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