the decoupling assumption in large stochastic … naughty example the dot is the starting point,...

Motivation Mean field and decoupling assumption Product-forms and decoupling assumption Conclusion

The decoupling assumption in large stochasticsystem analysis

Talk at ECLT

Andrea Marin1

1Dipartimento di Scienze Ambientali, Informatica e StatisticaUniversità Ca’ Foscari Venezia, Italy

(University of Venice, Italy) The decoupling assumption ECLT, 2016 1 / 29


Outline

1 Motivation

2 Mean field and decoupling assumption

3 Product-forms and decoupling assumption

4 Conclusion



Section 1

Motivation



What is the decoupling assumption?

It is normally underlying many analyses of stochastic systemsI Spreading of information in networksI Spreading of diseasesI Analysis of wireless and cabled communication networksI . . .

Why?I Without the decoupling assumption the system would be too

complicated to study



Running example

Example taken from the paper Discrete Markov chain approach tocontact-based disease spreading in complex networks by Gomezet al (2010), (130 citations)Goals of the paper:

I provide a model for contact-based disease spreadingI determine some values for the model parameters that characterise

the type of spreading (e.g., is it epidemic?)



The model

N nodes connected according to a adjacency matrix R = (rij)where 0 ≤ rij ≤ 1 is the probability of node i to be in contact withnode jEach node can be in one of the following two states:

I susceptible (S)I infected (I)

The edge of the graph is a connection along which the infectionspreadsAt each time slot each infected node makes λ (independent) trialsto transmits the disease to its neighbour with probability β per timeunitµ is the rate at which a node moves from infected to susceptible



Example of dynamics



What would we like to understand? Transient vs.Stationary analysis

Given an initial state, which is the probability of a certain(aggregated) state after n steps?

I Transient analysis, finite time horizonGiven an initial state, which is the probability of a certain(aggregated) state when n→∞?

I Stationary analysisI Does the system reach an equilibrium?I Does it depend on the initial state?

In the running example the authors focus on the stationaryanalysis

I Problem: in a network of N nodes we obtain a Markov chain of 2N

states whose stationary analysis has the computational cost ofO(23N)



The model

pi(t) probability of node i of being infectedβ is the intensity of infection spreadingµ recovery rateqi(t) probability of node i not being infected by any of itsneighbours

pi(t + 1) = (1− qi(t))(1− pi(t)) + (1− µ)pi(t) + µ(1− qi(t))pi(t)

qi(t) =

N∏j=1

(1− βrijpj(t))



The model

pi(t) probability of node i of being infectedβ is the intensity of infection spreadingµ recovery rateqi(t) probability of node i not being infected by any of itsneighbours

pi(t + 1) = (1− qi(t))(1− pi(t)) + (1− µ)pi(t) + µ(1− qi(t))pi(t)

qi(t) =

N∏j=1

(1− βrijpj(t))

Where is the decoupling assumption?



From the paper. . .

The formulation so far relies on the assumption that theprobabilities of being infected pi are independent randomvariables. This hypothesis turns out to be valid in the vastmajority of complex networks because the inherenttopological disorder makes dynamical correlations notpersistent.

Is that enough?Afterward the pi are computed as functions of β and µ by solving afixed point iteration scheme



Section 2

Mean field and decoupling assumption



Roadmap

We consider two analysis approaches:I Mean field modelsI Product-form models

We study the decoupling assumption for the two settings:I TransientI Stationary

Transient StationaryMean field ?? ??

Product-forms ?? ??



Mean field in a nutshell

Example: infection spreading1

N individuals who can be in one of the following three states:I D→ Dormant (infected but with no visible symptoms)I A→ Active (infected with visible symptoms)I S→ Susceptible

Discrete time setting

1Taken from A class of mean field interaction models for computer andcommunication systems, by Le Boudec et al.



Transition rules (from dormant)

D→ proportion dormant elements, A→ proportion of active elements,S→ proportion of susceptible elements

Recovering with probability δD

Activation with probability NDN−1N λ



Transition rules (from active)

Recovering with probability δA

Activation with probability DN

h+DN β



Transition rules (from susceptible)

Exogenous infection α0

Infection with probability rDN



Transition rules (from susceptible)

Exogenous activation α

Mean field: under some (mild) conditions, for N →∞ theprobabilistic model’s behaviour coincides almost surely with thetrajectory of the solution of ODE system for any finite time horizonProbabilistic→ deterministic



The ODE associated with the system

The drift for each variable D, A, S is given by:∆D = −DδD − 2D2λ− Aβ D(t)

h+D + S(α0 + rD)

∆A = 2D2λ+ Aβ Dh+D − AδA + Sα

∆S = DδD + AδA − S(α0 + rD)− Sα



Good news!

The decoupling assumption holds in transient regime!Consequences:

I We can focus on a single individualI Its behaviour is probabilistic but it interacts with a deterministic

environment as N →∞I The analysis corresponds to the transient analysis of a

time-inhomogeneous Markov chainI Idea: at each simulation step the environment changes the

transition probabilities of the Markov chain associated with a singleindividual (only three states!)

Transient StationaryMean field OK ??

Product-forms ?? ??



What about the stationary behaviour?

We can set the drift to 0 and look for a solution for D, S, AIn general the ODE may admit more fixed points

I In this case the decoupling assumption in stationary regime doesnot hold

Is proving the uniqueness of the fixed point enough?



A nice example

The dot is the starting point, while the cross is the fixed point solution.Taken from A class of mean field interaction models for computer and communication systems, by le Boudec et al.



A naughty example

The dot is the starting point, while the cross is the fixed point solution.Taken from A class of mean field interaction models for computer and communication systems, by le Boudec et al.



Not so good news

The decoupling assumption is valid in stationary regime ifI We have a unique fixed point of the ODEI All the trajectories of the system converge to the fixed pointI These properties depend on the specific ODE, hard to find general

results

This is usually extremely hard to prove

Transient StationaryMean field OK Sometimes, hard to prove

Product-forms ?? ??



Section 3

Product-forms and decoupling assumption



When is a stochastic model in product-form?

Setting: continuous timeN interacting individuals

I Sn the state space of nI Joint state space: S ⊆ S1 × S2 × · · · × SN

π(s) be the stationary probability of state s ∈ S:

π(s) ∝N∏

n=1

gn(sn)

gn(sn) is interpreted as the stationary probability of individual nisolated and re parameterised to take into account the interactionswith the other individualsProduct-form 6= stochastic independence since for t ∈ R, ingeneral:

π(s, t) 6=N∏

n=1

gn(sn, t)



Example: migration process

N individuals clustered in J coloniesEach colony has ni individuals, with

∑Ni=1 ni = N

State of the system n = (n1, . . . , nJ)

Tjk is the operator that moves one individual from colony j, withnj > 0 to colony nk

Colony connections are modelled by a graph with adjacencymatrix R = (rij), rij ∈ {0, 1}The migration process is regulated by the law:

q(n,Tjkn) = rijλjkΦj(nj)

with Φj(0) = 0.



Product-form

The stationary probability of observing state n is:

π(n) ∝J∏

j=1

αnjj∏nj

r=1 Φj(r)

αj is a non-trivial solution of αj∑

k λjkrjk =∑

k αkλkjrkj

We only need the graph to be irreducibleI No limiting assumptions on the structure or on the population

Transient StationaryMean field OK Sometimes, hard to prove

Product-forms NO OK



Section 4

Conclusion



Conclusion

The decoupling assumption is often needed to make the modelstractableHandling the decoupling assumption correctly is not trivialMean field vs. Product-forms:

I Mean field: less restrictions on the model, easy to handle thetransient, unclear when the limiting approximation is good.Stationary analyses must be handled carefully;

I Product-forms: models must fulfil some conditions, useful in thestationary regime, decoupling assumption holds, almost no resultsin the transient regime


the decoupling assumption in large stochastic … naughty example the dot is the starting point,...

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