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Mean Field Methods for Computer and Communication Systems

Mean Field Methods for Computer and Communication Systems. Jean-Yves Le Boudec EPFL Network Science Workshop Hong Kong 25-27 July 2012. Contents. Mean Field Interaction Model Convergence to ODE Finite Horizon: Fast Simulation and Decoupling assumption

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Mean Field Methods for Computer and Communication Systems

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  1. Mean Field Methods for Computer and Communication Systems Jean-Yves Le Boudec EPFL Network Science Workshop Hong Kong 25-27 July 2012 1

  2. Contents • Mean Field Interaction Model • Convergence to ODE • Finite Horizon: Fast Simulation and Decouplingassumption • Infinite Horizon: Fixed Point Method and Decouplingassumption 2

  3. 1 Mean Field Interaction Model 3

  4. Mean Field • A model introduced in Physics • interaction between particles is via distribution of states of all particle • An approximation method for a large collection of particles • assumes independence in the master equation • Why do we care in information and communication systems ? • Model interaction of manyobjects: • Distributedsystems, communication protocols, gametheory, self-organizedsystems 4

  5. A Few Examples Where Applied Never again ! E.L. 5

  6. Mean Field Interaction Model • Time is discrete (this talk) or continuous • N objects, N large • Object n has state Xn(t) • (XN1(t), …, XNN(t)) is Markov • Objects are observable onlythroughtheir state • “Occupancy measure”MN(t) = distribution of object states at time t 6

  7. Example: 2-Step Malware Mobile nodes are either `S’ Susceptible `D’ Dormant `A’ Active Time is discrete Transitions affect 1 or 2 nodes State space is finite = {`S’ , `A’ ,`D’} Occupancy measure isM(t) = (S(t), D(t), A(t)) with S(t)+ D(t) + A(t) =1 S(t) = proportion of nodes in state `S’[Benaïm and Le Boudec(2008)] Recovery D -> S Mutual upgrade D + D -> A + A Infection by active D + A -> A + A Recovery A -> S Recruitment by Dormant S + D -> D + D Direct infection S -> D Direct infection S -> A 7

  8. 2-Step Malware – Full Specification Recovery D -> S Mutual upgrade D + D -> A + A Infection by active D + A -> A + A Recovery A -> S Recruitment by Dormant S + D -> D + D Direct infection S -> D Direct infection S -> A • At every time step, pick one nodeunifatrandom • If nodeis in state : • Withprobamutate to S • Withproba, meetanothernode and bothmutate to • If nodeis in state : • Withproba change one node to • WithProbamutate to • If nodeis in state • Withprobameet a node and becomeinfected • Withprobabecomeinfected • Withprobabecomeinfected 8

  9. Simulation Runs, N=1000 nodes Node 1 State = D State = A State = S Node 2 Node 3 D(t) Proportion of nodes In state i=1 A(t) Proportion of nodes In state i=2 9

  10. Sample Runs with N = 1000 10

  11. The Importance of Being Spatial • Mobile node state = (c, t)c = 1 … 16 (position) t ∊ R+ (age of gossip) • Time iscontinuous • OccupancymeasureisFc(z,t) = proportion of nodesthatat location c and have age ≤ z[Age of Gossip, Chaintreau et al.(2009)] no class 16 classes Qqplots simulation vs meanfield 11

  12. What can we do with a Mean Field Interaction Model ? Large Nasymptotics, Finite Horizon fluid limit of occupancy measure (ODE) decouplingassumption (fast simulation) Issues When valid How to formulate the fluid limit Large t asymptotic Stationary approximation of occupancy measure Decoupling assumption Issues When valid 12

  13. E. L. 2. Convergence to ode 13

  14. To Obtain a Mean Field Limitwe Must MakeAssumptions about the IntensityI(N) • I(N) = (order of) expectednumber of transitions per object per time unit • A meanfieldlimitoccurswhenwere-scale time by I(N) i.e. one time slot i.e. weconsiderXN(t/I(N)) • I(N) = O(1): meanfieldlimitis in discrete time [Le Boudec et al (2007)]I(N) = O(1/N): meanfieldlimitis in continuous time [Benaïm and Le Boudec (2008)] (this talk) 14

  15. Intensity for this model is • In one time step, the number of objectsaffected by a transition is 0, 1 or 2; meannumber of affectedobjectsis • There are • Expectednumber of transitions per time slot per objectis 15

  16. The Mean Field Limit • Under verygeneral conditions (givenlater) the occupancymeasure converges, in law, to a deterministicprocess, m(t),called the meanfieldlimit • Finite State Space => ODE 16

  17. Mean Field Limit N = +∞ Stochastic system N = 1000 17

  18. Sufficient Conditions for Convergence • [Kurtz 1970], seealso [Bordenav et al 2008], [Graham 2000] • Sufficient condition verifiable by inspection: • probabilitiesatevery time slot have a limitwhen when I(N) = 1/N the condition istrue as soon as Second moment of number of objectsaffected in one timeslot • Similarresultwhenmeanfieldlimitis in discrete time [Le Boudec et al 2007] 18

  19. Example: Convergence to Mean Field • Number of transitions per time stepisbounded by 2, thereforethereis convergence to meanfield • The Meanfield limite is an ODE • One time step corresponds to Mean Field Limit N = +∞ Stochastic system N = 1000 19

  20. Formulating the Mean Field Limit • Drift = sum over all transitions ofproba of transition xDelta to system state MN(t) • Re-scale drift by intensity • Equation for meanfieldlimitis dm/dt = limit of rescaled drift • Can beautomatedhttp://icawww1.epfl.ch/IS/tsed drift = ODE = = = 20

  21. Convergence to Mean Field E. L. E.L. • Thus: For the finite state space case, most cases are verifiable by inspection of the model • For the general state space, thingsmaybe more complex(fluidlimitis not an ODE, e.g. [Chaintreau et al, 2009], [Gomez-Serrano et al, 2012]) 21

  22. 3. FINITE HORIZON :FastSimulation and Decouplingassumption 22

  23. The DecouplingAssumption • Often used in analysis of complexsystems • Saysthatobjects are asymptoticallymutuallyindependent(isfixed and • Whatis the relation to meanfield convergence ? 23

  24. The DecouplingAssumption • Often used in analysis of complexsystems • Saysthatobjects are asymptoticallymutuallyindependent(isfixed and • Whatis the relation to meanfield convergence ? • [Sznitman 1991] [For a meanfield interaction model: ]Decouplingassumption converges to a deterministiclimit • Further, if decouplingassumptionholds, state proba for anyarbitraryobject 24

  25. The TwoInterpretations of the Mean Field Limit At any time t Thus for and simulation step : Prob (node n is dormant) ≈ 0.48 Prob (node n is active) ≈ 0.19 Prob (node n is susceptible) ≈ 0.33 m(t) approximatesboth the occupancymeasure MN(t) the state probability for one objectat time t, drawnatrandomamongN 25

  26. Fast Simulation • The evolution for one object as if the otherobjectshad a state drawnrandomly and independentlyfrom the distribution m(t) • Is validover finite horizon whenevermeanfield convergence occurs • Can beused to perform «fast simulation», i.e., simulate in detailonly one or twoobjects, replace the rest by the meanfieldlimit (ODE) 26

  27. Wecanfast-simulateone node, and evencomputeits PDF atany time whereis the (transient) probability of a continuous time nonhomogeneous Markov process • Same ODE as meanfieldlimit, withdifferent initial condition pdf of node2 (initially in A state) occupancymeasure pdf of node 1 pdf of node 3 27

  28. 4. Infinite Horizon: Fixed Point Method and Decouplingassumption 28

  29. DecouplingAssumption in StationaryRegime • Stationary regime = for large • Here: • Prob(node n is dormant) ≈ 0.3 • Prob (node n is active) ≈ 0.6 • Prob (node n is susceptible) ≈ 0.1 • Decoupling assumptionsaysdistribution of prob for state of one objectiswith • We are interested in stationaryregime, i.ewe do 29

  30. Example: 802.11 Analysis, Bianchi’s Formula 802.11 single cell mi = proba one node is in backoff stage I = attempt rate  = collision proba See [Benaim and Le Boudec , 2008] for this analysis ODE for meanfieldlimit Solve for Fixed Point: Bianchi’s Fixed Point Equation [Bianchi 1998] 30

  31. ExampleWhereFixed Point MethodFails Same as before except for one parameter value : h = 0.1 instead of 0.3 The ODE does not converge to a unique attractor (limit cycle) The equationhas a unique solution (red cross) – but itisnot the stationaryregime ! 31

  32. When the Fixed Point MethodFails, DecouplingAssumptionDoes not Hold In stationaryregime, follows the limit cycle Assume you are in stationaryregime (simulation has run for a long time) and you observe that one node, say, is in state ‘A’ It is more likelythatis in region R Therefore, itis more likelythatsomeothernode, say, isalso in state ‘A’ Nodes are not independent – they are synchronized R h=0.1 32

  33. Example: 802.11 withHeterogeneousNodes • [Cho et al, 2010]Two classes of nodeswithheterogeneousparameters (restransmissionprobability)Fixed point equation has a unique solution, but thisis not the stationaryprobaThere is a limit cycle 33

  34. Whereis the Catch ? Decouplingassumptionsaysthatnodesm and n are asymptoticallyindependent There ismeanfield convergence for thisexample But wesawthatnodesmay not beasymptoticallyindependent … isthere a contradiction ? 34

  35. Markov chainisergodic • The decouplingassumptionmay not hold in stationaryregime, even for perfectlyregularmodels • A correct statementis: conditionallyindependentgiven the value of the mean field limit Mean Field Convergence ≠ mi(t) mj(t) mi(t) mj(t) 35

  36. Positive Result 1 [e.g. Benaim et al 2008] : DecouplingAssumptionHolds in StationaryRegime if meanfieldlimit ODE has a unique fixed point to which all trajectories converge Decouplingholds in stationaryregime Decouplingdoes not hold in stationaryregime 36

  37. Positive Result 2: In the Reversible Case, the Fixed Point MethodAlways Works • DefinitionMarkov Process with transition rates q(i,j)isreversibleiff 1. itisergodic 2. p(i) q(i,j) = p(j) q(j,i) for somep • Stationary points = fixed points • If processwithfiniteNisreversible, the stationarybehaviourisdeterminedonly by fixed points. 37

  38. A Correct Method in Order to Make the DecouplingAssumptions • 1. Writedynamical system equationsin transientregime • 2. Study the stationaryregime of dynamical system • if converges to unique stationary point m*thenmakefixed point assumption • elseobjects are coupled in stationaryregime by meanfieldlimitm(t) • Hard to predictoutcome of 2 (except for reversible case) 38

  39. StationaryBehaviour of Mean Field Limitis not predicted by Structure of Markov Chain • MN(t)is a Markov chain on SN={(a, b, c) ≥ 0, a + b + c =1, a, b, c multiples of 1/N} • MN(t)isergodic and aperiodic, for any value of • Depending on , thereis or is not a limit cycle for m(t) SN(for N = 200) h = 0.3 h = 0.1

  40. Conclusion • Meanfieldmodels are frequent in large scalesystems • Validity of approachisoften simple by inspection • Meanfieldisboth • ODE for fluidlimit • Fast simulation usingdecouplingassumption • Decouplingassumptionholdsatfinite horizon; may not hold in stationaryregime(except for reversible case) • Study the stationaryregime of the ODE ! (instead of computing the stationaryproba of the Markov chain) 40

  41. Thank You … 41

  42. References 42

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  44. [Gomez-Serrano et al, 2012] Gomez-Serrano J., Graham C. and Le Boudec J.-Y.The Bounded Confidence Model Of Opinion DynamicsMathematical Models and Methods in Applied Sciences, Vol. 22, Nr. 2, pp. 1150007-1--1150007-46, 2012. 2 44

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