1. 1 ENS, June 21, 2007 Jean-Yves Le Boudec, EPFL joint work with David McDonald, U. of Ottawa and Jochen Mundinger, EPFL …or an Art ? Is Mean Field a Technology…

2. 2  Most of the material in this presentation is based on J.-Y. Le BoudecJ.-Y. Le Boudec, D. McDonald and J. Mundinger A Generic Mean Field Convergence Result for Systems of Interacting Objects 4th International Conference on the Quantitative Evaluation of SysTems (QEST) 2007D. McDonaldJ. Mundinger also available at http://infoscience.epfl.ch/getfile.py?recid=108827&mode=best  this slide show http://icawww1.epfl.ch/PS_files/mean-field-leb-ens-june07.ppt

3. 3 Contents 1.Mean Field in communication and computer modeling 2.A Generic Model for a System of Interacting Objects 3.Convergence to the Mean Field 4.Fast Simulation 5.Full Scale Example: A Reputation System 6.Central Limit Theorem 7.Outlook

4. 4 Mean Field: What is it ?  Introduced in Physics  Markov process, continuous time, represents states of particles  Mean field is a model where interaction between particles is via some quantity derived from distribution of states of all particle  Mean field is also an approximation for the state of one particle where we assume independence in the master equation ( equivalent to the forward equation, called Chapman-Kolmogorov in queuing theory textbooks) state of particle n rate of transition r effect of transition r on particle n distribution of states independence assumption

5. 5 Mean Field: What is it ?  Mean field is also an approximation for the distribution of states where we assume determinism in the master equation  This is often called the “mean field” equation

6. 6 Convergence to Mean Field  Proving these approximation is usually done in the sense of convergence when some parameter gets large (e.g. number of particles)  The techniques for that are: convergence of one Markov process towards another.  Sources of techniques are

7. 7 A Few Examples Where Applied Never again ! E.L.

8. 8 Generic Approach to show convergence to mean field  [Ethier and Kurtz] 1.Find a scaling variable N and a family of Markov processes M N indexed by N 2.Show that the family M N is relatively compact in the sense of topology of weak convergence 3.Find the limit of the martingale problem for M N and show that it has a unique solution

9. 9 Contents 1.Mean Field in communication and computer modeling 2.A Generic Model for a System of Interacting Objects 3.Convergence to the Mean Field 4.Fast Simulation 5.Full Scale Example: A Reputation System 6.Central Limit Theorem 7.Outlook

10. 10 Mean Field Interaction Model  A Generic Model, with generic results  Our first cut at making mean field a technology  Time is discrete  N objects  Every object has a state in.  Mean field approx reduces size of model from N £ S to S  Informally: object n evolves depending only on Its own state How many other objects are in each state

11. 11  Model assumptions: X N n (t) : state of object n at time t M N i (t) = proportion of objects that are in state i M N is the “occupancy measure” R N (t) = “history” of occupancy measure Objects draws next state according to

12. 12 Two Mild Hypotheses g() is continuous

13. 13 Example 1: Robot Swarm  No memory: R N = M N  N objects = N robots  Robot has S = 2 possible states  Fits in framework: transitions for one robot depends on own state + how many are in search state  K N independent of N after proper re-scaling  Equation for mean field is  Mild hypotheses are true because K is continuous in M (in fact: linear)

14. 14 Example 2: ECN/TCP Gateways  N objects = TCP connections  State = sending rate (discrete)  Memory R = buffer at ECN gateway  Mean field equation  q(r) = proba of negative feedback when R=r  Mild assumptions are true if q() is continuous ECN received no ECN received

15. 15 Example 3: Heterogeneous (Multiclass Example)  Same as previous but introduce multiclass model  Aggressive connections, normal connection  State of an object = (c, i) c : class i : sending rate  Objects may change class or not  Also fits in our framework  Mean Field does not mean all objects are exchangeable !

16. 16 Contents 1.Mean Field in communication and computer modeling 2.A Generic Model for a System of Interacting Objects 3.Convergence to the Mean Field 4.Fast Simulation 5.Full Scale Example: A Reputation System 6.Central Limit Theorem 7.Outlook

17. 17  A slightly weaker form was proven in many of the references mentioned earlier, in particular  A close, continuous time cousin is in

18. 18 Practical Application  This replaces the stochastic system by a deterministic, dynamical system  This justifies the mean field equation in the large N regime  This can be used as a “fluid approximation” N i (t) = number of objects in state i

19. 19 Proof of Theorem  Based on The next theorem (fast simulation) A coupling argument An ad-hoc version of the strong law of large numbers The Glivenko Cantelli lemma

20. 20 Contents 1.Mean Field in communication and computer modeling 2.A Generic Model for a System of Interacting Objects 3.Convergence to the Mean Field 4.Fast Simulation 5.Full Scale Example: A Reputation System 6.Central Limit Theorem 7.Outlook

21. 21 Fast Simulation / Analysis of One Object  Assume we are interested in one object in particular E.g. distribution of time until a TCP connection reaches maximum rate  For large N, since mean field convergence holds, one may forget the details of the states of all other objects and replace them by the deterministic dynamical system  The next theorem says that, essentially, this is valid

22. 22 Fast Simulation Algorithm Returns next state for one object When transition matrix is K State of one specific object This is the mean field independence approximation Replace true value by deterministic limit

23. 23 Fast Simulation Result

24. 24 Practical Application  This justifies the mean field approximation (based on the independence assumption) for the stochastic state of one object as a large N asymptotic  Gives a method for fast simulation or analysis The state space for Y 1 has S states, instead of N £ S

25. 25 Contents 1.Mean Field in communication and computer modeling 2.A Generic Model for a System of Interacting Objects 3.Convergence to the Mean Field 4.Fast Simulation 5.Full Scale Example: A Reputation System 6.Central Limit Theorem 7.Outlook

26. 26 A Reputation System  My original motivation for this work  Illustrates the complete set of steps  System N objects = N peers Peers observe one subject and rate it Rating is a number in (0,1) Direct observations and spreading of reputation Confirmation bias + forgetting

27. 27 Operation of Reputation System: Forgetting  Z n (t) = reputation rating held by peer n  Subject is perceived as positive (with proba  ) or negative (with proba 1-  )  In case of direct positive observation  In case of direct negative observation  w is the forgetting factor, close to 1 (0.9 in next slides)

28. 28 Confirmation Bias  Peer also read other peer ratings  If overheard rating is z:   is the threshold of the confirmation bias

29. 29 Initially: peers have Z=0, 0.5 or 1  = 0.9 Every time step: direct obs p=0.01, meet liar proba 0.30, meet honest proba 0.69 Example of simulation: N=100 peers with maximal liars (always say Z=0)

30. 30 3 particular peers, one of each type  = 0.9

31. 31 The problem fits in our framework…  Assume discrete time  At every time step a peer Makes a direct observation Or overhears a liar Or overhears some honest peer Or does nothing  Object = honest peer  Transition depends on Own state Distribution of states of all other peers => Fits in our framework with memory R = occupancy measure M

32. 32 We would like to apply the mean field convergence result to analyze very large N  But model is continuous state space  Discretize reputation ratings ! Quantize Z n on L bits; replace Z n by X n = 2 L Z N, constrained to be integer Random rounding to avoid side effects; replace by where RANDROUND(2.7) = 2 with proba 0.3 and 3 with proba 0.7 E(RANDROUND(x)) = x

33. 33 Transition Matrix  Is deduced from the rules  The hypotheses apply (almost trivially; K is linear in M) Proba of meeting a honest peer other than self Converges uniformly in M to M N k (t) for large N

34. 34 Limiting reputation ratings: 0.9 and 0.1

35. 35 Different Parameters (few liars) Few liars Final ratings converge to true value Phase transition

36. 36 Different Initial Conditions

37. 37 The Model can serve different Liar Strategies  Strategy 1: liars always say Z= 0  Strategy 2: liar guesses your rating based on past experience Similar to strategy 1, except memory R = memory occupancy at step t-1  Strategy 3: liars know your rating and is as negative as you accept Similar to strategy 1, memory = occupancy measure M

38. 38 Liar Strategy 2 (infer) Liar Strategy 3 (side information) Peers starting at time 512

39. 39 Contents 1.Mean Field in communication and computer modeling 2.A Generic Model for a System of Interacting Objects 3.Convergence to the Mean Field 4.Fast Simulation 5.Full Scale Example: A Reputation System 6.Central Limit Theorem 7.Outlook

40. 40 A Central Limit Theorem Similar to Van Kampen’s Method  Case without memory and assuming K is differentiable:

41. 41 Contents 1.Mean Field in communication and computer modeling 2.A Generic Model for a System of Interacting Objects 3.Convergence to the Mean Field 4.Fast Simulation 5.Full Scale Example: A Reputation System 6.Central Limit Theorem 7.Outlook

42. 42 For the cases that fit in this framework…  We are closer to …  than to …

43. 43 Extend the modelling framework to: birth and death of objects transitions that affect several objects simultaneously enumerable but infinite set of states Further work E. L.