Correctness of Gossip-Based Membership under Message Loss

Maxim Gurevich, Idit Keidar

Technion

The Setting

•Many nodes – n▫10,000s, 100,000s, 1,000,000s, …

•Come and go▫Churn

•Fully connected network▫Like the Internet

•Every joining node knows some others▫(Initial) Connectivity

Membership: Each Node Needs To Know Some Live Nodes

•Applications▫Gossip partners ▫Unstructured overlay networks▫Gathering statistics

•Work best with random node samples▫Gossip algorithms converge fast▫Overlay networks are robust, good expanders▫Statistics are accurate

Membership Protocols

•Each node has a view ▫Set of node ids▫Supplied to the application▫Used by membership protocol for maintenance▫Modeled as a directed graph

u v

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v y w …

y

Desirable Properties

•Randomness…•Holy grail for samples: IID▫Each sample uniformly distributed▫Each sample independent of other samples

Avoid spatial dependencies among view entries Avoid correlations between nodes

▫Good load balance among nodes

What About Churn? Desirable Properties Cont’d

•Views should constantly evolve▫Remove failed nodes, add joining ones

•Views should evolve to IID from any state•Minimize temporal dependencies▫Dependence on the past should decay quickly ▫Useful for application requiring fresh samples

Do Existing Protocols Measure Up?

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Existing Work: Practical Protocols

•Studied only empirically▫Good load balance [Lpbcast, Jelasity et al 07] ▫Fast decay of temporal dependencies [Jelasity et al 07] ▫Induces spatial dependence

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v … w …

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zExample: Push protocol

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Existing Work: Analysis

•Analyzed theoretically [Allavena et al 05, Mahlmann et al 06]

▫ Uniformity, load balance, spatial independence ▫ Unrealistic assumptions

Atomic actions with bi-directional communication No message loss

▫ No bounds on decay of temporal dependencies

… … z …… … w …u v

w

v … w …

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zShuffle protocol

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Our Contribution: Bridge This Gap •Formally specify desirable properties outlined

above•A practical protocol▫Tolerates message loss, churn, failures▫No complex bookkeeping for atomic actions

•Formally prove the desirable properties▫Including under message loss

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Send & Forget Membership•The best of push and shuffle•Some view entries may be empty

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v … w … u w

u w

S&F: Message Loss

•Message loss▫Or no empty entries in v’s view

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u v

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S&F: Compensating for Loss

•Edges (view entries) disappear due to loss•Need to prevent views from emptying out•Keep the sent ids when too little ids in view

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u v

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S&F: Advantages over Other Protocols

•No bi-directional communication▫No complex bookkeeping▫Tolerates message loss

•Simple▫Amenable to formal analysis

Easy to implement

•Proving all desirable properties▫Analytical: degrees distribution w/out loss

Used in setting duplication threshold▫Markov 1: degree distribution with loss▫Markov 2: Markov Chain of reachable global states

IID samples, Temporal Independence

•Hold even under (reasonable) message loss!

Key Contribution: Analysis

Analytic Degree Distribution

•Similar (better) to that of a random graph•Validated by a more accurate Markov model

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0 10 20 30 40Node indegree

Binomial

S&F Analytical

S&F Markov

•Proving all desirable properties▫Analytical: degrees distribution w/out loss

Used in setting duplication threshold▫Markov 1: degree distribution with loss▫Markov 2: Markov Chain of reachable global states

IID samples, Temporal Independence

•Hold even under (reasonable) message loss!

Key Contribution: Analysis

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Node Degree Markov Chain

•Numerically compute the stationary distribution

Transitions without loss

Transitions due to loss

State corresponding to isolated node

outdegree0 2 4 6

inde

gree

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Results•Outdegree is bounded by

the protocol•Decreases with increasing

loss

• Indegree is not bounded• Low variance even under

loss•Typical overload at most 2x

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0 20 40 60 80Node outdegree

loss=0loss=0.01loss=0.05loss=0.1

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0 10 20 30 40Node indegree

loss=0loss=0.01loss=0.05loss=0.1

•Proving all desirable properties▫Analytical: degrees distribution w/out loss

Used in setting duplication threshold▫Markov 1: degree distribution with loss▫Markov 2: Markov Chain of reachable global states

IID samples, Temporal Independence

•Hold even under (reasonable) message loss!

Key Contribution: Analysis

Decay of Spatial Dependencies

• For uniform loss < 15%, dependencies decay faster than they are created

• 1 – 2loss rate fraction of view entries are independent▫ E.g., for loss rate of 3% more than 90% of entries are independent

u v

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uv

w

u does not delete the sent ids

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Temporal Independence

•Dependence on past views decays within O(log n view size) time

•Use “expected conductance”•Ids travel fast enough▫Reach random nodes in O(log n) hops▫Due to “sufficiently many” independent ids in views -

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Conclusions

•Formalized the desired properties of a membership protocol

•Send & Forget protocol▫Simple for both implementation and analysis

•Analysis under message loss▫Load balance▫Uniformity▫Spatial Independence▫Temporal Independence

Thank You