Fluid Limits for Gossip Processes

Vahideh Manshadi and Ramesh JohariDARPA ITMANET Meeting

March 5-6, 2009

Fluid limits for gossip processesV. Manshadi and R. Johari

Several goals:

(1) Extend fluid analysis to include heterogeneous random graphs.

(2) Get finer understanding of behavior when initial number of informed nodes is constant as N ! infinity.

(3) Extend the model to include link failures.

The simplicity of macroscopic models for information gossipcan be combined with the accuracy of microscopic stochastic models

MAIN RESULT:

We consider a random graph model where each nodehas d neighbors, and we consider a limit where thenumber of nodes N approaches infinity.We prove that the (random) sample path of themicro model converges to the (deterministic) pathof the corresponding macro model.

HOW IT WORKS:

We approximately characterize howinformation flows in the micro model between the sets of informed and uninformed nodes.This approximation is exact as N ! infinity.

ASSUMPTIONS AND LIMITATIONS:Our results currently only apply under specific topological assumptions.

Gossip is a simple model for communication between nodes:at random times, each node contacts a neighbor and relays its information.

Prior work has studied the time until all nodes acquire the information.Two versions of this model: a “micro” model and a “macro” model.

I(t)

Time t

“Macro”

I(t)

Time t

“Micro”

The micro model tracks exactly which nodes have the information.

The macro model is a mean field limit: what fraction of nodes have learned the information?

We connect these two models.

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Nodes that currently have

the info

Nodes that currently do not have the info

Micro and macro models of gossip processes have been available for several decades.

Unifying these will allow us to translate macro-level control insights to micro-level system designs.

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ACHIEVEMENT DESCRIPTION

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Problem Definition

N sensors form a network G

Initially, set I0 of nodes receive a piece of information.

At the timepoints of a Poisson process of rate ,each informed sensor contacts a neighbor selecteduniformly at random

If uninformed, that neighbor switches to informed with prob. p

Key questions:

How long does it take to inform all the sensors?

How does the network structure and connectivity affect the time until all nodes (or a fraction of nodes) acquire the information?

How does the size of I0 change the required time?

Related Work

Application of gossip protocols in sensor networks GCP (Gossip-based Code Propagation) for mobile wireless sensor networks

[Yann, Bertier, Fleury & Kermarrec 07]. Geographic Gossip: Efficient Averaging for Sensor Networks [Dimakis,

Sarwate & Wainright 08].

Microscopic model: preserve the combinatorial and probabilistic nature of the process. It takes O(log(n)) to make O(n) (all) nodes informed [R. Karp, C.

Schindelhauer, S. Shenker, & B. Vocking 00]. The constant factor depends on the network structure [Mosk-Aoyama &

Shah 06].

Macroscopic model: treat the process as continuous and deterministic A first order differential equation reasonably models the evolution [Bass 69]. S-shaped curve for the diffusion has been observed in several experiments Model also studied extensively in epidemiology (SI model)

Our Work

Our work rigorously connects the microscopic and macroscopic models using fluid limits.

Main result: We show that as number of nodes N ! 1,the microscopic model approaches the macroscopic model.

We prove this result in the case of a complete graph(including error terms) and random K-regular graphs.

I(t)

Time t

“Macro”

I(t)

Time t

“Micro”

For simplicity: assume a fully connected network

Let I(t) = # of informed sensors at time t

In a small period dt, approximately I(t) dt of informed nodes contact a random neighbor.

This neighbor is uninformed with probability (N - I(t))/N. So:

The solution is the logistic function.

We formally justify this heuristic argument(and an analog in the case of random regular graphs).

Heuristic Argument

Basic Approach

Directly studying the sample paths of the discrete stochastic model is not straightforward.

Instead we consider T(x, y):total time until y nodes are informed, given |I0| = x.

Convenient simplification:T(x, y) is a sum of independent exponential random variables.

We employ laws of large numbers to study the behavior ofT(x, y) as N grows large.

Time t

T(x,y)

x

y

Results: Complete graph

Theorem 1: For complete graph with n nodes and 0 < ®1,®2<1/2,

This result can be inverted to show that the sample path of the fraction of nodes that are informed converges to the discrete continuous model.

We can also provide a more refined analysis of TN,to reveal that if the initial set I0 stays constant of size x as N ! 1, then:

Results: Random Regular Graph

We now assume the network is a (uniformly) random K-regular graph.

This result matches the result for the complete graph.

Key insight in the proof:We are able to precisely analyze the number of edges present from nodes that are interested to nodes that are uninterested.

Theorem 2: For Gk and 0 < ®1,®2<1/2,

Future Work

Our goal: a stronger connection between macroscopic and microscopic models (for control, for analysis, etc.)

What about infinite graphs with specific degree distributions?

xi fraction of sensors have di links; Nodes with high degrees are more probable to be informed earlier.

Idea: analyze the degree distributions in the set of informed (uninformed) nodes and compute information flow (the size of the cut) between two sets.

What about other classes of random graphs? Sensors are scattered uniformly at random, the probability that any two nodes

share a link is p; this is the Random Geometric Graph model Site (bond) percolated graphs model the scenarios where some of the nodes

(links) in the network failIdea: Couple the link failure process to an equivalent contact failure in the original graph.