chess complex dynamics

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arXiv:1103.3681v1[physics.data-an]18Mar2011

Optimal Dimensionality Reduction of Complex Dynamics:

The Chess Game as Diffusion on a Free Energy Landscape.

Sergei V. KrivovInstitute of Molecular and Cellular Biology, University of Leeds, UK

(Dated: March 21, 2011)

Dimensionality reduction is ubiquitous in analysis of complex dynamics. The conventional dimen-sionality reduction techniques, however, focus on reproducing the underlying configuration space,rather than the dynamics itself. The constructed low-dimensional space does not provide completeand accurate description of the dynamics. Here I describe how to perform dimensionality reductionwhile preserving the essential properties of the dynamics. The approach is illustrated by analyzingthe chess game - the archetype of complex dynamics. A variable that provides complete and accu-rate description of chess dynamics is constructed. Winning probability is predicted by describingthe game as a random walk on the free energy landscape associated with the variable. The approachsuggests a possible way of obtaining a simple yet accurate description of many important complexphenomena. The analysis of the chess game shows that the approach can quantitatively describe thedynamics of processes where human decision-making plays a central role, e.g., financial and socialdynamics.

INTRODUCTION.

Complex processes are often described by a single vari-able to simplify their analyses. Examples include orderparameters in physics [1, 2], biomarkers in medicine [35], indexes and asset prices in economics [6, 7], citationcounts in bibliometrics [8, 9], and educational grades.To have descriptive and predictive power, or serve asan optimization target, the variable should completelyspecify the current state and future dynamics of the pro-cess. Construction of such variables is challenging. Con-ventional dimensionality reduction techniques, such asprincipal component analysis, and their generalizations[10, 11] often fail to produce such variables [2]. The tech-niques focus on compact representation of an ensemble of

configurations (the configuration space). The dynamicalinformation contained in the temporal sequence of theconfigurations (trajectory) is ignored.

If the future dynamics is completely specified by thecurrent value of the variable and does not depend onhistory or the values of other variables, the dynamics issaid to be Markovian. In this case the dynamics of thevariable reproduces the original dynamics, i.e., the pro-

jection preserves the dynamics. The stochastic dynamicsof the variable can be simply described as diffusion on afree energy profile and is completely specified by the freeenergy profile and the diffusion coefficient. Conversely,diffusive dynamics implies Markovianity.

Here I describe a method of constructing variables (theoptimal reaction coordinate hereafter) which preserve thedynamics upon the projection. The method originatedin the protein folding field, where the reaction coordi-nate and the associated free energy landscapes are usedto described in a simplified yet accurate way the com-plex dynamics of protein folding (see inset of Fig. 1)[12, 13]. The optimal reaction coordinate is constructedbased on the system dynamics. Given reaction coordi-nate time series X(it), the cut-based free energy pro-file (cFEP) can be constructed FC(x)/kT = lnZC(x),

where partition function ZC(x) equals half the numberof transitions performed by the reaction coordinate times

series through point x [14, 15] (for details see Appendix).The cFEP is complementary and superior to the conven-tional histogram based free energy profile as it is invariantto reaction coordinate rescaling, insensitive to statisticalnoise and capable of detecting sub-diffusion. Togetherthey determine the coordinate dependent diffusion coef-ficient D(x) and thus completely specify diffusive dynam-ics [14]. The optimal reaction coordinate is the one withthe highest cFEP [14, 15] with the following rationale.It generalizes the definition of the transition state as theminimum cut to any position on the reaction coordinate[16]. Projection on a bad reaction coordinate resultsin smaller barriers and faster kinetics due to overlappingof different parts of the configuration space. The optimalcoordinate exhibits the slowest kinetics [14]. The dynam-ics projected on this coordinate is closest to diffusive [15].A putative functional form of the reaction coordinate isproposed based on a general understanding of the pro-cess. For example, a linear combination of featuresthat could describe the process. For protein folding itcould be a weighted sum of distances between atoms ofa protein [15]. The coordinate is optimized (trained) ona sample of trajectories representing realizations of theprocess. The coefficients of the functional form are nu-merically optimized to make the cFEP along the coordi-nate the highest.

To emphasize the power and generality of the ap-proach the dynamics of the chess game is analyzed[17]. The chess is a model system of research in arti-ficial intelligence. Its complex dynamics is not gener-ated by a physical system and thus applicability of thefree energy landscape framework is not evident. Thegames played by the computer program GNUCHESS(http://www.gnu.org/software/chess) against itself areanalyzed here. No generality is lost, since computerssurpassed humans at chess when the Deep Blue won therematch with the World Chess Champion Garry Kas-
http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://arxiv.org/abs/1103.3681v1http://www.gnu.org/software/chesshttp://www.gnu.org/software/chesshttp://arxiv.org/abs/1103.3681v1


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0Z0Z0Z0ZS0Z0Z0Z00Z0Z0Z0ZZ0Z0Z0Z00Z0Z0Z0Zj0J0Z0Z00Z0Z0Z0ZZ0Z0Z0Z0

0Z0Z0Z0ZZ0Z0Z0Z00Z0Z0Z0ZZ0Z0Z0Z00Z0Z0Z0ZZ0Z0ZkZ00Z0Z0ZqZZ0Z0Z0ZK

rmblkansopopopop0Z0Z0Z0ZZ0Z0Z0Z00Z0Z0Z0ZZ0Z0Z0Z0POPOPOPOSNAQJBMR

white winsblack wins

Fre

eEnergy

reaction coordinate

state

denatured

state

transition

state

native

FIG. 1. (color online) Cartoon chess free energy land-scape. The game is described as a random walk (diffusion)on the free energy landscape (red line). Starting from themiddle the game continues until either the right (white wins)or the left (black wins) end of the profile has been reached.Lower barrier for white reflects that white has more chancesto win. The boards show representative p ositions for the re-gions on the landscape. The inset shows protein folding freeenergy landscape.

parov (http://www.research.ibm.com/deepblue). Theidea that the chess game can be described by a randomwalk may seem rather extravagant (Fig. 1). To gainan advantage, players devise precise sequences of moves,which are anything but random. However, in contrast tochecker [18] the chess are not solved yet. One can nottell the result of a chess game starting from any positionif played optimally. The result can only be guessed, sug-gesting applicability of stochastic description to the chessgame, which is corroborated by results presented below.

To keep the analysis one-dimensional only games thatend in a victory are considered. 10000 games, initiallypreprocessed (see Appendix) are used for the analysis.As a putative reaction coordinate the evaluation functionE(p), used in computer chess, is chosen. The functiongives a quantitative estimation of the value of a position(p) and is a weighted sum of various factors (the majorbeing the material factor) [19]. For example, a pawnhas a material value of 100 and a queen of 1100, so thatE(p) = 100(pw pb)+1100(Qw Qb) + ..., where pw, pband Qw, Qb are the number of white and black pawns andqueens on the board, respectively, and ... includes otherfactors describing more subtle properties of a position,

such as, board control, mobility, pawn structure, passedpawns, etc. Note that alternative functional forms ofreaction coordinate (e.g., the artificial neural networks)can be employed.

REACTION COORDINATE OPTIMIZATION.

The chess game has a non-equilibrium dynamics. Thegames proceed from the starting position to a checkmate

and never backwards. In this case the dynamics is com-pletely specified by Z+C(x) and Z

C(x) which measure theflux (number of transitions through point x) in the posi-tive and negative directions, respectively (Fig. 2a). Theirdifference manifests the non-equilibrium character of thedynamics and equals (for positive x) to the number ofgames won by white. The exponent shows that dy-namics along E(p) is sub-diffusive 0.3. The expo-nent measures how the amplitude of the random jumps(changes of the position) scales with time x t;for diffusive dynamics = 0.5 and x t. Thesub-diffusive dynamics indicates that the projected dy-namics is not Markovian (consequent displacements areanti-correlated) and that the putative reaction coordi-nate is not optimal, i.e., its value alone does not com-pletely specify the dynamics.

It has been shown that sub-diffusive dynamics in pro-tein folding is observed when a sub-optimal reaction coor-dinate is used for description [15]. Indeed, the evaluationfunction E(p) employed in computer chess is a poor re-action coordinate. It can distinguish between positions

where white (or black) has a clear advantage. Positionswith more subtle advantage or at highly dynamic phasesof the game (e.g., during an exchange of pieces) can notbe accurately evaluated [19]. In order to accurately evalu-ate a position, the computer performs an extensive searchover all possible continuations of the position to a signif-icant extent and selects the one that maximizes the min-imum gain [19]. Such brute force number-crunching is incontrast with a human way of playing chess with creativ-ity and intuition [20]. Here, instead, the coordinate is op-timized by making the FC higher. The numerical param-eters ofE(p) (e.g., the queens material value) were (iter-atively) randomly modified and kept if the modificationresulted in a higher FC (for details see Appendix). The

cFEPs for the optimized coordinate E(p) are marginallyhigher (by 0.3 around x=0, Fig. 2b) than that for thesub-optimal case (Fig. 2a). 0.5 indicates that thedynamics is diffusive and Markovian and that the op-timized reaction coordinate completely specifies the dy-namics, i.e., is the optimal reaction coordinate.

The chess game and the protein folding [15], both il-lustrate useful generic property of the cFEPs: the higheris the profile the more diffusive is the dynamics, i.e., thedynamics is not sub-diffusive per se. Consider two rea-sonably good reaction coordinates which differ locally butgive similar large-scale description of dynamics. At suffi-ciently large times scale (t2), when memory effects due

to sub-optimal projection [21] can be neglected and dy-namics is Markovian, the coordinates have similar cFEP.The sub-diffusion exponent can be estimated from thedistance between the profiles computed with the largetime step (t2) and the original (small) time step (t1):

(x) = 1 +ln ZC(x, t1) ln ZC(x, t2)

ln t1 ln t2 .

The higher is the cFEP (the smaller ZC) at the originaltime scale of t1, the smaller is the distance between the


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F/kT

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0.3 0.3

0.4 0.4

0.5 0.5

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lnZ

lnZ

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XXX

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FIG. 2. (color online) Free energy profiles of the chess game. a) Z+C

(dashed red), ZC

(dotted blue) and (solid black)

for the sub-optimal E(p) reaction coordinate. b) Z+C

(dashed red), ZC

(dotted blue) and (solid black) for the optimal E(p)reaction coordinate. c) The chess game is described as diffusion on the (equilibrium) cFEP. The cFEP is computed with the

free energy profile (solid red) and the Markov network (dashed blue) frameworks (from the cFEPs on panel b). E(p) and E(p)are transformed (rescaled) to X so that the diffusion coefficient D(x) equals to unity.

profiles, the larger is and more diffusive is the projecteddynamics.

THE EQUILIBRIUM FEP.

The Z+C(x) and Z

C(x) (Fig. 2b) computed from thenon-equilibrium trajectories do not represent the under-lying (equilibrium) FEP. The latter, however, can becomputed from them (Fig. 2c) (see Appendix). In or-der to emphasize the robustness of the results, they arerecomputed with the (complementary) Markov networkformalism (see Appendix), which describes the dynam-ics by a network of transitions between different states[22]. Though the profile computed with the Markov net-work shows some noise, the results obtained with both

frameworks are in very good agreement.The chess game (as a whole) is described in a simple,while accurate way as diffusion (with D(x) = 1) on theprofile. A game represents a particular realization of thestochastic diffusion process. It starts at x = 0 and con-tinues until either the right (white wins) or the left (blackwins) end of the profile has been reached. The relativelyflat region of the profile (|x| < 2.5) suggests that, initially,a trajectory (game) may switch many times between pos-itive and negative parts. It describes the constructive,search phase of the game, where the opponents are try-ing to get an advantage. After the barriers (|x| > 2.5)the profile is much steeper, making a return to the op-posite part very unlikely, meaning that the barriers arethe rate limiting step in the game dynamics. As soonas a decisive advantage has been gained (a barrier hasbeen overcame) the game strategy becomes much sim-pler: it is sufficient to exchange the pieces, while keepingthe advantage. The probability of overcoming a barriercan be roughly estimated as half of the probability of be-ing at the top of the barrier pi 0.5 exp(Fi/kT), whereFi is the barrier height [23]. The winning probability (ofwhite) P = pw/(pw+pb) = e

F/kT/(1+eF/kT) 0.6is in agreement with the number computed directly from

the games of 0.59; F/kT = Fw/kT Fb/kT 0.4(Fig. 2c).

THE PROBABILITY TO WIN AND GAMEANALYSIS.

A more stringent test is to compare the winning proba-bility P(x) for any position x, which is the probability toreach the right end of the profile, before reaching the leftend. It is analogous to the folding probability or com-mitor used in protein folding studies [24, 25]. Fig. 3ashows P(x) calculated directly from the trajectory, fromthe diffusive dynamics on the cFEP and with the Markovnetwork framework (for details see Appendix). They arein a very good agreement. P(x) computed with the sub-

optimal coordinate (Fig. 2a) notably disagree (Fig. 3b)because dynamics along this coordinate is neither dif-fusive nor Markovian. Naive approach to estimate thewinning probability by collecting statistics for every po-sition is impractical due to the sheer size of the configu-ration space (estimated as 1043 [19] ). Markovian coarse-graining of the space is essential, and is obtained here bythe projection onto the optimal coordinate. Once con-structed, the optimal reaction coordinate can be used toanalyze a chess game in simple terms by showing the evo-lution of the winning probability. As an example, gamebetween Garry Kasparov and Veselin Topalov played inHoogovens in 1999 is considered (Fig. 3c). Initially, blackequalizes the chances to win the game (at 10-th move).

Then, white increases the chances to win up to P = 0.7(at 23-rd move). The part of the game between 24-th and38-th moves (dashed line) which starts with the rook sac-rifice can not be analyzed due to the shortcomings of theevaluation function. More sophisticated variants of theevaluation function should make possible the analysis ofthe entire game.

The optimization procedure effectively tries to de-crease the size of each step, so that the chess dynamicsalong the optimal reaction coordinate consists of small


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FIG. 3. (color online) The probability to win. a) The probability (for white) to win the game starting from any position

computed along the optimal coordinate E(p): with diffusion on the cFEP (solid red) and with the Markov network framework(dashed blue), and directly from the played games (crosses). The inset shows the plot in logarithmic scale. b) that along thesuboptimal reaction coordinate E(p). c) The analysis of the game between Garry Kasparov and Veselin Topalov: probabilityto win vs the move number.

incremental changes (e.g., moves 1-23 on Fig. 3c). How-ever, the conclusion that a brilliant move, which alone

can change the course of a game is impossible, is wrong.Since the game is described as a random process, there isnon-zero Gaussian probability to have a large move along

the reaction coordinate p(x) ex2/2 (the diffusioncoefficient D = 1).

CONCLUDING DISCUSSION.

An optimization (learning) principle to simultaneouslyoptimize the reaction coordinate and the playing strategycan be suggested. ZC(x) (Fig. 2b, blue) which countsthe retrograde moves, is an indicator of the quality of

play. The quality consistently improves with increasing xsince the complexity of the game decreases. When movesare chosen according to the perfect winning strategyZC(x) = 0. Hence, FC(x) attains the upper bound(ZC(x) = 0) for the best playing strategy and the op-timal coordinate describing it. The principle can be usedto measure and optimize performance of stochastic al-gorithms, for example, to improve global optimizationheuristics [26].

The presented analysis can be improved in the follow-ing ways: the development of more sophisticated variantsof the evaluation function to treat the dynamic phases ofthe game; the construction of reaction coordinates tai-lored to different types of positions, or perhaps reactioncoordinate which is iteratively updated during the game;two dimensional free energy landscape to allow analysisof games ending in a draw.

Assuming that the chess game is a stochastic processthe optimal coordinate that provides accurate descriptionof the process has been constructed. The cut free energyprofile is a function of the reaction coordinate time se-ries alone. Detailed specification of the dynamics (eventhe rules of the game) is not necessary to perform theoptimization. It allows the approach to analyze phenom-

ena which numerous characteristics can be monitored,while construction of the complete dynamical model is

impractical. The analysis of the chess game shows thatthe approach can quantitatively describe the dynamics ofprocesses where human decision-making plays a centralrole, e.g., financial and social dynamics.

An interesting application is the construction of dis-ease biomarkers [5] , specifically for such difficult casesas cancer [3], ageing [27] or psychiatric disorders [4]. Anoptimal biomarker and the associated free energy land-scape that give an accurate description of the diseasedynamics can be constructed out of panel of monitoredparameters e.g., metabolomic, proteomic and genomicdata. Diffusive dynamics would indicate that the optimalbiomarker gives complete (Markovian) description, andsub-diffusive that some essential information is missing.Unlike conventional approaches, the proposed approachexplicitly treats the dynamical character of the process.

ACKNOWLEDGMENTS

This work was supported by an RCUK fellowship.

APPENDIX

The cut free energy profiles. Given reaction coor-

dinate (R( X)) and an ensemble of trajectories Xi(jt)sampled with interval t the ensemble of reaction coor-

dinate trajectories is defined as xi(jt) = R( Xi(jt)).The partition function of the conventional (histogram-based) free energy profile is estimated as

ZH(x) = Nx/x, (1)

where Nx is the number of trajectory points in bin x andx is the size of the bin. The partition function of thecut based free energy profile [14] at point x is estimated


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as half the number of transitions through that point, i.e.,

ZC(x) = 1/2i,j

{(xi(jt) x)(x xi(jt + t)},(2)

where {x} is the Heaviside step function.For Gaussian distribution of steps P(x)

exp(x2/(4Dt)) and assuming that FH(x) is approx-imately constant on the distance of the mean absolutedisplacement |x(t)|, one obtains [14]

ZC(x) = ZH(x)

D(x)t/; (3)

For non-Gaussian distribution of steps, observed here,assuming x2 = 2Dt, Eq. 3 is corrected by factor

ZC(x) = ZH(x)

D(x)t/ (4)

= |x|/

2x2/ (5)The correction factor is assumed to be coordinate inde-pendent and is computed from the entire trajectory. Thedistribution of steps is expected to converge to Gaussian

for more sophisticated evaluation functions, where eachmove changes a significant number of terms of the func-tion.

When dynamics is not equilibrium the net trajectoryflux is not zero and the detailed balance is not satis-fied. In this case positive and negative cut profiles thatmeasure the flow in the corresponding direction are in-troduced

Z+C(x) =i,j

(x xi(jt))(xi(jt + t) x)(6)

ZC(x) =i,j

(xi(jt) x)(x xi(jt + t)). (7)

Diffusion with constant flux J and gradientF(x)=ax. The flux for the steady state solution Pst(x)of the Smoluchowski equation is

J = D(x)eF(x)/x(PsteF(x)),

where = 1/kT and D(x) = D. Steady state distribu-tion is found as

Pst(x) = ZH(x) =J

Da Ceax,

where C is a constant defined by a boundary condition.

Distribution of displacements x during time step of tis

p(x, t) =1

4Dtexp[ (x + Dat)

2

4Dt].

Z+C, the number of transition from y < 0 to y > 0 equalsto

Z+C =

0

dyZH(y)

y

p(x, t)dx.

ZC, the number of transition from y > 0 to y < 0 equalsto

ZC =

0

dyZH(y)

y

p(x, t)dx.

Integrating over y one obtains

Z+C =

0

dxp(x, t)(Jx

Da + C1

eax

a )

ZC =

0

dxp(x, t)( JxDa

C1 eax

a).

The net flow equals to

Z+C ZC =

dxp(x, t)Jx

Da= Jt

ZC = (Z+C+ Z

C)/2 is found by expanding the exponents

ZC = ZH

Dt/(1 + O(Dt2

a2

))

The equilibrium FEP. F(x) can be found as

F(x)/kT = ln Pst(x) x J(x)dx

D(x)Pst(x).

Using Pst(x) = ZH(x), J(x)t = Z+C(x) ZC(x) and

D(x) = (ZC/ZH)22/t, one obtains

F(x)/kT = ln ZH(x) x 2(Z+C(x) ZC(x))ZH(x)dx

Z2C(x)(8)

The natural coordinate. A reaction coordinate (x)

with variable diffusion coefficient can be transformed tothe natural coordinate (y) with the diffusion coefficientequals to unity by numerically integrating [14]

dy = 1

t/Zh(x)/Zc(x)dx. (9)

In this case the diffusive dynamics is completely specifiedby the free energy profile only.

The sub-diffusion exponent. In case of sub-diffusive dynamics, the mean absolute displacementscales with time as |x(t)| t, where < 0.5.The coordinate dependent exponent is determined bycomparing ZC(x) at two different sampling intervals [15]

(x) = 1 + ln ZC(x, t1) ln ZC(x, t2)ln t1 ln t2 . (10)

Reaction coordinate optimization. The optimalreaction coordinate is defined as the one that has cutbased free energy profile FC(x) highest for every value(x) of the coordinate [1416]. It generalizes the defini-tion of the transition state as the minimum cut to anyposition on the reaction coordinate [16]. Projection ona bad reaction coordinate results in smaller barriersand faster kinetics due to overlapping of different parts


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of the configuration space. The optimal coordinate ex-hibits the slowest kinetics [14]. The dynamics projectedon this coordinate is closest to diffusive [15].

The coordinate as a whole is optimized by numericallymaximizing

Z1C (x)ZH(x)dx. For a flexible form of the

reaction coordinate where ZC(x) for all x can be con-sidered as independent the functional attains the maxi-mum when all ZC(x) are minimal (all FC(x) are maxi-

mal). For a less flexible reaction coordinate the optimummay be a compromise solution where different parts ofthe coordinate are optimized to a different degree. An-other optimization functional max

Z2C (x)ZH(x)dx

eF(x)/kT/D(x)dx, which assigns more weight to thehigher part of the profile, gives very similar results. Forthe case of over-damped Langevin dynamics

eF(p)/kT/D(p)dp

attains the maximum when p is the committor function(pcomm( X)) or the folding probability, an alternative def-

inition of the optimal reaction coordinate [28].The reaction coordinate is a weighted sum of differentcomponents of the evaluation function E(p) =

aifi(p),

where ai are the weights, fi(p) are the components of theevaluation function and p denotes the position. Start-ing with the initial set of the weights a0i (that of theGNUCHESS program) the coordinate is iteratively im-proved by randomly changing the weights ai and accept-ing the change if the value of the functional is increased.

The cFEP is invariant with respect to local changesof scale of the evaluation function (gauge invariant), itdepends only on the relative order of the points. Thismakes the construction of the optimal coordinate mucheasier, since a putative reaction coordinate E(p) shouldonly reproduce the relative order, not the absolute valueof the goodness of the positions.

The Markov network framework. The Markovnetwork describing the dynamics is constructed by parti-tioning the reaction coordinate into bins of size 0.005and computing the transition probabilities as pij =nij/

i nij, where nij is the number of transitions from

bin j to bin i. The reaction coordinate was convertedto the natural coordinate to ensure uniform partition.The equilibrium FEP was constructed by computing theequilibrium network neqij

neqij = pijpeqj , (11)

where peqj are the equilibrium populations

peqi =j

pijpeqj , (12)

and the sub-networkpij is the strongly connected compo-nent of the network, i.e., there is a path in each directionbetween any two nodes of the sub-network. In particular,the terminal (checkmate) nodes are discarded. ZC and

ZH are computed as

ZC(x) = 1/2ij

neqij {(x(i) x)(x x(j))} (13)

ZH(x) =

ij,x(j)=x

neqij , (14)

where x(i) and x(j) are the positions of the clusters i and

j, respectively.The probability to win (the committor or the fold-

ing probability in the analysis of protein folding dy-namics) measures probability to reach point x = b be-fore reaching point x = a. For the diffusion on a one-dimensional free energy profile it can be estimated [29]as

pcomm(x) =

xa

eF(x)/kT/D(x)dx/

ba

eF(x)/kT/D(x)dx,

which can be expressed as

pcomm(x) = x

a Zh(x)Z2c (x)dx

ba Zh(x)Z2c (x)dx(15)

It can be estimated from the Markov network as

pcommi =j

pjipcommj (16)

with pcomm10 = 1 and pcomm01 = 0, where 1-0 and 0-1 are the

nodes corresponding to the states where white or blackhas won, respectively. It can be estimated directly fromthe trajectories as pcommi = n

wi /(n

wi + n

bi ), where n

wi and

nbi are the number of times trajectory visiting bin i endsin whites or blacks victory, respectively [30].

Preprocessing of the chess game trajectories.

The ensemble of the chess game trajectories consists of10000 games played by the GNUCHESS program againstitself with default parameters. The positions duringhighly dynamic phases of the game (e.g., during exchangeof pieces) can not be accurately evaluated with the em-ployed evaluation function [19]. To alleviate these prob-lems the trajectories were processed as follows. Initially,every continuous sequence of exchanges of pieces is cutoff from the game trajectories. The trajectories wereprojected to the evaluation function coordinate by com-puting the evaluation function for every second positionwith white turn to move (every fourth ply; ply is a sin-gle move either by white or black). The trajectories thenwere transformed (normalized) to the natural coordinate.To avoid highly dynamic phases of the game other thanthe continuing sequence of exchanges of pieces, the stepswith change in the value of the reaction coordinate largerthan the threshold value of 5 (along the natural coordi-nate) were discarded. The first two moves of every gamewere discarded, to remove the very early developmentphase of the game and keep the game dynamics homo-geneous. More sophisticated variants of the evaluationfunction should make these preprocessing steps unneces-sary.


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