Correctness of Gossip-Based Membership under Message Loss Maxim GurevichIdit Keidar Technion.

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

2

Membership: Each node needs to know some live nodes

•Each node has a view ▫Set of node ids▫Supplied to the application▫Constantly refreshed

•Typical size – log n

3

Applications

•Applications▫Gossip-based algorithm▫Unstructured overlay networks▫Gathering statistics

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

4

Modeling Membership Views

•Modeled as a directed graph

u v

w

v y w …

y

5

Modeling Protocols: Graph Transformations

•View is used for maintenance•Example: push protocol

… … w …… … z …u v

w

v … w …

w

z

6

Desirable Properties?

•Randomness▫View should include random samples

•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

7

What About Churn?

•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

8

Global Markov Chain

•A global state – all n views in the system•A protocol action – transition between global

states•Global Markov Chain G

u v u v

9

Defining Properties Formally

•Small views▫Bounded dout(u)

•Load balance▫ Low variance of din(u)

•From any starting state, eventually(In the stationary distribution of MC on G)▫Uniformity

Pr(v u.view) = Pr(w u.view) ▫Spatial independence

Pr(v u. view| y w. view) = Pr(v u. view) ▫Perfect uniformity + spatial independence load balance

10

Temporal Independence

•Time to obtain views independent of the past•From an expected state▫Refresh rate in the steady state

•Would have been much longer had we considered starting from arbitrary state▫O(n14) [Cooper09]

11

Existing Work: Practical Protocols

•Tolerates asynchrony, message loss•Studied only empirically ▫Good load balance [Lpbcast, Jelasity et al 07] ▫Fast decay of temporal dependencies [Jelasity et al 07] ▫ Induce spatial dependence

Push protocol

u v

w

u v

w

w

z z

12

v … z …

Existing Work: Analysis

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

▫ Uniformity, load balance, spatial independence ▫Weak bounds (worst case) on temporal independence

•Unrealistic assumptions – hard to implement ▫ Atomic actions with bi-directional communication▫ No message loss

… … z …… … w …u v

w

v … w …

w

zShuffle protocol

z

*

13

Our Contribution : Bridge This Gap

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

•Formally prove the desirable properties▫Including under message loss

14

… …

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

u v

w

v … w … u w

u w

15

S&F: Message Loss

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

u v

w

u v

w

16

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▫ Push-like when views are too small

u v

w

u v

w

17

S&F: Advantages over Other Protocols

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

•Simple▫Without unrealistic assumptions▫Amenable to formal analysis

Easy to implement

18

•Degree distribution▫Closed-form approximation without loss▫Degree Markov Chain with loss

•Stationary distribution of MC on the global graph G▫Uniformity▫Spatial Independence▫Temporal Independence

•Hold even under (reasonable) message loss!

Key Contribution: Analysis

19

Degree Distribution without loss•In all reachable graphs:▫dout(u) + 2din(u) = const▫Better than in a random graph – indegree bounded

•Uniform stationary distribution on reachable states in G

•Combinatorial approximation of degree distribution▫The fraction of reachable graphs with specified node

degree▫Ignoring dependencies among nodes

20

Degree Distribution without Loss: Results

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

0

0.05

0.1

0.15

0.2

0 10 20 30 40Node indegree

Binomial

S&F Analytical

S&F Markov

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0

0.05

0.1

0.15

0.2

0 20 40 60 80Node outdegree

Binomial

S&F Analytical

S&F Markov

Setting Degree Thresholds to Compensate for Loss

•Note: dout(u) + 2din(u) = const invariant no longer holds – indegree not bounded

22


23



…

Degree Markov Chain

•Given loss rate, degree thresholds, and degree distributions

• Iteratively compute the stationary distribution

Transitions without loss

Transitions due to loss

State corresponding to isolated node

outdegree0 2 4 6

inde

gree

0

1

2

3

…

…

…

…

…

…

…

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Results• Outdegree is bounded by the

protocol• Decreases with increasing loss

• Indegree is not bounded by the protocol

• Still, its variance is low, even under loss

• Typical overload at most 2x

0

0.05

0.1

0.15

0.2

0.25

0 20 40 60 80Node outdegree

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

0

0.05

0.1

0.15

0.2

0.25

0 10 20 30 40Node indegree

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

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Uniformity

•Simple!•Nodes are identical•Graphs where uv isomorphic to graphs

where uw•Same probability in stationary distribution

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Decay of Spatial Dependencies

•Assume initially > 2/3 independent good expander

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

u v

w

uv

w

u does not delete the sent ids

…

…

u w

29

Decay of Spatial Dependencies: Results

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

are independent

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Temporal Independence

•Start from expected state▫Uniform and spatially independent views

•High “expected conductance” of G•Short mixing time▫While staying in the “good” component

32

Temporal Independence: Results•Ids travel fast enough▫Reach random nodes in O(log n) hops▫Due to “sufficiently many” independent ids in views

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

33

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

34

Thank You

Correctness of Gossip-Based Membership under Message Loss Maxim GurevichIdit Keidar Technion.

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