ข Others show similar results

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Why is connection preference

close to the critical value of grid

dimension in the real world !?

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Our Proposal

โ€ข Game-theoretic formation of navigable small world

โ€“ strong theoretical and empirical support

โ€“ Navigable small world network is not only one equilibrium, but is the only one tolerating both collusions and random perturbations

โ€“ Surprising connection with relationship reciprocityโ€ข New insight: balance between connection reciprocity connection distance leads to

network navigability!

โ€ข Other earlier attempts [Mathias&Gopalโ€™01, Clauset&Mooreโ€™03, Sandberg&Clarkeโ€™06, Chaintreau et al.โ€™08, Hu et al.โ€™11]

โ€“ Use node or link dynamics, mostly by simulation, some theoretical results on approximate settings for the navigability, none connects to reciprocity

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Game Theoretic Model

โ€ข Players: ๐‘›๐‘˜ nodes in a ๐‘˜-dimensional grid

โ€ข Strategies: connection preference ๐‘Ÿ๐‘ข โˆˆ [0, +โˆž) of node ๐‘ข

โ€“ ๐‘ข has a long-range link to ๐‘ฃ with probability Pr ๐‘ข โ†’ ๐‘ฃ 1

๐‘‘ ๐‘ข,๐‘ฃ ๐‘Ÿ๐‘ข

โ€“ Indicate the preference of ๐‘ข in connecting to local or remote nodes

โ€“ For convenience, we discretize ๐‘Ÿ๐‘ข โˆˆ {0, ๐›พ, 2๐›พ, 3๐›พ, โ€ฆ }, ๐›พ --- granularity

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Payoff function: Distance-Reciprocity Tradeoff

โ€ข First attempt: average routing distance as payoff

โ€“ Random network (๐‘Ÿ๐‘ข โ‰ก 0) seems to be the equilibrium

โ€ข Novel payoff function: distance reciprocity tradeoff

๐œ‹๐‘ข ๐‘Ÿ๐‘ข, ๐ซโˆ’๐‘ข =

๐‘ฃโ‰ ๐‘ข

๐‘๐‘ข ๐‘ฃ, ๐‘Ÿ๐‘ข ๐‘‘ ๐‘ข, ๐‘ฃ ร—

๐‘ฃโ‰ ๐‘ข

๐‘๐‘ข ๐‘ฃ, ๐‘Ÿ๐‘ข ๐‘๐‘ฃ ๐‘ข, ๐‘Ÿ๐‘ฃ

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Average grid distance of a long-

range link --- prefer faraway nodes

to get diverse information

Average probability that the long-

range link is reciprocated --- prefer

mutual relationship

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Theoretical Analysis

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Uniform Nash Equilibria

โ€ข Theorem 1. For sufficiently large ๐‘›, If everyone else plays the

same strategy (๐ซโˆ’๐‘ข โ‰ก ๐‘ ), the best response of ๐‘ข is

๐ต๐‘ข ๐ซโˆ’๐‘ข โ‰ก ๐‘  = ๐‘˜ if ๐‘  > 00 if ๐‘  = 0

โ€ข Corollary 2. There are only two uniform Nash equilibria:

โ€“ Navigable small-world network (๐ซ โ‰ก ๐‘˜)

โ€“ Random small-world network (๐ซ โ‰ก 0)

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Intuition (Proof Sketch)

โ€ข ๐ซโˆ’๐‘ข โ‰ก ๐‘ , ๐‘  > 0: everyone else (slightly) prefers local nodes

โ€“ If ๐‘Ÿ๐‘ข < ๐‘˜, ๐‘ขโ€™s long-rang links achieve good distance but poor reciprocity

โ€“ If ๐‘Ÿ๐‘ข > ๐‘˜, ๐‘ขโ€™s long-rang links achieve good reciprocity but poor distance

โ€“ If ๐‘Ÿ๐‘ข = ๐‘˜, ๐‘ขโ€™s long-rang links achieve best balance between distance and

reciprocity

โ€ข ๐ซโˆ’๐‘ข โ‰ก ๐‘ , ๐‘  = 0: everyone else connects uniformly to other nodes

โ€“ Reciprocity is a constant regardless of ๐‘Ÿ๐‘ขโ€“ Thus, set ๐‘Ÿ๐‘ข = 0 to achieve the largest average grid distance

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Stability of Navigable Small World ---

Collusion Toleration

โ€ข What if a group of players (instead of one player) want to collude

and deviate together for better payoff?

โ€ข Theorem 3. Navigable small-world network (๐ซ โ‰ก ๐‘˜) is a strong

Nash equilibrium for sufficiently large ๐‘›.

โ€“ Strong Nash: no collusion group of any size could successfully deviate

from the equilibrium without someone in the group got hurt in payoff.

โ€“ Reason: In any strategy profile, if ๐‘Ÿ๐‘ข โ‰  ๐‘˜, ๐‘ขโ€™s payoff is strictly worse than

its payoff in the navigable small world.

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Stability of Navigable Small World ---

Random Perturbation Toleration

โ€ข What if perturbations occur at random players, without increasing payoff constraint?

โ€ข Theorem 4. In the navigable small-world network (๐ซ โ‰ก ๐‘˜), even if every node has an independent probability of 1 โˆ’ ๐‘›โˆ’๐œ€ (for small ๐œ€ > 0) to be perturbed to an arbitrary strategy, with high probability every player ๐‘ข wants to set ๐‘Ÿ๐‘ข = ๐‘˜ as its best strategy after the perturbation.

โ€ข Intuition: a small portion of randomly distributed nodes holding ๐‘Ÿ๐‘ข = ๐‘˜ is enough to pull everyone to ๐‘Ÿ๐‘ข = ๐‘˜.

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Instability of Non-navigable Equilibria ---

Not Tolerating Collusions from a Small Group

โ€ข Does any other Nash equilibrium tolerate Collusion? --- NO!

โ€ข Theorem 5. No other equilibrium tolerates the collusion of 2๐‘›โˆ’๐œ€

(for small ๐œ€ > 0) fraction of players.

โ€ข Intuition: Dual aspect of Theorem 4 --- a small portion of evenly

distributed nodes collude and set ๐‘Ÿ๐‘ข = ๐‘˜ is enough to pull

everyone to ๐‘Ÿ๐‘ข = ๐‘˜.

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Instability of Non-navigable Equilibria ---

Not Tolerating Random Perturbations

โ€ข Does any other Nash equilibrium tolerate random perturbation?

โ€“ No if perturbed players could set strategy to ๐‘˜ (by Theorem 4)

โ€ข What if the target strategy set after perturbation does not contain ๐‘˜?

โ€ข Theorem 6. In the random small world (๐ซ โ‰ก 0), for an arbitrary finite

target strategy set ๐‘† after the perturbation (๐›ฝ = max ๐‘† > 0), if every

๐‘ข is perturbed to every ๐‘  โˆˆ ๐‘† โˆ– {0} with probability at least ๐‘›โˆ’

๐‘˜โˆ’1 ๐œ€

๐‘˜+๐›ฝ

(for small ๐œ€ > 0), then with high probability the best strategy for every

๐‘ข after the perturbation is ๐‘Ÿ๐‘ข = ๐‘˜.

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Implication of Theoretical Analysis

โ€ข Navigable small world is the only stable state of the system

โ€“ Once in it, any size of collusion, or large random perturbation cannot

shake the system out of navigable small world

โ€“ If the system temporarily gets stuck at other states (other equilibria)

โ€ข Small size collusion can bring the system back to navigable small world

โ€ข Small size random perturbation can also bring the system back to navigable

small world

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Grid size: 100 x 100

Empirical Evaluation

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Stability of NE under Perturbation

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โ€ข At time step 0, each player is perturbed independently with probability ๐‘.

โ€ข At time step ๐‘ก > 0, every player picks the best strategy based on the strategies of others in the previous step.

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DRB Game with Limited Knowledge (1)

โ€ข Scenario 1: knowing friendsโ€™ strategies.

โ€ข At every step ๐‘ก โ‰ฅ 0, each player ๐‘ข creates ๐‘ž out-going long-range links based on her current strategy, and learns the (noisy) connection preferences of these ๐‘žlong-range contacts, then infer othersโ€™ connection preferences.

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DRB Game with Limited Knowledge (2)

โ€ข Scenario 2: No information about othersโ€™ strategies.

โ€ข a player creates a certain number of links with the current strategy, and computes the payoff by multiplying the average link distance and the percentage of reciprocal links.

โ€ข at each step each player only has one chance to slightly modify her current strategy. If the new strategy yields better payoff, the player would adopt the new strategy.

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Conclusion & Future Work

โ€ข The first model connecting reciprocity with navigability

Distance x Reciprocity Navigability

โ€“ Navigable small world is the only stable system state

โ€“ Strong theoretical and empirical support

โ€ข Future workโ€“ Non-uniform population distribution

โ€“ Arbitrary base graph

โ€“ Other more general long-range link distribution than power-law

โ€“ Integrating with node mobility and link dynamics

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Thanks, and questions?

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