The game of go as a complex network The game of go as a complex network Bertrand Georgeot, Olivier Giraud, Vivek Kandiah supported by EC FET Open project NADINE B.G. and O. Giraud, Europhysics Letters 97 68002 (2012) Quantware group Laboratoire de Physique Théorique, IRSAMC, UMR 5152 du CNRS Université Paul Sabatier, Toulouse Bertrand Georgeot (CNRS Toulouse) The game of go as a complex network ECT workshop, July 2012 1 / 19
Networks • Recent field: study of complex networks • Tools and models have been created • Many networks are scale-free, with power-law distribution of links • Difference between directed and non directed networks • Important examples from recent technological developments: internet, World Wide Web, social networks... • Can be applied also to less recent objects • In particular, study of human behavior: languages, friendships... Bertrand Georgeot (CNRS Toulouse) The game of go as a complex network ECT workshop, July 2012 2 / 19
Games • Network theory never Goban applied to games • Games represent a privileged approach to human decision-making • Can be very difficult to modelize or simulate = ⇒ While Deep Blue famously beat the world chess champion Kasparov in 1997, no computer program has beaten a very good go player even in recent times. Bertrand Georgeot (CNRS Toulouse) The game of go as a complex network ECT workshop, July 2012 3 / 19
Rules of go • White and black stones alternatively put at intersections of 19 × 19 lines • Stones without liberties are removed • Handicap stones can be placed • Aim of the game: construct protected territories • total number of legal positions ∼ 10 171 , compared to ∼ 10 50 for chess Bertrand Georgeot (CNRS Toulouse) The game of go as a complex network ECT workshop, July 2012 4 / 19
Databases • We use databases of expert games in order to construct networks from the different sequences of moves, and study the properties of these networks • Databases available at http://www.u-go.net/ • Whole available record, from 1941 onwards, of the most important historical professional Japanese go tournaments: Kisei (143 games), Meijin (259 games), Honinbo (305 games), Judan (158 games) • To increase statistics and compare with professional tournaments, 4000 amateur games were also used. Bertrand Georgeot (CNRS Toulouse) The game of go as a complex network ECT workshop, July 2012 5 / 19
Vertices of the network ”plaquette” ⇒ square of 3 × 3 intersections • We identify plaquettes related by symmetry • We identify plaquettes with colors swapped = ⇒ 1107 nonequivalent Examples of plaquettes plaquettes with empty centers = ⇒ vertices of our network Bertrand Georgeot (CNRS Toulouse) The game of go as a complex network ECT workshop, July 2012 6 / 19
Zipf’s law • Zipf’s law: empirical law 0 observed in many natural 0 -0.5 -0.5 distributions (word log F(n) -1 -1 frequency, city sizes...) -1.5 log F(n) -1.5 -2 0 0.5 1 1.5 2 2.5 3 • If items are ranked log n -2 according to their -2.5 frequency, predicts a -3 0 0.5 1 1.5 2 2.5 3 3.5 log n power-law decay of the Normalized integrated frequency vs the rank. frequency distribution of 1107 • integrated distribution of moves. Thick dashed line is 1107 moves clearly follows y = − x . Inset: same for a Zipf’s law, with an positions on the board exponent ≈ 1 . 06 Bertrand Georgeot (CNRS Toulouse) The game of go as a complex network ECT workshop, July 2012 7 / 19
Sequences of moves • we connect vertices corresponding to moves a and b if b follows a in a game at a distance ≤ d . 5 4 • Each choice of d defines a different network. log f(n) 3 • Left: frequency distribution 2 for sequences of the 1107 1 moves with d = 4. 0 0 1 2 3 4 5 6 log n Algebraic decrease visible, exponent from ≈ 1 (short Integrated frequency distribution sequences) to ≈ 0 . 7 (long of sequences of moves f ( n ) for sequences). (from top to bottom) two to seven successive moves (all = ⇒ Sequences of moves follow databases together), plotted Zipf’s law (cf languages) against the ranks of the moves. = ⇒ Exponent decreases as longer sequences reflect individual strategies Bertrand Georgeot (CNRS Toulouse) The game of go as a complex network ECT workshop, July 2012 8 / 19
Sequences of moves Four possible definitions: 0 • C1: positions on the board, 4 b follows a if b is played log P(d) -1 immediately after a log f(n) -2 • C2:positions on the board, 2 0 0.5 1 b follows a if b is played log d after a at distance d = 4 • C3: sequence of vectors 0 0 2 4 6 log n between successive Integrated frequency positions with d = 4 distribution of sequences of • C4: as before moves for two (continuous) and = ⇒ move sequences, even three (dashed lines) successive long ones, are well hierarchized moves, cases C1 (black), C2 by our initial definition (red), C3 (green), C4 (blue). = ⇒ amateur database departs Inset: distribution of distances from all professional ones, between moves P ( d ) . All playing more often at shorter professional tournaments are distances different from amateur games. Bertrand Georgeot (CNRS Toulouse) The game of go as a complex network ECT workshop, July 2012 9 / 19
Link distributions 3 • Tails of link distributions P in P out log P in , log P out very close to a power-law 2 1 / k γ with exponent γ = 1 . 0 3 P out 2 for the integrated 1 P in 1 distribution. 0 -3 -2 -1 0 • The results are stable in the 0 -4 -3 -2 -1 0 log(k/k max ) sense that the exponent does not depend on the Normalized integrated database considered. distribution of ingoing links P in (solid) and outgoing links P out = ⇒ network displays the (dashed), Thick solid line is scale-free property y = − x . = ⇒ symmetry between ingoing Inset: P in (solid curves) and P out and outgoing links is a (dashed curves), d = 2 (black), peculiarity of this network 3 (red), 4 (green), 5 (blue) and 6 (violet). Bertrand Georgeot (CNRS Toulouse) The game of go as a complex network ECT workshop, July 2012 10 / 19
Directed network: Google algorithm Weighted adjacency matrix 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 3 1 1 0 0 0 0 0 1 3 6 7 3 2 1 H = 0 0 0 1 1 1 3 1 0 0 0 0 0 0 5 4 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 Ranking pages { 1 , . . . , N } according to their importance. PageRank vector p = stationary vector of H : Bertrand Georgeot (CNRS Toulouse) The game of go as a complex network ECT workshop, July 2012 11 / 19
Computation of PageRank p = H p ⇒ p = stationary vector of H : can be computed by iteration of H . To remove convergence problems: Replace columns of 0 (dangling nodes) by 1 N : H → matrix S 1 0 0 0 0 0 0 7 1 1 0 0 0 0 0 3 7 1 1 1 0 0 0 0 3 7 2 1 1 In our example, H = 0 0 1 1 1 . 3 7 1 1 0 0 0 0 0 7 2 1 0 1 0 0 0 0 7 1 0 0 0 0 0 0 7 To remove degeneracies of the eigenvalue 1, replace S by G = α S + ( 1 − α ) 1 N Bertrand Georgeot (CNRS Toulouse) The game of go as a complex network ECT workshop, July 2012 12 / 19
Ranking vectors • The PageRank algorithm gives the PageRank vector, with amplitudes p i , with 0 ≤ p i ≤ 1 • PageRank is based on ingoing links • One can define a similar vector based on outgoing links (CheiRank) • HITS algorithm: Authorities (ingoing links) and Hubs (outgoing links) • Other eigenvalues and eigenvectors of G reflect the structure of the network Bertrand Georgeot (CNRS Toulouse) The game of go as a complex network ECT workshop, July 2012 13 / 19
Ranking vectors -1 • Clustering coefficient detects local connected -2 PageRank clusters. CheiRank log(rank) -3 • Here depends on the 0.8 Hubs number of games n g -4 0.6 Authorities CC included, but almost not 0.4 -5 on the database. 2000 4000 n g 0 0.5 1 1.5 2 2.5 3 • For large n g , it goes to an log i asymptotic value which Ranking vectors of G . Top bundle: seems larger than 0 . 7 PageRank. Second bundle: (higher CC than WWW CheiRank. Third bundle: Hubs. ≈ 0 . 11) Fourth bundle: Authorities. • Ranking vectors follow an Straight dashed line is y = − x . algebraic law Inset: Clustering coefficient as a function of the number of games • Symmetry between n g included to construct the distributions of ranking network; blue squares: vectors based on ingoing professional tournaments; circles: links and outgoing links. amateur games. Bertrand Georgeot (CNRS Toulouse) The game of go as a complex network ECT workshop, July 2012 14 / 19
PageRank vs CheiRank • Left: correlation between the PageRank and the 1000 CheiRank for the five databases considered. • Strong correlation between K* 500 these rankings based respectively upon ingoing and outgoing links. 0 = ⇒ Strong correlation between 0 500 1000 K moves which open many K* vs K where K (resp. K*) is possibilities of new moves the rank of a vertex when and moves that can follow ordered according to PageRank many other moves. vector (resp CheiRank) for = ⇒ However, the symmetry is amateur (violet stars) and professional (other) databases. far from exact Bertrand Georgeot (CNRS Toulouse) The game of go as a complex network ECT workshop, July 2012 15 / 19
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