NASA Workshop on Collectives Ames Lab, 6 August 2002 Complex System Management:

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 … + 5 … + 1 … + 4 … + 3 … + 2 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 endogenous vs. exogenous … + 5 … + 1 … + 4 … + 3 … + 2

killer app: ‘designer system’ I system’s time evolution S (t) L(t) = L … + 5 … + 1 … + 4 … + 3 … + 2 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 … + 5 … + 1 … + 4 … + 3 … + 2 e.g. avoid ‘dangerous’ large changes

Complex Systems: Tails of the Unexpected Typically Levy-like Sits somewhere between Lorentzian and Gaussian, but hard to tell since • finite dataset • non-stationarity Fat tails etc. are ‘obvious’ from statistics but … temporal correlations (e.g. system crashes) do not show up! Distribution of increments of S (t )  big problem for standard risk analysis

don’t enter the game at time t Binary game Challet & Zhang In general, define w(t) according to the game of interest limited global resource level histories SO . . . . WHAT’S THE GAME ? 0 e.g. sell S 1 e.g. buy N history at time t . . . . 1 0 strategies history at time t +1 . . . . 0 1 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+L0sin w t time t time t

Global information m (t) for m=2 Stochastic perturbations from coin-tossing agents Periods of entirely deterministic behaviour Deterministic map of binary game evolution • Binary games behave as a stochastically perturbed deterministic system • Replace stochastic term from coin-tossing agents by its mean • Jefferies, Hart & NFJ Phys. Rev. E 65, 016105 (2002)

random matrix  initial strategy allocation  quenched disorder Deterministic map of binary game evolution [ PRE 65, 016105 (2002) ] • ‘ 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 • r confidence level • m (t) global information m{0,1,..P-1} P = 2m • a m ( t) response of strategies to m ( t ) ; aR {-1,1} • Y symmetrized strategy allocation tensor • Deterministic game defined by mapping equations: • Binary El Farol Game: w(t) = L(t) V(t) - n1 (t) • MG: L(t)=0.5 w(t) > 0  1 wins w (t) < 0  0 wins strategy R s = 2 strategy R’ 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  0for • stochastic strategy use • mixed-ability populations e.g. MG crowd - anticrowd pairs execute uncorrelated random walks sum of variances •  … also works for generalized games coin-toss walk step-size # of walks

GCMG m =3 GCMG m =10 Jefferies & NFJ cond-mat/0207523 Design of generalized binary games $G11 m =10 $G11 m =3 dynamical properties very sensitive to game’s microstructure $G13 m =10 $G13 m =3

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 demand described by: time during crash Assume: crash length: participating ‘crash’ nodes • Expected demand (and volume) during crash are thus given by:

Hart & NFJ cond-mat/0207588 Physica A (2002) in press Convergence of ‘parallel-world’ trajectories prior to crash system’s evolution  : 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

Optimal collectives of autonomous defectse.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 … + 5 … + 1 … + 4 … + 3 … + 2 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 defectse.g. nanodevice output, robot action, . . . Challet & NFJ, PRL 89, 028701 (2002) Tumer and Wolpert (2002) average error over all components med <> N = 10 random cost approach N = 20 <> unconstrained, analog N devices constrained, analog N devices

Optimal collectives of autonomous defectse.g. nanodevice output, robot action, . . .[Challet & NFJ, PRL (2002)] <> <> MG with agents accounting for their impact 2 strategies per agent unconstrained, analog N devices

Optimal collectives of autonomous defectse.g. nanodevice output, robot action, . . .[Challet & NFJ, PRL (2002)] N binary components Each component has I input bits Can perform F different logical operations, hence P = F 2I transformations f = probability that component i systematically gives wrong output  = fraction of component sets with at least one perfect subset 0.2 simple enumeration  0.25 0.3 & sorting f unconstrained, binary N devices

Optimal collectives of autonomous defectse.g. nanodevice output, robot action, . . .[Challet & NFJ, PRL (2002)] Optimum: average over 10,000 samples  = fraction of component sets with at least one perfect subset 0.2 simple enumeration   Majority Game: average over 300 samples, 500P iterations 2 components/agent 0.25 0.3 & sorting f Majority Game constrains the system to M=N/2 Possible improvement with Grand Canonical Majority Game GCMajG ? unconstrained, binary N devices

win lose Winning by losinglosing game + losing game = winning gameunsafe + unsafe = safe GAME A rotate randomly by GAME B rotate randomly by Randomly playing Games A and B

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

-1 +1 -1 -1 +1 +1 +1 -1 J. Parrondo et al. PRL (1999) Generalization to 2 history-dependent games: R. Kay & NFJ cond-mat/0207386 Application to quantum computing: C.F. Lee & NFJ quant-ph/0203043

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)

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’: • apply ‘no free lunch’ • minimize the ‘risk’ by choosing a suitable risk-management strategy

no risk management mission unsuccessful mission successful probability change in ‘wealth’ of system

risk management … but assume no frictioni.e. it ‘costs’ nothing to intervene 3 interventions 30 interventions probability change in ‘wealth’ of system change in ‘wealth’ of system standard deviation of ‘wealth’ distribution time between interventions

risk management … and frictioni.e. it ‘costs’ something to intervene 30 interventions 3 interventions probability change in ‘wealth’ of system change in ‘wealth’ of system standard deviation of ‘wealth’ distribution there is an ‘optimal’ time-delay between interventions time between interventions

NASA Workshop on Collectives Ames Lab, 6 August 2002 Complex System Management: