It’s Your Move


Have you ever wondered what would happen if you could map the interactions between two people?  Specifically, tying one person’s actions to the direct cause of another person’s actions, which in turn is the stimulus for a reaction to the previous action and so on.   Now, let’s watch the behavior over time and document each decision.  What you will have is an empirical and dynamic data set that could be used to study and analyze decision-making behavior over time.  Further, if we designate the actions as nodes in a network and the relationships of the results of the actions as the links, you would have an empirical longitudinal network, which could be used to study network behavior over time.

The problem, however, is that obtaining this type of data is extremely hard especially when you wish to involve social interaction.  In other words, we find many non-human examples of longitudinal networks, such as crystal formation and decay, chemical structures and bonding analysis, or even the habits of honeybees in the wild; but generating similar social networks or at least such networks that involve human decisions has proven to be challenging—until now.

Capitalizing on an idea proposed by Marc Anthony Johnson, a former USMA cadet, and presented in its early stage of development at SUNBELT 2008, a group of cadets and faculty members have resurrected this idea and are pushing forward to develop the ultimate longitudinal dataset.  What is it, you ask? Simply put, it is the direct influence network generated by a game of chess.

A chessboard is an 8 by 8 grid of alternatively colored black and white squares labeled on the x-axis from A to H and the y-axis from 1 to 8.  Two sets of pieces are assembled on the board opposite each other, one white, and the other black.  White pieces occupy the rows 1 and 2 while the black pieces occupy rows 7 and 8.  There are six different types of pieces; pawn, rook, knight, bishop, queen and king.  Each team possesses 8 pawns, 2 rooks, 2 knights, 2 bishops, 1 queen and 1 king.  Each type of piece can only move in a certain pattern through the grid, and each player can only move once per turn.  From a social network perspective, chess games offer less stochastic behavior exhibited in many social networks because the game is controlled by rules and strict limitations on movement.  We can consider it a deterministic model since these keep the uncertainty of the model relatively low.

Additionally, a chess game is a wealth of data.  Each game consists of roughly 40 turns.  A turn is defined as white moves, then black moves.  That means that one chess game can yield 40 different networks divided into equally spaced time-turn increments.  Databases found on the Internet consist of thousands of games from novices to masters.  It would not be a stretch to conclude that analyzing chess games in this way that could yield a flood of longitudinal datasets to study.  Below are but a few of the networks extracted from a single game.

By mapping each move into a network, we built a chess network.  Circled below are the two pieces in the network that correspond to the move made on the board.

Finally, Using network measures like closeness, betweenness, eigenvector centrality and density; it may be possible to analyze longitudinal networks in a more effective way.  Figure 1 depict changes that can be seen when using a CUSUM algorithm and graphing the changes.  Figure 2 is a chart of the network density, and figure 3 graphs the Eigenvector Centrality of the two Kings.  Interestingly, correlations are seen after just looking at one network and hopefully these observations can lead to deeper insight into the mathematics of networks.

Interested? If you would like to join us on the project don’t hesitate to contact me at or .  Exciting things are about to happen!

About mathgru

LTC Johnson's interests are focused on applying graph theory and stochastic differential equations to model challenging problems in social network analysis. He has previously developed finite element solutions of partial differential equations to simulate the scattering properties of buried IEDs and investigated numerical methods to solve ordinary differential equations in particular Predictor-Corrector Implicit Methods and Explicit Newmark-Beta algorithms for first and second order differential equations. He is a passionate mathematician and continuously seek way to apply his craft to solve battlefield problems.
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