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					                NASA Workshop on Collectives
                  Ames Lab, 6 August 2002

      Complex System Management:
Hoping for the Best by Coping with the Worst

               Hoping for the best
        . . . but coping with the worst

                         Neil F. Johnson
                   n.johnson@physics.ox.ac.uk
            Department of Physics, Oxford University, U.K.

             Collaborators on several of the projects discussed:
P. Jefferies, D. Lamper, M. Hart, R. Kay, P.M. Hui & Damien Challet
                                 Outline
•   Collectives & complex systems: design issues
       global outcomes: best-case vs. OK-case vs. worst-case
       static vs. dynamical

•   Dynamical collectives: multi-agent models
      generalized binary games with time-dependent global resources
      deterministic vs. stochastic formalism
      undesirable outcomes -- system crashes & their control
                            -- fate, or just bad luck ?
      immunization of a complex system/collective

•   Static collectives: optimal collectives of autonomous defects
       near-perfect combinations of defective components
       defective component + defective component = working device

•   Winning by losing: optimal collectives of autonomous games
      successful combinations of unsuccessful games
      lose + lose = win, failure + failure = success, unsafe + unsafe = safe

•   Topology of collectives: network-based multi-agent games
•   Risk management in collectives & complex systems
                                  Topic
•   Collectives & complex systems: design issues
       global outcomes: best-case vs. OK-case vs. worst-case
       static vs. dynamical

•   Dynamical collectives: multi-agent models
      generalized binary games with time-dependent global resources
      deterministic vs. stochastic formalism
      undesirable outcomes -- system crashes & their control
                            -- fate, or just bad luck ?
      immunization of a complex system/collective

•   Static collectives: optimal collectives of autonomous defects
       near-perfect combinations of defective components
       defective component + defective component = working device

•   Winning by losing: optimal collectives of autonomous games
      successful combinations of unsuccessful games
      lose + lose = win, failure + failure = success, unsafe + unsafe = safe

•   Topology of collectives: network-based multi-agent games
•   Risk management in collectives & complex systems
•    Complex Systems
    •  Many degrees of freedom with internal frustration, feedback,
       history-dependence, adaptation, evolution, non-stationarity,
       non-equilibrium, memory, single realization, exogenous effects
    •  Collectives, multi-agent systems, forward and inverse problems
    •  Mix of deterministic and stochastic behavior

•   The Right Stuff
    System’s evolution can be optimized, controlled, managed. Robust

•   The Wrong Stuff
    System has a bad day . . .
    Heads down wrong path, leading to dangerous values, fluctuations,
    crashes. Endogenous and exogenous factors.
    Instabilities. Spontaneous secondary mission.              Brittle

•   The Good Stuff
    System behaves OK, not great but not bad
    Avoids bad scenarios, e.g. system crash

     PLAN B may be ‘best’ e.g. lowest risk
   Consider the global performance S(t) of a collective/complex system

    Examples [Workshop website, Tumer & Wolpert]:
       throughput in a data network
       total scientific information gathered by a constellation of deployable
        instruments
       GDP growth in a human economy
       percentage of available free energy exploited by an ecosystem

•   The Right Stuff: optimize/maximize global performance S(t)
        mission successful

•    The Good Stuff:
             S(t) less/more than Scritical for all time t, or time-window T
           <S(t)> less/more than Scritical for all time t, or time-window T
       Var[ S(t) ] less/more than critical for all time t, or time-window T
       < [ S(t) ]n > less/more than X for any n        etc….
    mission reasonably successful … not a disaster
    –                                                       mission not a
         disaster !
      real-world static system
system’s time evolution S (t)

                             ideal response L(t) = L
                             actual response L + 

                                              time
               …+1     …+2      …+3       …+4    …+5




e.g. minimize error by adjusting initial ‘quenched disorder’
 real-world dynamical system
system’s time evolution S (t)



                                global resource level L(t)
                                deterministic vs. stochastic
                                continuous vs. discrete
                                known vs. unknown
              …+1     …+2          …+3         …+4      …+5
                                endogenous vs. exogenous
 killer app: ‘designer system’ I
system’s time evolution S (t)



                                 L(t) = L



              …+1      …+2      …+3      …+4    …+5




    e.g. minimize ‘noise’, typical fluctuation size,
    hence optimize winnings, efficiency, use of global resource
killer app: ‘designer system’ II
system’s time evolution S (t)




                                           time
              …+1     …+2       …+3    …+4     …+5




                  e.g. avoid ‘dangerous’ large changes
Complex Systems: Tails of the Unexpected
 Example of Fat Tails
                                                                                                  x2
                                         C                                           Typically Levy-like
                                                                                           1       
 px                                                                         p x Sits somewhere between Lorentzian
                                                                                                     2 2
                                           2 2 (Lorentzian)                                      e x (Gaussian) (0.1)
                                      x C 
                                       2
                                                                                         2 x 2
                                                                                     and Gaussian, but hard to tell since
  Distribution of increments of S (t )                                                     •     finite dataset
        14
                                                                                           •     non-stationarity
    probability density p[  S/S ]




                                     12

                                     10                                              Fat tails etc. are ‘obvious’ from
                                                                                     statistics but …
                                      8                                              temporal correlationsGaussian
                                      6                                                                      do not show up!
                                                                                     (e.g. system crashes) Lorentzian

                                      4

                                      2

                                      0
                                      -0.15     -0.1   -0.05        0         0.05      0.1     0.15
                                                               re turn S/S



 ….but the variance (and hence volatilit y) is infinite for the Lorentzian
                                           big problem for standard risk analysis
                                  Topic
•   Collectives & complex systems: design issues
       global outcomes: best-case vs. OK-case vs. worst-case
       static vs. dynamical

•   Dynamical collectives: multi-agent models
      generalized binary games with time-dependent global resources
      deterministic vs. stochastic formalism
      undesirable outcomes -- system crashes & their control
                            -- fate, or just bad luck ?
      immunization of a complex system/collective

•   Static collectives: optimal collectives of autonomous defects
       near-perfect combinations of defective components
       defective component + defective component = working device

•   Winning by losing: optimal collectives of autonomous games
      successful combinations of unsuccessful games
      lose + lose = win, failure + failure = success, unsafe + unsafe = safe

•   Topology of collectives: network-based multi-agent games
•   Risk management in collectives & complex systems
                                        In general, define w(t) according to the game of interest
                                               don’t enter the game
    Binary game  f n t,n t ;n t  1,n t resource levelXt
           m   wt 
                                               Challet & Zhang
                                                      at time t
                                                  1; ;Lt,
                                                                                limited global
   2 histories        1     0      1         0

                                                         SO . . . . WHAT’S THE GAME ?
      11       10   01   00
      0        0    0     0                                                        exces s demand
      0        0    0     1
                               S
                                                       n0          0             Dt   n1 t  n0 t
       .       .    .     .
                                                             e.g. sell                       ?
                                                                                 V t,Dt  wt 
                                                                                            
      0        0    1     1
                                                                                 wt   0  1 wins
                                                                   1
       .       .    .     .


      0        1    1     1
                                                       n1                        wt   0  0 wins
       .       .    .     .                                  e.g. buy

                                         N
      1        1    1     1

                                                         n 0 t   n 1 t  V t   N
      2m
  2                      history at time t
strategies
                         ....10                                            history at time t +1

                                                                            ....01
                        agent memory m = 2
 Binary version of El Farol Game with time-dependent resource level
                      (i.e. seating capacity) L(t)
correlation between L(t)   and A(t)             system ‘learns’




                                                                  frequency w




                                       system ‘confused’


                                      attendance A (t)
                                      L(t)=L+L0 sin w t


                    time t                                                  time t
Deterministic map of binary game evolution

   Binary games behave as a stochastically perturbed deterministic system



       Global information  (t) for m=2
       3


       2


       1


       0
           32            64             96            12 8           16 0   19 2


                     Stochastic perturbations
                     from coin-tossing agents

                       Periods of entirely deterministic behaviour

   Replace stochastic term from coin-tossing agents by its mean

                 Jefferies, Hart & NFJ         Phys. Rev. E 65, 016105 (2002)
Deterministic map of binary game evolution
   ‘ attendance ’ = ‘ demand ’ A ( t ) = n1 (t) - n0 (t) = D ( t ) [not always true!]
                        ‘ volume ’   V ( t ) = n1 (t) + n0 (t)
       S (t)       strategy score vector                             [ PRE 65, 016105 (2002) ]
       r          confidence level
        (t)     global information {0,1,..P-1}     P = 2m
       a ( t ) response of strategies to  ( t ) ; aR {-1,1}
                 symmetrized strategy allocation tensor                strategy R
                                                                                   3   0   4       0   0   5   2   0   0   3

   Deterministic game defined by mapping equations:
                                                          s=2                      0

                                                                                   4
                                                                                       0

                                                                                       7
                                                                                           0

                                                                                           0
                                                                                                   7

                                                                                                   6
                                                                                                       0

                                                                                                       0
                                                                                                           1

                                                                                                           1
                                                                                                               0

                                                                                                               0
                                                                                                                   7

                                                                                                                   0
                                                                                                                       0

                                                                                                                       4
                                                                                                                           0

                                                                                                                           0
      Binary El Farol Game: w(t) = L(t) V(t) - n1 (t)                             0   0   0       0   8   0   3   2   0   0
      MG: L(t)=0.5 w(t) > 0  1 wins w (t) < 0  0 wins                           1   5   0       0   0   4   0   0   0   4




                                                                     strategy R’
                                                                                   0   4   0       7   0   6   0   3   0   0

                                                                                   3   0   0       0   1   0   0   1   0   7

                                                                                   0   0   1       0   3   0   0   2   3   0

                                                                                   0   2   0       3   0   2   7   0   4   0

                                                                                   4   0   7       0   4   3   0   3   0   0


                                                                             random matrix                     
                                                                              initial strategy allocation
                                                                              quenched disorder
                                                                                               1
                                                                                        2                    T
                                                                                                                       
In general, success & payoff may not be so simple to define  w(t) complicated functional form
Crowd - Anticrowd effect                                         J. Phys. A: Math. Gen. 32, L427 (1999)
                                                                  Physica A 298, 537 (2001)
                                       large crowds   >> 0  wastage
                                       but   0 for
e.g. MG                                    • stochastic strategy use
                                           • mixed-ability populations

                                                                             crowd - anticrowd
                                                 coin-toss                   pairs execute
          volatility                                                         uncorrelated
                                                                             random walks
              
                                                                             sum of variances
                                                                             
                                                                             … also works for
                                          memory m                           generalized games
            Memory m     2m+1 << N.s      2m+1 ~ N.s     2m+1 >> N.s
              Crowd      large            medium         ~1
               size
            Anticrowd    small            medium         ~0
               size
           Net crowd Ğ
           anticrowd
                         large            small          small                    walk step-size
            pair size    >> 1                            ~1
            # crowd -    ~ 2m             ~ 2m           < 2m
            anticrowd                                                             # of walks
              pairs      << N             <N             ~N
GCMG   m =3   GCMG   m =10        Jefferies & NFJ
                              cond-mat/0207523
                             Design of
                             generalized
                             binary games

$G11   m =3   $G11   m =10
                                   dynamical
                                   properties
                                 very sensitive
                                   to game’s
                                 microstructure
$G13   m =3   $G13   m =10
Jefferies & NFJ, cond-mat/0201540
 Lamper & NFJ, PRL 017902 (2002)
                                                                          Jefferies & NFJ, cond-mat/0201540
                                                                           Lamper & NFJ, PRL 017902 (2002)

           Anatomy of a system crash
       During persistence      Dt      sgnSR  r     sgnSR  r   
            demand described by:           R a  1
                                                R              R a  1
                                                                    R

                      time during crash



      Assume: SR a  1 ~ N S 1,
                   R
                                              
      crash length:             ma x   S 1, *   c *
                                          *        *               participating ‘crash’ nodes
       Expected demand (and volume) during crash are thus given by:
 Dt                                                  V t    
 c  r                                                c  r   
                                                                                   
erf               
                                                       2  erf              
                                                                                   
            2                                                      2      
                                                                             
                     
         c  r                                          c  r    
 erf              
                                                        erf             
        2                                              2  
                                               Hart & NFJ cond-mat/0207588
                                                     Physica A (2002) in press

      Convergence of ‘parallel-world’ trajectories prior to crash
system’s evolution




                                                                           
                                                  2   j i
                                            1           ( x  x ) P( x j )P( xi )
                                                                   2

                                                    i  j                   


                                             :    spread of paths
                                             indicates role of
                                             ‘fate’ vs. ‘bad luck’
                                Hart & NFJ cond-mat/0207588
                                      Physica A (2002) in press

Immunizing against system crash




        Protecting the system

Can reduce chances of system crash, by
   forcing earlier down-movements

       system gets immunized
                                  Topic
•   Collectives & complex systems: design issues
       global outcomes: best-case vs. OK-case vs. worst-case
       static vs. dynamical

•   Dynamical collectives: multi-agent models
      generalized binary games with time-dependent global resources
      deterministic vs. stochastic formalism
      undesirable outcomes -- system crashes & their control
                            -- fate, or just bad luck ?
      immunization of a complex system/collective

•   Static collectives: optimal collectives of autonomous defects
       near-perfect combinations of defective components
       defective component + defective component = working device

•   Winning by losing: optimal collectives of autonomous games
      successful combinations of unsuccessful games
      lose + lose = win, failure + failure = success, unsafe + unsafe = safe

•   Topology of collectives: network-based multi-agent games
•   Risk management in collectives & complex systems
         Optimal collectives of autonomous defects
               e.g. nanodevice output, robot action, . . .
        Challet & NFJ, PRL 89, 028701 (2002)    Tumer and Wolpert (2002)


         output

                                        ideal output L(t) = L
                                        actual output L + 

                                                            time
                        …+1       …+2          …+3      …+4     …+5


N defective devices with a distribution of errors

Combine a subset M < N to form high performance (i.e. low-error) collective:

unconstrained, analog               constrained, analog
unconstrained, binary               constrained, binary
Optimal collectives of autonomous defects
       e.g. nanodevice output, robot action, . . . error over
                                           average
Challet & NFJ, PRL 89, 028701 (2002)              all components
                                       Tumer and Wolpert (2002)
                                             <>               med


                                             N = 10




                                               N = 20     random cost
                                                            approach
    <>




unconstrained, analog                       constrained, analog
     N devices                                  N devices
Optimal collectives of autonomous defects
      e.g. nanodevice output, robot action, . . .
               [Challet & NFJ, PRL (2002)]




   <>                                       <>




                                             MG with
                                             agents accounting
                                             for their impact

unconstrained, analog
                                             2 strategies per agent
     N devices
     Optimal collectives of autonomous defects
               e.g. nanodevice output, robot action, . . .
                          [Challet & NFJ, PRL (2002)]
                                                 N binary components

    0.2                                          Each component has I input bits
                    simple enumeration           Can perform F different logical

    0.25                                         operations, hence
                                                 P = F 2I transformations

     0.3                      & sorting          f = probability that component i
           f                                     systematically gives wrong output


                                                  = fraction of component sets
                                                 with at least one perfect subset




       unconstrained, binary
            N devices
     Optimal collectives of autonomous defects
             e.g. component sets
     = fraction ofnanodevice output, robot action, . . .
                                                     Optimum: average
                         [Challet &
    with at least one perfect subsetNFJ, PRL (2002)] over 10,000 samples



    0.2
                   simple enumeration
                                        
    0.25                                               Majority Game:
                                                       average over
                                                       300 samples,
     0.3                     & sorting                 500P iterations
           f                                           2 components/agent




                                             Majority Game constrains
                                             the system to M=N/2
                                             Possible improvement with
                                             Grand Canonical Majority Game
          unconstrained, binary              GCMajG ?
               N devices
                                  Topic
•   Collectives & complex systems: design issues
       global outcomes: best-case vs. OK-case vs. worst-case
       static vs. dynamical

•   Dynamical collectives: multi-agent models
      generalized binary games with time-dependent global resources
      deterministic vs. stochastic formalism
      undesirable outcomes -- system crashes & their control
                            -- fate, or just bad luck ?
      immunization of a complex system/collective

•   Static collectives: optimal collectives of autonomous defects
       near-perfect combinations of defective components
       defective component + defective component = working device

•   Winning by losing: optimal collectives of autonomous games
      successful combinations of unsuccessful games
      lose + lose = win, failure + failure = success, unsafe + unsafe = safe

•   Topology of collectives: network-based multi-agent games
•   Risk management in collectives & complex systems
                   Winning by losing
           losing game + losing game = winning game
                     unsafe + unsafe = safe



                    GAME A
                                           2
win                 rotate randomly by   n         pwin  1 3  0.5
                                            3
                   GAME B
lose                                       2
                   rotate randomly by    m         pwin  3 7  0.5
                                            7

  Randomly playing Games A and B                pwin  11 21  0.5
              Colin                                          Nash
                                                             Colin
             A         B                               A              B
       A     (2, 2)       (-2, 9/4)            A       (-1/2, -1/2) (1, -1)
Rose                                  Rose
       B     (9/4, -2) (-1, -1)                B        (-1, 1)        (0, 0)


                         Switching randomly                          Pareto
                       between 2 ‘losing’ games
                         gives ‘winning’ game



                                      Colin
                                  A            B
                      A           (3/4, 3/4)   (-1/2, 5/8)
           Rose
                      B           (5/8, -1/2) (-1/2, -1/2)
         Game A                     Game B
                                                                  J. Parrondo et al. PRL (1999)
                                                          +1
                                    state 1          p1
           change in capital at     ( -1, -1 )                     Generalization to
           timestep t                               1 p1
                                                                   2 history-dependent
           +1 if win                                      -1
           -1 if lose                                              games:
                                                          +1       R. Kay & NFJ
                                   state 2           p2            cond-mat/0207386
                        +1         ( -1, +1 )
                                                    1 p2
                        p
+1 or -1                                                  -1
                      1 p                                         Application to
                                                          +1
 ( -1, -1 )             -1         state 3           p3            quantum computing:
 ( -1, +1 )                        ( +1, -1 )       1 p3          C.F. Lee & NFJ
 ( +1, -1 )                                               -1       quant-ph/0203043
 ( +1, +1 )
                                                         +1
                                  state 4           p4
                                  ( +1, +1 )        1 p4                -1 +1
 change in capital in                                              p1
                                                         -1                       p2
 previous two outcomes

           ( t-2, t-1)                           1 p1         -1 -1              +1 +1      p4
                                                                         p 3 1 p 2
                                                                1 p 3            1 p 4
                                                                         +1 -1
                                  Topic
•   Collectives & complex systems: design issues
       global outcomes: best-case vs. OK-case vs. worst-case
       static vs. dynamical

•   Dynamical collectives: multi-agent models
      generalized binary games with time-dependent global resources
      deterministic vs. stochastic formalism
      undesirable outcomes -- system crashes & their control
                            -- fate, or just bad luck ?
      immunization of a complex system/collective

•   Static collectives: optimal collectives of autonomous defects
       near-perfect combinations of defective components
       defective component + defective component = working device

•   Winning by losing: optimal collectives of autonomous games
      successful combinations of unsuccessful games
      lose + lose = win, failure + failure = success, unsafe + unsafe = safe

•   Topology of collectives: network-based multi-agent games
•   Risk management in collectives & complex systems
                Network games




Eguiluz & Zimmerman, PRL 85, 5659 (2001)  power-law tails
Zheng, NFJ et al., Eur. Phys. J. B 27, 213 (2002)
Analytics using generating function  tune power-law exponent
Herding  like-minded agents form clusters
          power-law distribution of cluster sizes & signal S(t)
                                  Topic
•   Collectives & complex systems: design issues
       global outcomes: best-case vs. OK-case vs. worst-case
       static vs. dynamical

•   Dynamical collectives: multi-agent models
      generalized binary games with time-dependent global resources
      deterministic vs. stochastic formalism
      undesirable outcomes -- system crashes & their control
                            -- fate, or just bad luck ?
      immunization of a complex system/collective

•   Static collectives: optimal collectives of autonomous defects
       near-perfect combinations of defective components
       defective component + defective component = working device

•   Winning by losing: optimal collectives of autonomous games
      successful combinations of unsuccessful games
      lose + lose = win, failure + failure = success, unsafe + unsafe = safe

•   Topology of collectives: network-based multi-agent games
•   Risk management in collectives & complex systems
  Risk management in collectives
 borrow terminology from finance [c.f. Hogg, Huberman]
 avoid standard local-in-time stochastic p.d.e. approach
 allow for non-Gaussian, non-stationary distributions, temporal correlations
 include friction due to communication/intervention costs
     variation of global `wealth’:
                              T /


                              i 0
                                          
     WT  C0  T   i [Si ] Si 1  Si   i  
     apply ‘no free lunch’

            WT  PRISK   varWT 
     minimize the ‘risk’ by choosing a suitable risk-management strategy

                      varWT 
                                       0
                        S,t   *
                             no risk management
                                                               mission              mission
                                                               unsuccessful         successful
               0.12

                0.1

               0.08
probability
        Prob




               0.06

               0.04

               0.02

                 0
                      -204   -181   -158   -136   -113   -90     -67   -45    -22   1
                                               Change in Wealth
                                           change in ‘wealth’ of system
                     risk management … but assume no friction
                          i.e. it ‘costs’ nothing to intervene
                             30 30 re-hedges
                                interventions                                                                  3 interventions
                                                                                                                   3 re-hedges
              0.25                                                                              0.07
               0.2                                                                              0.06
probability




                                                                                                0.05
              0.15
   Prob




                                                                                         Prob
                                                                                                0.04
               0.1                                                                              0.03
                                                                                                0.02
              0.05
                                                                                                0.01
                0                                                                               0.00
                 -17.2 -14.6 -12.1 -9.5 -7.0 -4.4 -1.9 0.7 3.3 5.8 8.4                              -61.1 -53.8 -46.4 -39.1 -31.8 -24.5 -17.2 -9.9 -2.6 4.8 12.1
                     change in ‘wealth’ of system
                                 Change in Wealth                                                              Change in of
                                                                                                   change in ‘wealth’ Wealthsystem

                                    standard deviation of ‘wealth’ distribution
                                                  Standard Deviation

                                                       10.00
                                                        8.00
                                            St. Dev.




                                                        6.00
                                                        4.00
                                                        2.00
                                                        0.00
                                                               0.0   2.0   4.0   6.0   8.0      10.0   12.0   14.0   16.0

                                                         time between interventions
                                                                    Trading time
                                  risk management … and friction
                               i.e. it ‘costs’ something to intervene
                                     interventions
                                 30 30 re-hedges                                                                        3 interventions
                                                                                                                            3 re-hedges

              0.04                                                                                     0.07
              0.03                                                                                     0.06
probability




              0.03                                                                                     0.05




                                                                                                Prob
    Prob




              0.02                                                                                     0.04
              0.02                                                                                     0.03
              0.01                                                                                     0.02
              0.01                                                                                     0.01
              0.00                                                                                     0.00
                  -87.1 -80.3 -73.5 -66.7 -59.9 -53.1 -46.3 -39.5 -32.8 -26.0 -19.2                        -68.5 -62.2 -55.9 -49.6 -43.3 -37.0 -30.6 -24.3 -18.0 -11.7 -5.4
                     change in ‘wealth’ of system
                                Change in Wealth                                                               change in ‘wealth’ of system
                                                                                                                          Change in Wealth

                                           standard deviation of ‘wealth’ distribution
                                                          Standard Deviation

                                                                 14
                                                                 12                                                                                      there is an
                                                                 10
                                                                                                                                                          ‘optimal’
                                                      St. Dev.




                                                                  8
                                                                  6                                                                                      time-delay
                                                                  4
                                                                                                                                                           between
                                                                  2
                                                                  0                                                                                     interventions
                                                                      0.0   2.0   4.0   6.0   8.0       10.0   12.0   14.0   16.0

                                                                 time between interventions
                                                                            Trading time

				
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posted:12/8/2011
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