The decoupling assumption in large stochastic system analysis Talk at ECLT

Transcription

1 The decoupling assumption in large stochastic system analysis Talk at ECLT Andrea Marin 1 1 Dipartimento di Scienze Ambientali, Informatica e Statistica Università Ca Foscari Venezia, Italy (University of Venice, Italy) The decoupling assumption ECLT, / 29

2 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, / 29

3 Section 1 Motivation (University of Venice, Italy) The decoupling assumption ECLT, / 29

4 What is the decoupling assumption? It is normally underlying many analyses of stochastic systems Spreading of information in networks Spreading of diseases Analysis of wireless and cabled communication networks... Why? Without the decoupling assumption the system would be too complicated to study (University of Venice, Italy) The decoupling assumption ECLT, / 29

5 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: 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, / 29

6 The model N nodes connected according to a adjacency matrix R = (r ij ) where 0 r ij 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: 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, / 29

7 Example of dynamics (University of Venice, Italy) The decoupling assumption ECLT, / 29

8 Example of dynamics (University of Venice, Italy) The decoupling assumption ECLT, / 29

9 Example of dynamics (University of Venice, Italy) The decoupling assumption ECLT, / 29

10 Example of dynamics (University of Venice, Italy) The decoupling assumption ECLT, / 29

11 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? Transient analysis, finite time horizon Given an initial state, which is the probability of a certain (aggregated) state when n? 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 Problem: in a network of N nodes we obtain a Markov chain of 2 N states whose stationary analysis has the computational cost of O(2 3N ) (University of Venice, Italy) The decoupling assumption ECLT, / 29

12 The model p i (t) probability of node i of being infected β is the intensity of infection spreading µ recovery rate q i (t) probability of node i not being infected by any of its neighbours p i (t + 1) = (1 q i (t))(1 p i (t)) + (1 µ)p i (t) + µ(1 q i (t))p i (t) q i (t) = N (1 βr ij p j (t)) j=1 (University of Venice, Italy) The decoupling assumption ECLT, / 29

13 The model p i (t) probability of node i of being infected β is the intensity of infection spreading µ recovery rate q i (t) probability of node i not being infected by any of its neighbours p i (t + 1) = (1 q i (t))(1 p i (t)) + (1 µ)p i (t) + µ(1 q i (t))p i (t) q i (t) = N (1 βr ij p j (t)) j=1 (University of Venice, Italy) The decoupling assumption ECLT, / 29

14 The model p i (t) probability of node i of being infected β is the intensity of infection spreading µ recovery rate q i (t) probability of node i not being infected by any of its neighbours p i (t + 1) = (1 q i (t))(1 p i (t)) + (1 µ)p i (t) + µ(1 q i (t))p i (t) q i (t) = N (1 βr ij p j (t)) j=1 (University of Venice, Italy) The decoupling assumption ECLT, / 29

15 The model p i (t) probability of node i of being infected β is the intensity of infection spreading µ recovery rate q i (t) probability of node i not being infected by any of its neighbours p i (t + 1) = (1 q i (t))(1 p i (t)) + (1 µ)p i (t) + µ(1 q i (t))p i (t) q i (t) = N (1 βr ij p j (t)) j=1 (University of Venice, Italy) The decoupling assumption ECLT, / 29

16 The model p i (t) probability of node i of being infected β is the intensity of infection spreading µ recovery rate q i (t) probability of node i not being infected by any of its neighbours p i (t + 1) = (1 q i (t))(1 p i (t)) + (1 µ)p i (t) + µ(1 q i (t))p i (t) q i (t) = N (1 βr ij p j (t)) j=1 (University of Venice, Italy) The decoupling assumption ECLT, / 29

17 The model p i (t) probability of node i of being infected β is the intensity of infection spreading µ recovery rate q i (t) probability of node i not being infected by any of its neighbours p i (t + 1) = (1 q i (t))(1 p i (t)) + (1 µ)p i (t) + µ(1 q i (t))p i (t) q i (t) = N (1 βr ij p j (t)) j=1 Where is the decoupling assumption? (University of Venice, Italy) The decoupling assumption ECLT, / 29

18 From the paper... The formulation so far relies on the assumption that the probabilities of being infected p i 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 p i are computed as functions of β and µ by solving a fixed point iteration scheme (University of Venice, Italy) The decoupling assumption ECLT, / 29

19 Section 2 Mean field and decoupling assumption (University of Venice, Italy) The decoupling assumption ECLT, / 29

20 Roadmap We consider two analysis approaches: Mean field models Product-form models We study the decoupling assumption for the two settings: Transient Stationary Transient Stationary Mean field???? Product-forms???? (University of Venice, Italy) The decoupling assumption ECLT, / 29

21 Mean field in a nutshell Example: infection spreading 1 N individuals who can be in one of the following three states: 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, / 29

22 Transition rules (from dormant) D proportion dormant elements, A proportion of active elements, S proportion of susceptible elements Recovering with probability δ D Activation with probability NDN 1 N λ (University of Venice, Italy) The decoupling assumption ECLT, / 29

23 Transition rules (from active) Recovering with probability δ A Activation with probability DN β h+d N (University of Venice, Italy) The decoupling assumption ECLT, / 29

24 Transition rules (from susceptible) Exogenous infection α 0 Infection with probability rd N (University of Venice, Italy) The decoupling assumption ECLT, / 29

25 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, / 29

26 The ODE associated with the system The drift for each variable D, A, S is given by: D = Dδ D 2D 2 λ Aβ D(t) h+d + S(α 0 + rd) A = 2D 2 λ + Aβ D h+d Aδ A + Sα S = Dδ D + Aδ A S(α 0 + rd) Sα (University of Venice, Italy) The decoupling assumption ECLT, / 29

27 Good news! The decoupling assumption holds in transient regime! Consequences: 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!) Transient Stationary Mean field OK?? Product-forms???? (University of Venice, Italy) The decoupling assumption ECLT, / 29

28 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 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, / 29

29 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, / 29

30 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, / 29

31 Not so good news The decoupling assumption is valid in stationary regime if 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 Transient Stationary Mean field OK Sometimes, hard to prove Product-forms???? (University of Venice, Italy) The decoupling assumption ECLT, / 29

32 Section 3 Product-forms and decoupling assumption (University of Venice, Italy) The decoupling assumption ECLT, / 29

33 When is a stochastic model in product-form? Setting: continuous time N interacting individuals Sn the state space of n Joint state space: S S1 S 2 S N π(s) be the stationary probability of state s S: N π(s) g n (s n ) n=1 g n (s n ) 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 stochastic independence since for t R, in general: N π(s, t) g n (s n, t) n=1 (University of Venice, Italy) The decoupling assumption ECLT, / 29

34 Example: migration process N individuals clustered in J colonies Each colony has n i individuals, with N i=1 n i = N State of the system n = (n 1,..., n J ) T jk is the operator that moves one individual from colony j, with n j > 0 to colony n k Colony connections are modelled by a graph with adjacency matrix R = (r ij ), r ij {0, 1} The migration process is regulated by the law: with Φ j (0) = 0. q(n, T jk n) = r ij λ jk Φ j (n j ) (University of Venice, Italy) The decoupling assumption ECLT, / 29

35 Product-form The stationary probability of observing state n is: π(n) J j=1 α n j j nj r=1 Φ j(r) α j is a non-trivial solution of α j k λ jkr jk = k α kλ kj r kj We only need the graph to be irreducible No limiting assumptions on the structure or on the population Transient Stationary Mean field OK Sometimes, hard to prove Product-forms NO OK (University of Venice, Italy) The decoupling assumption ECLT, / 29

36 Section 4 Conclusion (University of Venice, Italy) The decoupling assumption ECLT, / 29

37 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: 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, / 29