1. Simulated dataset from Ph.D. work of Alexander Statnikov September 2007

2. Background

3. Graphical Notation

4. T X 10 X6X6 X7X7 X8X8 T T X 26 X 27 X 28 X 29 X 15 X 11 In total, there are 72 equivalent local neighborhoods of T in the network. T X1X1 X2X2 X3X3 X 11 X 12 X 13 X 14 X5X5 X9X9 X 18 X 16 X 15 X 17 X 19 X 20 X 21 T X 22 X 23 X 24 X 25 Basic Network Structure X 30

5. Example of TIE relations (1/2) X2X2 X3X3 X 11 T 0 1 2 0 1 2 0 2 0 1 2 333 1 0 1 2 3 X1X1 3 The probability distribution of discrete random variables X 1, X 2, X 3, X 11, and T is represented graphically below. Red arrows denote nonzero conditional probabilities. Notice that variables X 1, X 2, X 3, X 11, and T are not deterministically related to each other. The following TIE relations hold in the data: TIE T (X 1, X 2 )TIE T (X 1, X 3 )TIE T (X 1, X 11 ) TIE T (X 2, X 3 )TIE T (X 2, X 11 )TIE T (X 3, X 11 ) TIE X11 (X 1, X 2 )TIE X11 (X 1, X 3 )TIE X11 (X 2, X 3 )

6. Example of TIE relations (2/2) X 12 X 13 X 14 1 2 0 1 2 0 1 0 The following TIE relations hold in the data: TIE T (X 12, X 13 )TIE T (X 12, X 14 )TIE T (X 13, X 14 ) 2 0 1 T 3 2 The probability distribution of discrete random variables T, X 12, X 13, and X 14 is represented graphically below. Red arrows denote nonzero conditional probabilities. Notice that variables T, X 12, X 13, and X 14 are not deterministically related to each other.

7. Add more equivalent local neighborhoods, so that not all of them are of the minimal size. Add more variables that can potentially be false-positives. These variables can be both connected & unconnected to T. Extending Basic Network Structure