Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
The decoupling assumption in large stochastic
system analysis
Talk at ECLT
Andrea Marin1
1 Dipartimento
di Scienze Ambientali, Informatica e Statistica
Università Ca’ Foscari Venezia, Italy
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
1 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Outline
1
Motivation
2
Mean field and decoupling assumption
3
Product-forms and decoupling assumption
4
Conclusion
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
2 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Section 1
Motivation
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
3 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
What is the decoupling assumption?
It is normally underlying many analyses of stochastic systems
I
I
I
I
Spreading of information in networks
Spreading of diseases
Analysis of wireless and cabled communication networks
...
Why?
I
Without the decoupling assumption the system would be too
complicated to study
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
4 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Running example
Example taken from the paper Discrete Markov chain approach to
contact-based disease spreading in complex networks by Gomez
et al (2010), (130 citations)
Goals of the paper:
I
I
provide a model for contact-based disease spreading
determine some values for the model parameters that characterise
the type of spreading (e.g., is it epidemic?)
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
5 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
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 with
node j
Each node can be in one of the following two states:
I
I
susceptible (S)
infected (I)
The edge of the graph is a connection along which the infection
spreads
At each time slot each infected node makes λ (independent) trials
to transmits the disease to its neighbour with probability β per time
unit
µ is the rate at which a node moves from infected to susceptible
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
6 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Example of dynamics
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
7 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Example of dynamics
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
7 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Example of dynamics
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
7 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Example of dynamics
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
7 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
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 horizon
Given an initial state, which is the probability of a certain
(aggregated) state when n → ∞?
I
I
I
Stationary analysis
Does the system reach an equilibrium?
Does it depend on the initial state?
In the running example the authors focus on the stationary
analysis
I
Problem: in a network of N nodes we obtain a Markov chain of 2N
states whose stationary analysis has the computational cost of
O(23N )
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
8 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
The model
pi (t) probability of node i of being infected
β is the intensity of infection spreading
µ recovery rate
qi (t) probability of node i not being infected by any of its
neighbours
pi (t + 1) = (1 − qi (t))(1 − pi (t)) + (1 − µ)pi (t) + µ(1 − qi (t))pi (t)
qi (t) =
N
Y
(1 − βrij pj (t))
j=1
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
9 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
The model
pi (t) probability of node i of being infected
β is the intensity of infection spreading
µ recovery rate
qi (t) probability of node i not being infected by any of its
neighbours
pi (t + 1) = (1 − qi (t))(1 − pi (t)) + (1 − µ)pi (t) + µ(1 − qi (t))pi (t)
qi (t) =
N
Y
(1 − βrij pj (t))
j=1
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
9 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
The model
pi (t) probability of node i of being infected
β is the intensity of infection spreading
µ recovery rate
qi (t) probability of node i not being infected by any of its
neighbours
pi (t + 1) = (1 − qi (t))(1 − pi (t)) + (1 − µ)pi (t) + µ(1 − qi (t))pi (t)
qi (t) =
N
Y
(1 − βrij pj (t))
j=1
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
9 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
The model
pi (t) probability of node i of being infected
β is the intensity of infection spreading
µ recovery rate
qi (t) probability of node i not being infected by any of its
neighbours
pi (t + 1) = (1 − qi (t))(1 − pi (t)) + (1 − µ)pi (t) + µ(1 − qi (t))pi (t)
qi (t) =
N
Y
(1 − βrij pj (t))
j=1
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
9 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
The model
pi (t) probability of node i of being infected
β is the intensity of infection spreading
µ recovery rate
qi (t) probability of node i not being infected by any of its
neighbours
pi (t + 1) = (1 − qi (t))(1 − pi (t)) + (1 − µ)pi (t) + µ(1 − qi (t))pi (t)
qi (t) =
N
Y
(1 − βrij pj (t))
j=1
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
9 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
The model
pi (t) probability of node i of being infected
β is the intensity of infection spreading
µ recovery rate
qi (t) probability of node i not being infected by any of its
neighbours
pi (t + 1) = (1 − qi (t))(1 − pi (t)) + (1 − µ)pi (t) + µ(1 − qi (t))pi (t)
qi (t) =
N
Y
(1 − βrij pj (t))
j=1
Where is the decoupling assumption?
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
9 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
From the paper. . .
The formulation so far relies on the assumption that the
probabilities of being infected pi are independent random
variables. This hypothesis turns out to be valid in the vast
majority of complex networks because the inherent
topological disorder makes dynamical correlations not
persistent.
Is that enough?
Afterward the pi are computed as functions of β and µ by solving a
fixed point iteration scheme
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
10 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Section 2
Mean field and decoupling assumption
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
11 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Roadmap
We consider two analysis approaches:
I
I
Mean field models
Product-form models
We study the decoupling assumption for the two settings:
I
I
Transient
Stationary
Mean field
Product-forms
(University of Venice, Italy)
Transient
??
??
The decoupling assumption
Stationary
??
??
ECLT, 2016
12 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Mean field in a nutshell
Example: infection spreading1
N individuals who can be in one of the following three states:
I
I
I
D → Dormant (infected but with no visible symptoms)
A → Active (infected with visible symptoms)
S → Susceptible
Discrete time setting
1
Taken from A class of mean field interaction models for computer and
communication systems, by Le Boudec et al.
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
13 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
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
(University of Venice, Italy)
The decoupling assumption
NDN −1
λ
N
ECLT, 2016
14 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Transition rules (from active)
Recovering with probability δA
Activation with probability
(University of Venice, Italy)
The decoupling assumption
DN
β
h+DN
ECLT, 2016
15 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Transition rules (from susceptible)
Exogenous infection α0
Infection with probability rDN
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
16 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Transition rules (from susceptible)
Exogenous activation α
Mean field: under some (mild) conditions, for N → ∞ the
probabilistic model’s behaviour coincides almost surely with the
trajectory of the solution of ODE system for any finite time horizon
Probabilistic → deterministic
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
17 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
The ODE associated with the system
The drift for each variable D, A, S is given by:

D(t)
2

∆D = −DδD − 2D λ − Aβ h+D + S(α0 + rD)
D
∆A = 2D2 λ + Aβ h+D
− AδA + Sα


∆S = DδD + AδA − S(α0 + rD) − Sα
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
18 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Good news!
The decoupling assumption holds in transient regime!
Consequences:
I
I
I
I
We can focus on a single individual
Its behaviour is probabilistic but it interacts with a deterministic
environment as N → ∞
The analysis corresponds to the transient analysis of a
time-inhomogeneous Markov chain
Idea: at each simulation step the environment changes the
transition probabilities of the Markov chain associated with a single
individual (only three states!)
Mean field
Product-forms
(University of Venice, Italy)
Transient
OK
??
The decoupling assumption
Stationary
??
??
ECLT, 2016
19 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
What about the stationary behaviour?
We can set the drift to 0 and look for a solution for D, S, A
In general the ODE may admit more fixed points
I
In this case the decoupling assumption in stationary regime does
not hold
Is proving the uniqueness of the fixed point enough?
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
20 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
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.
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
21 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
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.
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
22 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Not so good news
The decoupling assumption is valid in stationary regime if
I
I
I
We have a unique fixed point of the ODE
All the trajectories of the system converge to the fixed point
These properties depend on the specific ODE, hard to find general
results
This is usually extremely hard to prove
Mean field
Product-forms
(University of Venice, Italy)
Transient
OK
??
Stationary
Sometimes, hard to prove
??
The decoupling assumption
ECLT, 2016
23 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Section 3
Product-forms and decoupling assumption
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
24 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
When is a stochastic model in product-form?
Setting: continuous time
N interacting individuals
I
I
Sn the state space of n
Joint state space: S ⊆ S1 × S2 × · · · × SN
π(s) be the stationary probability of state s ∈ S:
π(s) ∝
N
Y
gn (sn )
n=1
gn (sn ) is interpreted as the stationary probability of individual n
isolated and re parameterised to take into account the interactions
with the other individuals
Product-form 6= stochastic independence since for t ∈ R, in
general:
N
Y
π(s, t) 6=
gn (sn , t)
n=1
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
25 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Example: migration process
N individuals clustered in J colonies
Each colony has ni individuals, with
PN
i=1 ni
=N
State of the system n = (n1 , . . . , nJ )
Tjk is the operator that moves one individual from colony j, with
nj > 0 to colony nk
Colony connections are modelled by a graph with adjacency
matrix R = (rij ), rij ∈ {0, 1}
The migration process is regulated by the law:
q(n, Tjk n) = rij λjk Φj (nj )
with Φj (0) = 0.
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
26 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Product-form
The stationary probability of observing state n is:
π(n) ∝
n
J
Y
αj j
Qnj
j=1
r=1 Φj (r)
P
P
αj is a non-trivial solution of αj k λjk rjk = k αk λkj rkj
We only need the graph to be irreducible
I
No limiting assumptions on the structure or on the population
Mean field
Product-forms
(University of Venice, Italy)
Transient
OK
NO
Stationary
Sometimes, hard to prove
OK
The decoupling assumption
ECLT, 2016
27 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Section 4
Conclusion
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
28 / 29
Motivation
Mean field and decoupling assumption
Product-forms and decoupling assumption
Conclusion
Conclusion
The decoupling assumption is often needed to make the models
tractable
Handling the decoupling assumption correctly is not trivial
Mean field vs. Product-forms:
I
I
Mean field: less restrictions on the model, easy to handle the
transient, unclear when the limiting approximation is good.
Stationary analyses must be handled carefully;
Product-forms: models must fulfil some conditions, useful in the
stationary regime, decoupling assumption holds, almost no results
in the transient regime
(University of Venice, Italy)
The decoupling assumption
ECLT, 2016
29 / 29