Minimizing Seed Set Selection with Probabilistic Guarantee ... fileKleinberg’s Small-World...
Transcript of Minimizing Seed Set Selection with Probabilistic Guarantee ... fileKleinberg’s Small-World...
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Milgramโ67: Six Degrees of Separation
โข 296 People in Omaha, NE, were given a
letter, asked to try to reach a stockbroker in
Boston, MA, via personal acquaintances
โข 20% reached target
โข average number of โhopsโ in the completed
chains = 6.5
โข Why are chains so short?
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Watts & Strogatzโ98: Small-World Model
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โข Propose two important features of
the small-world networks
โ Low diameter
โ High clustering
โข Propose a random rewiring model
โข But one feature of the Milgram
experiment is missing!
Kleinbergโ00: Navigable Small World
โข Notice the feature of efficient
decentralized navigation in Milgramโs
experiment --- navigability
โ Subjects only use local information to
navigate the network
โข Adjust the rewiring model of
Watts&Strogatz
โข Prove the navigability of the model at a
critical parameter setting
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Kleinbergโs Small-World Model
โข Put ๐๐ people on a ๐-dimensional grid
โข Connect each to its immediate grid neighbors
โข Add one directed long-range link per node
โ Node ๐ข connects to ๐ฃ with probability
Pr ๐ข โ ๐ฃ 1
๐ ๐ข, ๐ฃ ๐
โ ๐ โ [0,+โ) is connection preference: โข ๐ close to โ: prefers to connect to nodes in the vicinity
โข ๐ close to 0: prefer to connect to faraway nodes equally as neighboring nodes
โข ๐ = 0: reduces to the Watts & Strogatz model (random network)
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Decentralized Routing in Kleinbergโs Model
โข Decentralized greedy routing:โ given a target ๐ก, every node ๐ข routes the message for ๐ก to ๐ขโs
neighbor (local or long-range contact) closest to ๐ก in grid distance
โข Main result:โ ๐ = ๐: routing is efficient ๐(log2 ๐) --- navigable network
โ ๐ < ๐ or ๐ > ๐: routing is not efficient ฮฉ ๐๐ for some ๐ related to ๐.
โข Intuition:โ ๐ > ๐: long-range links are too close to move towards the target
โ ๐ < ๐: long-range links are too random to zoom into the target
โ ๐ = ๐: right balance between fast moving towards and zooming into the target
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๐ > ๐
๐ < ๐
๐ = ๐What is the parameter in real networks?
Empirical Validation
โข Liben-Nowell et al. โ05:
โ fractional dimension ๐ผ for non-uniform population distribution
๐ค: ๐ ๐ข,๐ค โค ๐ ๐ข, ๐ฃ = ๐ โ ๐ ๐ข, ๐ฃ ๐ผ
โ When ๐ = ๐ผ, the network is navigable
โ LiveJournal dataset (495K nodes): ๐ผ โ 0.8, ๐ = 1.2
โข Ours:
โ Renren dataset (10 mil nodes)
โ ๐ผ โ 1.0, ๐ = 0.9
โข 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
โข 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 reciprocityโข New insight: balance between connection reciprocity connection distance leads to
network navigability!
โข Other earlier attempts [Mathias&Gopalโ01, Clauset&Mooreโ03, Sandberg&Clarkeโ06, Chaintreau et al.โ08, Hu et al.โ11]
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
โข Players: ๐๐ nodes in a ๐-dimensional grid
โข Strategies: connection preference ๐๐ข โ [0, +โ) of node ๐ข
โ ๐ข has a long-range link to ๐ฃ with probability Pr ๐ข โ ๐ฃ 1
๐ ๐ข,๐ฃ ๐๐ข
โ Indicate the preference of ๐ข in connecting to local or remote nodes
โ For convenience, we discretize ๐๐ข โ {0, ๐พ, 2๐พ, 3๐พ, โฆ }, ๐พ --- granularity
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Payoff function: Distance-Reciprocity Tradeoff
โข First attempt: average routing distance as payoff
โ Random network (๐๐ข โก 0) seems to be the equilibrium
โข Novel payoff function: distance reciprocity tradeoff
๐๐ข ๐๐ข, ๐ซโ๐ข =
๐ฃโ ๐ข
๐๐ข ๐ฃ, ๐๐ข ๐ ๐ข, ๐ฃ ร
๐ฃโ ๐ข
๐๐ข ๐ฃ, ๐๐ข ๐๐ฃ ๐ข, ๐๐ฃ
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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
Theoretical Analysis
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Uniform Nash Equilibria
โข Theorem 1. For sufficiently large ๐, If everyone else plays the
same strategy (๐ซโ๐ข โก ๐ ), the best response of ๐ข is
๐ต๐ข ๐ซโ๐ข โก ๐ = ๐ if ๐ > 00 if ๐ = 0
โข Corollary 2. There are only two uniform Nash equilibria:
โ Navigable small-world network (๐ซ โก ๐)
โ Random small-world network (๐ซ โก 0)
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Intuition (Proof Sketch)
โข ๐ซโ๐ข โก ๐ , ๐ > 0: everyone else (slightly) prefers local nodes
โ If ๐๐ข < ๐, ๐ขโs long-rang links achieve good distance but poor reciprocity
โ If ๐๐ข > ๐, ๐ขโs long-rang links achieve good reciprocity but poor distance
โ If ๐๐ข = ๐, ๐ขโs long-rang links achieve best balance between distance and
reciprocity
โข ๐ซโ๐ข โก ๐ , ๐ = 0: everyone else connects uniformly to other nodes
โ Reciprocity is a constant regardless of ๐๐ขโ Thus, set ๐๐ข = 0 to achieve the largest average grid distance
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Stability of Navigable Small World ---
Collusion Toleration
โข What if a group of players (instead of one player) want to collude
and deviate together for better payoff?
โข Theorem 3. Navigable small-world network (๐ซ โก 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 ๐๐ข โ ๐, ๐ขโs payoff is strictly worse than
its payoff in the navigable small world.
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Stability of Navigable Small World ---
Random Perturbation Toleration
โข What if perturbations occur at random players, without increasing payoff constraint?
โข Theorem 4. In the navigable small-world network (๐ซ โก ๐), even if every node has an independent probability of 1 โ ๐โ๐ (for small ๐ > 0) to be perturbed to an arbitrary strategy, with high probability every player ๐ข wants to set ๐๐ข = ๐ as its best strategy after the perturbation.
โข Intuition: a small portion of randomly distributed nodes holding ๐๐ข = ๐ is enough to pull everyone to ๐๐ข = ๐.
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Instability of Non-navigable Equilibria ---
Not Tolerating Collusions from a Small Group
โข Does any other Nash equilibrium tolerate Collusion? --- NO!
โข Theorem 5. No other equilibrium tolerates the collusion of 2๐โ๐
(for small ๐ > 0) fraction of players.
โข Intuition: Dual aspect of Theorem 4 --- a small portion of evenly
distributed nodes collude and set ๐๐ข = ๐ is enough to pull
everyone to ๐๐ข = ๐.
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Instability of Non-navigable Equilibria ---
Not Tolerating Random Perturbations
โข Does any other Nash equilibrium tolerate random perturbation?
โ No if perturbed players could set strategy to ๐ (by Theorem 4)
โข What if the target strategy set after perturbation does not contain ๐?
โข Theorem 6. In the random small world (๐ซ โก 0), for an arbitrary finite
target strategy set ๐ after the perturbation (๐ฝ = max ๐ > 0), if every
๐ข is perturbed to every ๐ โ ๐ โ {0} with probability at least ๐โ
๐โ1 ๐
๐+๐ฝ
(for small ๐ > 0), then with high probability the best strategy for every
๐ข after the perturbation is ๐๐ข = ๐.
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Implication of Theoretical Analysis
โข 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)
โข Small size collusion can bring the system back to navigable small world
โข 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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โข At time step 0, each player is perturbed independently with probability ๐.
โข At time step ๐ก > 0, every player picks the best strategy based on the strategies of others in the previous step.
DRB Game with Limited Knowledge (1)
โข Scenario 1: knowing friendsโ strategies.
โข At every step ๐ก โฅ 0, each player ๐ข creates ๐ out-going long-range links based on her current strategy, and learns the (noisy) connection preferences of these ๐long-range contacts, then infer othersโ connection preferences.
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DRB Game with Limited Knowledge (2)
โข Scenario 2: No information about othersโ strategies.
โข 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.
โข 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
โข 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
โข 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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