1. Graphs and Matrices Spring 2012 Mills College Dan Ryan Lecture Slides by Dan Ryan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.Dan RyanCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License

2. Four Representations of a Graph G={V,E} NODE List A B D E B A C D C A B D D A B C E A EDGE List AB AD AE BA BC BD CA CB CD DA DB DC EA A C B D E MATRIX

3. Matrix 10001 11220 12000 30100 01111 ROWSCOLUMNS A matrix ELEMENT The MAIN DIAGONAL SQUARE MATRIX RECTANGULAR MATRIX

4. Nomenclature: nrows by mcols 10 11 12 30 01 1000 1122 100 112 120 1 3 0 5x2 matrix 2x4 matrix 3x1 matrix 3x3 matrix

5. Matrix

6. Sometimes we say “Vector” Each row (or column) in a matrix can be thought of as an ordered set of numbers describing an object. For a graph matrix, a row or a column is a “vector” of a vertex’s connections The fourth vertex is connected to the first, second, and third, but not the fifth

7. Transpose The transpose of a matrix is a swapping of its rows and columns

8. Transposing a Matrix transpose i j

9. Matrix Arithmetic

10. Example: Convert 2-Mode to 1-Mode(s) 1.Write the rectangular incidence matrix I A 123 BDC ABCD 11011 20110 31111 GROUPS PEOPLE A “groups by people” matrix

11. Example: Convert 2-Mode to 1-Mode(s) 2.Compute transpose of I, I T ABCD 11011 20110 31111 123 A101 B011 C111 D101 A “groups by people” matrix A “people by groups” matrix

12. When we multiply matrices… Columns of first factor = rows of second factor (People x Groups) X (Groups x People) = (People x People) (Groups x People) X (People x Groups) = (Groups x Groups) I T 101 011 111 101 I 1011 0110 1111 A GxG 313 122 324 I T 101 011 111 101 I 1011 0110 1111 A PxP 2122 1221 2232 2122 =x = x

13. Degree - 1 0 0 0 1 - 0 1 0 0 0 - 1 1 0 1 1 - 1 0 0 1 1 -

14. Mean(average) Degree

15. Degree and Edges

16. Adjacency Matrix

17. Matrix Arithmetic See also: http://www.jtaylor1142001.net/calcjat/Contents/CMatrix.htm#MatrixMulthttp://www.jtaylor1142001.net/calcjat/Contents/CMatrix.htm#MatrixMult

18. Directed Unweighted Undirected Weighted Undirected Unweighted Directed Weighted dichotomize symmetrize E A B C D 2 4 5 1 1 - 4 0 0 0 0 - 0 1 0 0 0 - 0 0 0 1 5 - 1 0 0 2 0 - E A B C D E A B C D 2 4 5 1 1 E A B C D - 1 0 0 0 0 - 0 1 0 0 0 - 0 0 0 1 1 - 1 0 0 1 0 - - 4 0 0 0 4 - 0 1 0 0 0 - 5 2 0 1 5 - 1 0 0 2 1 - - 1 0 0 0 - 0 1 0 - 1 1 - 1 -

19. Paths If there is a path from vertex A to vertex B… …then there is a sequence of edges connecting them E.g., if there is a 2-path from A to C then there is some vertex B such that there is an edge AB and an edge BC AC B Path A to C?

20. Put another way An edge from vertex j to vertex i means A ij =1 If there is a 2-path from j to i then there is a k such that A ik =1 and A kj =1. …i…j…k…  0011000 i01110 1 0  0110100 j0110100  0000000 k001 1 001  0100000 j i k

21. A ik =1 and A kj =1 That means: if there is a path from vertex j to vertex k to vertex i THEN there is a k such that A ik A kj =1 …i…j…k…  -000000 i0-100 1 0  00-1010 j000-000  0010-00 k100 1 0-0  010000- j i k a b c There is an edge from k to i There is an edge from j to k

22. Another way to see it… If vertex j is connected to vertex k AND Vertex I is connected to k THEN Row I will have a 1 in position k AND Column j will have a 1 in position k

23. Elements of Product Matrix are Vector Products 21101 = 13110 11201 01010 10102 01100 10101 11000 00001 01010 01100 X 10101 11000 00001 01010 i j A 2 ij is how many times the row version and the column version have 1s in the same place

24. …i…j…k…  -000000 i0-100 1 0  00-1010 j000-000  0010-00 k100 1 0-0  010000- Are there other paths from j to i? j i k

25. Look for vertices between j and i …i…j………  0011000 i0111010  0110100 j0110100  0000000  0011001  0100000 j i k

26. Example: Convert 2-Mode to 1-Mode(s) 2.Compute transpose of I, I T A 123 BDC ABCD 11011 20110 31111

27. Degree Distribution “Distribution of X” = pattern of different values X takes on More specifically: the frequency of each value 4 2 3 3 5 1 2 4 4 4 4 4 5 5 2 2 2 2 2 2 1 3 3 3 1 || 2 |||| ||| 3 |||| 4 |||| | 5 ||| 3 5 1 2 4 4 4 4 4 5 5 2 2 2 2 2 2 2 1 3 3 3 3 4

28. Diagonal Elements of A 2 are Vertex Degree 01100 A= 10101 11000 00001 01010 21101 A2=A2= 13110 11201 01010 10102

29. Why? 21101 = 13110 11201 01010 10102 A2A2 01100 10101 11000 00001 01010 A 01100 X 10101 11000 00001 01010 A Degree equals number of “out and back” paths of length 2

30. How about A 3 ? 24311 = 42404 34211 10102 14120 A3A3 01100 X 10101 11000 00001 01010 A 21101 13110 11201 01010 10102 A2A2

31. What do these represent? 24311 = 42404 34211 10102 14120 A3A3 01100 X 10101 11000 00001 01010 A 21101 13110 11201 01010 10102 A2A2

32. PRACTICE:COMPUTE A 2 FROM WXYZ W -100 TO X 0-11 Y 10-0 Z 0010 A W Z X Y FROM WXYZ W 0011 TO X 1010 Y 0100 Z 1000 A2A2

33. PRACTICE:COMPUTE A 3 FROM WXYZ W -100 TO X 0-11 Y 10-0 Z 0010 A W Z X Y FROM WXYZ W 0011 TO X 1010 Y 0100 Z 1000 A2A2 FROM WXYZ W 1010 TO X 1100 Y 0011 Z 0100 A3A3

35. …i…j…k…  -000000 i0-100 1 0  00-1010 j000-000  0010-00 k100 1 0-0  010000- Are there other paths from j to i? j i k