nasa workshop on collectives ames lab, 6 august 2002 complex system management:

NASA Workshop on CollectivesAmes 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. [email protected]

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







• 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 StuffSystem’s evolution can be optimized, controlled, managed. Robust

• The Wrong StuffSystem 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 badAvoids 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 !

system’s time evolution S (t)

… + 1 … + 3 … + 4 … + 5… + 2

real-world static system

e.g. minimize error by adjusting initial ‘quenched disorder’

time

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

… + 1 … + 3 … + 4 … + 5… + 2

global resource level L(t)deterministic vs. stochasticcontinuous vs. discreteknown vs. unknownendogenous vs. exogenous

real-world dynamical system


… + 1 … + 3 … + 4 … + 5… + 2

e.g. minimize ‘noise’, typical fluctuation size,hence optimize winnings, efficiency, use of global resource

L(t) = L


killer app: ‘designer system’ I

time

… + 1 … + 3 … + 4 … + 5… + 2

e.g. avoid ‘dangerous’ large changes


killer app: ‘designer system’ II

Example of Fat Tails

€

p x( ) =

C

x

2

+ C

2

π

2

(Lorentzia )n

€

p x( ) =

1

2 π x

2

e

−

x

2

2 x

2

(Gaussian) (0.1)

0

2

4

6

8

10

12

14

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

return Δ S/S

probability density p[

ΔS/S

]

GaussianLorentzian

….but the variance (and hence volatilit y) is infinite for the Lorentzian

Distribution of increments of S (t )

big problem for standard risk analysis

Complex Systems: Tails of the UnexpectedTypically Levy-likeSits 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!

Topic







history at time t +1

. . . . 0 1

1e.g. buy

Binary game Challet & Zhang

0e.g. sell

€

n0

€

n1

S

N

history at time t

. . . . 1 0

agent memory m = 2

strategies

11 10 01 00

0 0 0 0

0 0 0 1

. . . .

0 0 1 1

. . . .

0 1 1 1

. . . .

1 1 1 1

€

22m

limited global resource level

€

n0 t( ) + n1 t( ) = V t( ) < N

don’t enter the gameat time t

€

2mhistories

€

excess demand

D t( ) = n1 t( )− n0 t( )

V t( ),D t( ) ? ⏐ → ⏐ w t( )

w t( ) > 0⇒ 1 wins

w t( ) < 0⇒ 0 wins

€

w t( ) = f n1 t( ),n0 t( );n1 t −1( ),n0 t −1( );K ;L t( ), X t( )[ ]

In general, define w(t) according to the game of interest

SO . . . . WHAT’S THE GAME ?

frequency w

time t time t

attendance A (t)

L(t)=L+L0 sin w t

system ‘confused’

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’

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)

0

1

2

3

32 64 96 128 160 192

Stochastic perturbationsfrom coin-tossing agents

Periods of entirely deterministic behaviour

Global information (t) for m=2

Deterministic map of binary game evolution

[ PRE 65, 016105 (2002) ]

3 0 4 0 0 5 2 0 0 3

0 0 0 7 0 1 0 7 0 0

4 7 0 6 0 1 0 0 4 0

0 0 0 0 8 0 3 2 0 0

1 5 0 0 0 4 0 0 0 4

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

strategy R

str

ate

gy

R’

€

Ψ=1

2 Ω + Ω T[ ]

random matrix initial strategy allocation quenched disorder

€

Ω

s = 2

In general, success & payoff may not be so simple to define w(t) complicated functional form

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 r confidence level (t) global information {0,1,..P-1} P = 2m

a ( t ) response of strategies to ( t ) ; aR {-1,1} Ψ 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

memory m

volatility

Memory m 2m+1 << N.s 2m+1 ~ N.s 2m+1 >> N.s

Crowd

sizelarge medium ~ 1

Anticrowdsize

small medium ~ 0

Net crowd –anticrowdpair size

large>> 1

small small

~ 1

# crowd -anticrowd

pairs

~ 2m

<< N

~ 2m

< N

< 2m

~ N

coin-toss

Crowd - Anticrowd effect

crowd - anticrowd pairs executeuncorrelatedrandom walks

sum of variances … also works for

generalized games

walk step-size

# of walks

e.g. MG

large crowds >> 0 wastage but 0 for

• stochastic strategy use• mixed-ability populations

J. Phys. A: Math. Gen. 32, L427 (1999) Physica A 298, 537 (2001)

$G11 m =3 $G11 m =10

$G13 m =3 $G13 m =10

GCMG m =3 GCMG m =10

dynamicalproperties

very sensitive to game’s

microstructure

Jefferies & NFJcond-mat/0207523

Design of generalizedbinary games

Jefferies & NFJ, cond-mat/0201540Lamper & NFJ, PRL 017902 (2002)

During persistence demand described by:

time during crash

Assume:

crash length: participating ‘crash’ nodes

Expected demand (and volume) during crash are thus given by:

€

τmax = S−1,μ *

μ *∑ = cμ*

μ *∑

€

D t +τ[ ]∝ sgn SR − r − τ[ ]R∋a R

μ =−1

∑ − sgn SR − r +τ[ ]R∋a R

μ =1

∑

Anatomy of a system crash

€

D t + τ[ ] ∝

erf−cμ + r + τ

2σ

⎡

⎣ ⎢

⎤

⎦ ⎥

−erfcμ + r −τ

2σ

⎡

⎣ ⎢

⎤

⎦ ⎥

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

€

V t +τ[ ] ∝

2 − erf−cμ + r + τ

2σ

⎡

⎣ ⎢

⎤

⎦ ⎥

−erfcμ + r −τ

2σ

⎡

⎣ ⎢

⎤

⎦ ⎥

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

€

SR∋aR

μ= −1

~ N S−1

,σ[ ]

Jefferies & NFJ, cond-mat/0201540Lamper & NFJ, PRL 017902 (2002)

)()()(21 2

ii j

jij xPxPxx∑ ∑ ⎥⎦

⎤⎢⎣

⎡−=λ

Convergence of ‘parallel-world’ trajectories prior to crash

Hart & NFJ cond-mat/0207588Physica A (2002) in press

syste

m’s

evolu

tion

λ : spread of paths

indicates role of ‘fate’ vs. ‘bad luck’

Protecting the system

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

system gets immunized

Immunizing against system crash

Hart & NFJ cond-mat/0207588Physica A (2002) in press

Topic







output

… + 1 … + 3 … + 4 … + 5… + 2

Optimal collectives of autonomous defectse.g. nanodevice output, robot action, . . .

Challet & NFJ, PRL 89, 028701 (2002) Tumer and Wolpert (2002)

time

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

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, analogunconstrained, binary constrained, binary

unconstrained, analogN devices

constrained, analogN devices


Challet & NFJ, PRL 89, 028701 (2002) Tumer and Wolpert (2002)

<>

<>

med

random costapproach

average error overall components

N = 10

N = 20


[Challet & NFJ, PRL (2002)]

<>

unconstrained, analogN devices

MG with agents accounting for their impact

2 strategies per agent

<>

N binary components

Each component has I input bitsCan perform F different logical operations, henceP = F 2I transformations

f = probability that component i

systematically gives wrong output

= fraction of component sets

with at least one perfect subset



f

unconstrained, binaryN devices

0.2

0.25

0.3

simple enumeration

& sorting



unconstrained, binaryN devices

0.2

0.25

0.3

f

Optimum: averageover 10,000 samples

Majority Game: average over 300 samples, 500P iterations2 components/agent

= fraction of component sets

with at least one perfect subset

Majority Game constrainsthe system to M=N/2Possible improvement with Grand Canonical Majority GameGCMajG ?

simple enumeration

& sorting

Topic







GAME Arotate randomly bywin

lose

€

n2π3

GAME Brotate randomly by

€

m2π7

€

pwin = 13 < 0.5

Randomly playing Games A and B

€

pwin =1121 > 0.5

€

pwin = 37 < 0.5

Winning by losinglosing game + losing game = winning game

unsafe + unsafe = safe

A B

A (2, 2) (-2, 9/4)

B (9/4, -2) (-1, -1)

Rose

ColinA B

A (-1/2, -1/2) (1, -1)

B (-1, 1) (0, 0)

Rose

Colin

A B

A (3/4, 3/4) (-1/2, 5/8)

B (5/8, -1/2) (-1/2, -1/2)

Rose

Colin

Pareto

Nash

Switching randomly between 2 ‘losing’ games

gives ‘winning’ game

+1 -1

-1 +1

-1 -1 +1 +1

€

p1

€

1− p1

€

1− p3

€

p3

€

1− p2

€

p2

€

p4

€

1− p4

+ 1

- 1

+ 1

- 1

+ 1

- 1

+ 1

- 1

+ 1

- 1

+1 or -1

( -1, -1 )

( -1, +1 )

( +1, -1 )

( +1, +1 )

( -1, -1 )( -1, +1 )

( +1, -1 )

( +1, +1 )

change in capital inprevious two outcomes

( t-2, t-1)

Game A Game B

change in capital attimestep t+1 if win-1 if lose

st ate 1

st ate 2

st ate 3

st ate 4

€

p1

€

1 − p1

€

p2

€

1 − p2

€

1 − p3

€

1 − p4

€

p3

€

p4

€

p

€

1 − p

Generalization to 2 history-dependentgames:R. Kay & NFJcond-mat/0207386

Application to quantum computing:C.F. Lee & NFJquant-ph/0203043

J. Parrondo et al. PRL (1999)

Topic







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







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

€

ΔWT = C0 − ΡT + φiτ [Siτ ] S i +1( )τ − Siτ( )i =0

T /τ

∑ +κ iτ

€

ΔWT = PRISK = λ var ΔWT[ ]

€

∂var ΔWT[ ]∂φ S, t[ ]

φ =φ *

= 0

no risk management

0

0.02

0.04

0.06

0.08

0.1

0.12

-204 -181 -158 -136 -113 -90 -67 -45 -22 1

Change in Wealth

Prob

change in ‘wealth’ of system

pro

bab

ilit

y

missionsuccessful

mission unsuccessful

risk management … but assume no frictioni.e. it ‘costs’ nothing to intervene

30 re-hedges

0

0.05

0.1

0.15

0.2

0.25

-17.2 -14.6 -12.1 -9.5 -7.0 -4.4 -1.9 0.7 3.3 5.8 8.4

Change in Wealth

Prob

3 re-hedges

0.000.010.020.030.040.050.060.07

-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

Prob

Standard Deviation

0.00

2.00

4.00

6.00

8.00

10.00

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0

Trading time

St. Dev.

30 interventions 3 interventions

standard deviation of ‘wealth’ distribution

change in ‘wealth’ of system change in ‘wealth’ of system

time between interventions

pro

bab

ilit

y

30 re-hedges

0.000.010.010.020.020.030.030.04

-87.1 -80.3 -73.5 -66.7 -59.9 -53.1 -46.3 -39.5 -32.8 -26.0 -19.2

Change in Wealth

Prob

3 re-hedges

0.000.010.020.030.040.050.060.07

-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

Prob

Standard Deviation

02468

101214

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0

Trading time

St. Dev.

30 interventions 3 interventions

pro

bab

ilit

y

risk management … and frictioni.e. it ‘costs’ something to intervene

change in ‘wealth’ of system change in ‘wealth’ of system

standard deviation of ‘wealth’ distribution

time between interventions

there is an ‘optimal’ time-delay between

interventions

nasa workshop on collectives ames lab, 6 august 2002 complex system management:

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