Gossip algorithms : “infect forever” dynamics Low-level objectives: – One-to-all: Disseminate...
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Transcript of Gossip algorithms : “infect forever” dynamics Low-level objectives: – One-to-all: Disseminate...
Gossip algorithms :“infect forever” dynamics
• Low-level objectives:– One-to-all: Disseminate rumor from source node to all nodes of
network
– All-to-all: Each node initially holds specific rumor, to be spread to all nodes
• Applications– One-to-all: announce change in topology (new node arrival) in ad hoc
network; spread content (data chunk) of interest to all nodes in P2P network
– All-to-all: monitoring global state of network (eg sensors spreading warnings about abnormal temperature…)
Types of gossip algorithms• Synchronization modes:
– Synchronous (slotted time, simultaneous operations by each node)– Asynchronous (continuous time, single node wakes up & performs
operation)
• Type of operation: contact neighbor node to– Push all known rumors– Pull rumors known by contacted node– Push-Pull: do both
• Neighbor selection:– Uniform at random among neighbors, i.i.d. over node wake-up events– Round-robin– …
Push on complete graph with uniform neighbor selection: broadcast time
• Slotted time: Meaning: <
• More generally if node pushes to neighbors in each slot then [Pittel 87] Then
Intuition: Initial phase:
nb of reached nodes after t slots until it reaches Lasts for slots
Intermediate phase: reaches in slots Final phase:
nb of unreached nodes reduces by factor in slots, shrinking from to
Push on complete graph with uniform neighbor selection: broadcast time
• Continuous time: each node wakes up after random timer with Exp(1) distribution expires
Instants of a node’s awakenings: Poisson process with intensity 1Then
Proof elements: whiteboard [using properties of Exp variables and Poisson processes]
Further results: broadcast time for push-pull satisfies
in all-to-all scenario with push, for constant Cwith high probability (w.h.p.)
A fortiori satisfies same bound (possibly with smaller constant C)
Non-complete, possibly sparse graphs: conductance, isoperimetric constant and
expanders Graph conductance of
where and min over <(: degree of node )
Isoperimetric (also known as Cheeger, or edge-expansion) constant:
where min over <Special case: for regular graph ,
Graph with is an expanderInterest in graphs both sparse (low degrees) and with high expansion:-Epidemics spread very quickly despite graph being sparse-random walks forget quickly initial point (hence can sample quickly from stationary distribution)Example: hypercube on nodes satisfies Q: determine isoperimetric constant of line-graph
Performance on graphs with conductance
Assume regular graph and continuous time: nodes wake up after expiration of Exp(1) timers. Then for any fixed
w.h.p., Proof elements: whiteboard [Coupling of Markov processes]
Assume discrete slotted time, graph not necessarily regular. Then for some universal constant C:
w.h.p., [Giakkoupis 2011; bound known to be tight for specific graphs]
Graph conductance characteristic of gossip performance,for uniform neighbor selection
Extension (1)Beating dependency
Can non-uniform neighbor selection achieve faster than dissemination?
Yes: [Hauepler 2014] proposes deterministic gossip algorithm succeeding in slotted time where : graph diameter
Beats for
Extension (2)Competing epidemic disseminations
Context: P2P system for live streaming dissemination (such as PPLive)Users want to obtain sequence of rumors (=data packets) injected by source node,with low delay Upload bandwidth constraint: only 1 rumor can be pushed by any node in one time
Local scheduling decision: which packet to push?
?
Sender’s packets
Receiver’s packets
???
1 2 4 5 7 8
1 2 3
9
Favors overall system performance: createspotential for new transmissions from receiver
An example strategy: uniform random peer, latest « chunk » push
? ?
Sender’s chunks
Receiver’s chunks
Last chunk
??????
Fraction of reached nodes
Time
1 2 4 5 7 8
10
Allows streaming at 63% of optimal rate with optimal delay, (by performing source coding at source node, creating redundancy in disseminated chunks)
[Bonald-Massoulie-Mathieu et al, 2008]
uniform random peer, latest « chunk » pushPerformance with complete graph
Each node receives fraction 1-1/e 63% of all chunks In order-optimal ( ) time